Exact Solution of the Schrodinger and Faddeev
Equations for Few Body Systems
Thesis Submitted for the Degree
“Doctor of Philosophy”
by
Nir Barnea
Submitted to the Senate of the Hebrew University
June 1997
This work was carried out
under the supervision of Professor V. B. Mandelzweig
Abstract
In this thesis, conducted in the field of few-body physics, a new approach for the study of small
nuclear, molecular, and atomic systems has been developed and applied. This approach is based
on the hyperspherical harmonic method.
In the hyperspherical method the N -body system, with 3(N − 1) internal degrees of freedom,
is described by a single hyper-radial coordinate and a set of 3N − 4 angular coordinates. This is
done in complete analogy to the 2-body problem, where after removing the center of mass motion,
one rewrites the relative vector using polar coordinates and expands the wave function in terms
of the spherical harmonics, YLM . The hyperspherical harmonic functions are the eigenfunctions of
the angular part of the 3N − 4 dimensional Laplace operator.
Currently there exist two major representations for the hyperspherical harmonic functions, distinguished by their symmetry properties under exchange of particles. In the non-symmetrized
version, the functions realize irreducible representations, irreps, of the 3N − 3 dimensional rotation
group O(3N − 3), but have no particular symmetry under particles permutations. For any num-
ber of particles and any value of the total angular momentum they are expressed through linear
combinations of products of Jacobi polynomials and ordinary spherical functions. This version is
obviously inconvenient if identical particles are involved, as we are interested in states with well
defined permutational symmetry.
In the more sophisticated version, the functions, which we call rotational harmonics, realize
irreps of O(3N − 3) together with irreps of O(N − 1) and with irreps of the symmetric group SN ,
which is a sub-group of O(N − 1). Unfortunately explicit construction of these functions turns out
to be a very difficult problem even for the three particles case. The 3-body functions for L = 0
were found by Smith and by Whitten & Smith at the end of the 60’s. Recently, I have derived an
explicit analytic expressions for the 3-body functions in case of L = 1, and for the corresponding
matrix elements.
Consequently, one would like to start with the non-symmetric version of the hyperspherical
harmonics and to find a way to construct functions with adapted permutational symmetry. In
3
principle this can be achieved, in the simplest way, by using the Young symmetrizer, or more
efficiently by recursive construction of the symmetrized hyperspherical functions using the transposition class-sum operator to identify the irreps of the symmetric group. In practice it turns out
that even for 4 particles these methods runs into difficulties.
In this thesis I present a unique solution to the long standing problem of constructing rotational
hyperspherical harmonics. This solution leads to a new approach to the symmetrization problem using the orthogonal group O(N −1) and the group sub-group chain O(3N −3) ⊃ O(3)⊗O(N − 1) ⊃
O(3) ⊗ SN . Comparing this approach with other methods one finds that this algorithm is much
more efficient. Furthermore, for large N the construction of the relevant functions is almost independent of N . The use of the rotational hyperspherical harmonics also simplifies the calculation of
matrix elements, which together with the symmetrization method are the mathematical basis for
the solution of the Schroedinger equation. I demonstrate this method by applying it to calculate
the ground states of atomic and nuclear four body systems.
In addition the hyperspherical harmonic method has been applied for the study of the Faddeev
equations, considering the problems of identical and non-identical spinless particles in three dimensional space, interacting via harmonic oscillator two-body potentials. I have solved these problems
analytically for any excitation and for an arbitrary value of the total angular momentum. These
solutions are the only examples of general analytic solution of the Faddeev equation in three dimensional space, and it is a unique example of a solution of the Faddeev equations for non-identical
particles.
4
Contents
1 Introduction
8
1.1
About this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.2
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3
Scientific background and research objectives . . . . . . . . . . . . . . . . . . . . .
9
2 The Hyperspherical Harmonic method
15
2.1
The hyperspherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2
The Laplace operator in hyperspherical coordinates . . . . . . . . . . . . . . . . . . 20
2.3
The hyperspherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4
The Raynal-Revai and the T-coefficients . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.1
The Raynal-Revai coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.2
The hyperspherical recoupling coefficients - the T-coefficients . . . . . . . . . 31
3 Construction of Hyperspherical functions Symmetrized with respect to the Or5
thogonal Group
4
34
3.1
The Orthogonal Parentage Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2
Construction of 3-Body Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1
The 3-Body Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.2
Explicit Construction of 3-Body Functions for L = 0, 1 . . . . . . . . . . . . 41
3.3
The Hyperspherical Gel’fand-Zetlin states . . . . . . . . . . . . . . . . . . . . . . . 46
3.4
The computational Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5
Shortcut for large N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Permutational Symmetry
4.1
4.2
68
The reduction problem O(N − 1) ↓ SN
. . . . . . . . . . . . . . . . . . . . . . . . 69
Symmetrized Hyperspherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Matrix Elements of Two Body Potentials Between symmetrized Hyperspherical
States
5.1
78
Kinematic rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.1.1
The relative coordinate rotation - “type” A . . . . . . . . . . . . . . . . . . 82
5.1.2
The transposition rotation - “type” B . . . . . . . . . . . . . . . . . . . . . 83
5.1.3
The 900 rotation - “type” C
. . . . . . . . . . . . . . . . . . . . . . . . . . 84
6
5.1.4
The general procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2
The representation matrices of the rotation (and reflection) operators
5.3
Matrix elements of arbitrary two-body operators . . . . . . . . . . . . . . . . . . . 87
5.4
Explicit derivation of the matrix elements of a central two-body interaction . . . . 89
5.5
Analytic expression for the matrix element . . . . . . . . . . . . . . . . . . . . . . . 91
6 Application of the Method for Physical Systems
6.1
6.2
. . . . . . . 85
94
Analytic Solution of Faddeev Equation . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1.1
Identical Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1.2
Non-Identical Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
The N -body Schroedinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2.1
4-Nucleons interacting via an s-wave potential . . . . . . . . . . . . . . . . . 110
6.2.2
3H like Li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2.3
Ground State of the Lithium Atom . . . . . . . . . . . . . . . . . . . . . . . 114
7 Conclusions
117
A The Orthogonal Group and the Gel’fand-Zetlin States
120
B
127
The mass weighted Jacobi coordinates for the N-body problem
7
Chapter 1
Introduction
8
1.1
About this Thesis
This thesis is in the field of few-body physics and was conducted under the supervision and in
collaboration with professor Victor Mandelzweig (thesis adviser) and also in collaboration with Dr.
Akiva Novoselsky. Major parts of the research described in this thesis were done during my army
service, as a part of my studies for a masters degree in physics. This studies were later widen into
a doctoral research. Most of the results of this research are summarized in a series of papers, see
Refs. [Bar90, Bar91, Bar92, Bar94, Nov95, Bar96, Bar97, Bar97a, Bar97b]. My aim in writing
this thesis was to give a clear review of the few-body calculation method I developed together with
my collaborators, starting from the mathematical basis of the hyperspherical harmonic approach
through the symmetrization process, and ending with few physical applications of the method.
1.2
Outline
The thesis is organized as follows, in Chapter 2 I review the “tree” method for the construction of
the nonsymmetric hyperspherical harmonics. In Chapter 3 the new algorithm for the construction
of the rotational harmonics is presented, as well as the explicit expressions available for the 3-body
system. Chapter 4 deals with the reduction problem O(N − 1) ↓ SN , using a modification of the
Novoselsky, Katriel and Gilmore [Nov88] recursive algorithm. The construction of hyperspherical
harmonics with definite permutational symmetry is presented in Section 4.2 summarizing the results
of Chapters 3 & 4. Analytic evaluation of 2-body matrix elements needed for physical applications
is presented in Chapter 5. Few examples for the application of the method for physical systems
are given in Chapter 6.
1.3
Scientific background and research objectives
The method of hyperspherical harmonics, introduced in 1935 by Zernike and Brinkman [Zer35]
and reintroduced in the 60’s by Delves [Del59], Simonov [Sim66], Zickendraht [Zic65] and Smith
9
[Smi60], has revealed itself in the last three decades as a powerful tool for the study of few-body
systems. The hyperspherical functions were applied to atomic three and four-body systems like
the He & Li atoms [Haf87, Haf89, Wan95], to nuclear systems like the triton and the α particle
[Viv92, Viv95] and even to hadronic calculations [Muk93].
In complete analogy to the 2-body problem, where after removing the center of mass motion,
one rewrites the inter particles vector using polar coordinates, namely radius and two angles, and
expand the wave function in terms of the spherical harmonics which span the 2-dimensional sphere.
In the N -body problem the 3(N − 1) internal degrees of freedom are expressed in terms of a single
hyper-radial coordinate and a set of 3N − 4 angular coordinates. The 3N − 4 dimensional sphere is
then spanned by hyperspherical harmonic functions which are the eigenfunctions of the generalized
angular momentum operator.
Currently there exist two major representations for hyperspherical harmonic functions distinguished by their symmetry properties under exchange of the particles. In the non-symmetrized
version the functions, also known as the “tree” functions, realize irreducible representations of
the 3N − 3 dimensional rotation group O(3N − 3) (which is a symmetry group of the noninteracting N-body Hamiltonian ) but have no particular symmetry under particles permutations.
For any number of particles and any value of the total angular momentum L they are expressed
through linear combinations of products of Jacobi polynomials and ordinary spherical functions
[Vil65, Vil68, Vil95]. The matrix elements of two body potentials between such hyperspherical
states could be calculated analytically (see for example [Kri90] and references therein). However
this version is obviously inconvenient if identical particles are involved.
In the more sophisticated version [Sur67] which I will refer to as the rotational hyperspherical
harmonics, or rotational harmonics, the functions realize irreducible representations of O(3N − 3)
together with irreducible representations of the group O(N − 1) of kinematic rotations. Kinematic
rotations [Fab83] are transformations from one set of Jacobi coordinates to another set of relative
coordinates. As there are N − 1 Jacobi coordinates in the N -body problem, this rotations coincide
with the N − 1 dimensional orthogonal group O(N − 1), that in our context might be referred to
as the kinematic group. The N -particles permutation group SN is a subgroup of the kinematic
group O(N − 1). Unfortunately the construction of the rotational hyperspherical harmonic basis
10
turns out to be a very difficult problem [Mos84, Cas84, Cha84] even for the three particles case.
The 3-body rotational harmonics for L = 0 were found by Smith [Smi60] and by Whitten and
Smith [Whi68] who have shown that they could be expressed through the sum of products of two
Wigner D-functions, depending on spatial and kinematic variables respectively. The corresponding
matrix elements for the case of the Coulomb, Gaussian and harmonic-oscillator two body potentials
were calculated analytically by Whitten [Whi68]. They where generalized for arbitrary two body
operators by Mandelzweig [Man80], who has shown that such matrix elements can be expressed
through sum of the products of Clebsh-Gordan coefficients and explicitly written functions of radial
variable, which are uniquely determined by the operator’s power series expansion. Recently we have
derived [Bar90, Bar91] an explicit analytic expressions for the 3-body rotational harmonics in case
of L = 1, and for the corresponding matrix elements. Nevertheless, the extension of the method for
L > 1 or for systems with more then 3 particles is extremely difficult, and it seems quite impossible
to derive analytic expressions for these functions.
Consequently, one would like to start with the non-symmetric version of the hyperspherical
harmonics and to find a way to construct hyperspherical functions with adapted permutational
symmetry. This problem has been considered extensively, starting in the late 60’s, with the works
of Nyiri & Smorodinskii (see [Nyi69] and references therein), who made an attempt to transform the
nonsymmetric hyperspherical functions to symmetric functions. Raynal & Revai [Ray70] who derived explicit expression for the kinematic rotation brackets in the 3-body case, Kil’dyushov [Kil72]
who found an explicit method to apply the Young symmetriser to the nonsymmetric hyperspherical
harmonics and others (see [Efr95] and references therein). Recently this problem was reconsidered
by Clark & Greene [Cla80] and by Wang et al. [Wan95] in the context of atomic physics, taking
advantage of the infinite mass nuclei, and by Efros [Efr95] who suggested a computational boosting
using a numerical approach to the evaluation of the transformation brackets.
Few years ago Novoselsky and Katriel [Nov94] have formulated a new algorithm for the construction of hyperspherical functions with well defined symmetry with respect to particles permutation.
In this method the hyperspherical functions belonging to well defined irreducible representations
(irreps) of the symmetric group are constructed recursively. Assuming that the (N − 1)-particle
symmetrized hyperspherical functions have already been constructed in terms of the Jacobi coordinates. The N ’th Jacobi coordinate state is coupled to these states, obtaining N -particle hyper-
11
spherical functions with well defined angular momentum and hyperspherical angular momentum
quantum numbers. Invariant subspaces of these functions, determined by common values of angular
momentum, hyperspherical angular momentum and the permutational symmetry irrep of the first
(N − 1)-particles, are constructed. The transposition class-sum operator of the symmetric group
is then diagonalized within these invariant subspaces. The obtained eigenvalues uniquely identify
the irreps of the symmetric group whereas the eigenvectors are the hyperspherical coefficients of
fractional parentage (hscfps).
Developing a computer code that implements the Novoselsky and Katriel formalism described
above, which is one of the most sophisticated approaches to the symmetrization problem. We found
out that, unfortunately, even for 4 particles, as the value of the hyperspherical angular momentum
K (equivalent to the power of the polynomials) increases we obtain enormous number of hscfps.
For example, if we build all the possible 4 particles states, there are 471 hscfps for K = 4, 5805
hscfps for K = 6, 49675 hscfps for K = 8, and 313127 for K = 10. Therefore, one is forced to
look for new ideas and methods in order to find a generally valid efficient procedure applicable for
systems with more then three particles.
The main objective of this thesis is to summarize the new approach I, along with my collaborators, have developed. Starting from the construction of the symmetrized hyperspherical functions,
through the necessary methodology needed for physical calculations and ending with some physical
examples. The highlight of this new approach is, that instead of attempting to symmetrize the
nonsymmetric hyperspherical harmonics, we construct the rotational harmonics realizing irreps of
O(N − 1), and then we symmetrize them with respect to particles permutations. This procedure
enabled us to split the big spaces, which are invariant under permutations, into smaller subspaces
invariant under the appropriate subgroup. We used the Gel’fand and Zetlin (GZ) approach [Gel50]
to completely specify the irreps of O(N ) according to the canonical chain of the group-subgroup
O(N ) ⊃ O(N − 1) ⊃ · ⊃ O(3) ⊃ O(2). This scheme for classification of N -body states was
proposed many years ago by Surkov [Sur67] and by Louck & Galbraith [Lou72].
From a physicist point of view, the orthogonal group and the GZ basis plays already an important role in the theoretical microscopic approach to nuclear collective models [Row80, Fil74].
A problem of 3N degrees of freedom of N nucleons can be described by 9 collective degrees of
12
freedom and 3N − 9 internal degrees of freedom which may be associated with the manifold
O(N − 1)/O(N − 4). Looking for wave functions with a definite irrep of a dynamical group
Sp(6N ), where Sp(6) × O(N − 1) is one of its subgroups, Moshinsky [Mos84] have noted that the
collective effects can be introduced by the constraint that the N -body wave function is restricted
to a given irrep of the orthogonal group O(N − 1). Moshinsky and Quesene have shown in [Mos71]
that the irrep of the group O(N − 1) determines that of the group Sp(6) and thus these two groups
are “complementary”. Therefore, some of the microscopic collective models of the nuclei are related
to the Sp(6) group and others to the O(N − 1) group. The main difficulty in applying these ideas
to physics, is the problem of constructing basis functions which belong to a definite GZ state. This
problem was a great obstacle in the development of the orthogonal group approach to the N -body
problem [Mos84, Cas84, Cha84] and in further development of Surkov’s approach [Sur67].
Thus, in order to realize this new approach we had to develop a new method [Bar96, Bar97] for
the construction of irreducible representations of the orthogonal group. We have used this method
to present a new recursive algorithm for constructing rotational hyperspherical harmonics, and used
a modification of the Novoselsky, Katriel, Gilmore algorithm [Nov88, Nov89] to solve the O(N −1) ↓
SN reduction problem, obtaining symmetrized hyperspherical functions. This new method present a
unique solution to the long standing problem of constructing rotational hyperspherical harmonics.
More then that, comparing the cpu time and memory needed for constructing hyperspherical
functions with definite permutational symmetry in the original Novoselsky & Katriel algorithm
[Nov94] with the algorithm presented in this thesis one finds that the current algorithm is much
better.
The application of the hyperspherical functions for the solution of the Schroedinger equation,
requires the calculation of the matrix elements of the kinetic and potential energy operators between different hyperspherical states. We have derived [Nov95] the matrix elements of two-body
operators between the permutational symmetry adapted hyperspherical functions presented by
Novoselsky and Katriel [Nov94]. Actually, for identical particles, only the matrix element of the
two-body operator between the last two particles has to be calculated. This matrix element can
be calculated using the hyperspherical cfps, the Raynal-Revai coefficients [Ray70], [Ray73], the
angular momentum recoupling coefficients - the 6j coefficients, and the hyperspherical recoupling
coefficients - the T-coefficients [Kil72], [Kil73]. A detailed description of the method for this cal-
13
culation is given in Ref. [Nov95]. Although this method is complete, it requires many calculation
steps for each matrix element, involving the Raynal-Revai and the T-coefficients. This results
from the fact that the two-body interaction depends on the relative coordinate between the two
particles whereas the hyperspherical functions are expressed in terms of the Jacobi coordinates.
The transformation from Jacobi coordinates to the relative coordinates, although requires simple
rotations, is difficult to implement.
This difficulty is greatly reduced by using the new approach we introduced earlier, since we
do not have to use the Raynal-Revai and the T-coefficients. We just have to use the appropriate
representation matrices of the orthogonal group. In Ref. [Bar97b] we develop this idea and present
the explicit way to calculate the matrix elements.
In order to verify the method, we have used it for calculating the ground state of atomic and
nuclear four body systems. In addition we have applied the hyperspherical harmonic method
for the study of the Faddeev equations, we have considered the problems of identical and nonidentical spinless particles in three dimensional space, interacting via harmonic oscillator two-body
potentials. We have solved [Bar92, Bar94] these problems analytically for any excitation and for an
arbitrary value of the total angular momentum. These solutions are the only examples of general
analytic solution of the Faddeev equation in three dimensional space, and it is a unique example
of a solution of the Faddeev equations for non-identical particles.
14
Chapter 2
The Hyperspherical Harmonic method
15
The study of nuclear systems composed of N nucleons have led to the construction of the
hyperspherical harmonics, which are harmonic polynomials in 3(N − 1) dimensional space. The
hyperspherical harmonics can be constructed in various ways, nevertheless the only general method
to construct N -body hyperspherical functions which belong to definite irreducible representation of
the group O(3) of spatial rotations, is the “tree” method proposed by Vilenkin et al. [Vil65, Vil68]
about thirty years ago. In this graphic method, each hyperspherical coordinate system is set in
correspondence to a definite “tree” diagram. for example, the coordinates
x1 = ρcos(α)cos(β)cos(δ)
(2.1)
x2 = ρcos(α)cos(β)sin(δ)
x3 = ρcos(α)sin(β)
x4 = ρsin(α)cos(γ)
x5 = ρsin(α)sin(γ)
correspond to the “tree” diagram in Fig. 2. The “tree” diagram have the following interpretation,
Figure 2.1: “Tree” diagram
x5
¡
@
@ γ ¡
@¡
s
@
@
@
x4
@
@
x3
@
@
@
@
@ α ¡
@¡
s
x2
@
@ δ ¡
@
@¡
s
@
¡
@ β ¡
@¡
s
¡
¡
¡
¡
¡
x1
segments joining “tree” nodes or a node with a coordinate are called propagators. With each node
we associate a certain angle φ; propagator extending to the right (left) upward from a node is set in
accordance with the cosine (sine) of the angle φ. Each coordinate xi is represented by the product
of all the propagators from the vertex of the diagram to the coordinate xi . It is well known that
the n-dimensional Laplace operator admits the separation of variables in polyspherical coordinates.
Therefore the hyperspherical functions associated with each node and with the “tree” expanding
from that node upwards can be written as a product of left “tree” hyperspherical function, right
“tree” hyperspherical function and a function of φ the angle associated with that node. The rules
for obtaining the hyperspherical functions from the “tree” are given in Ref. [Vil65]. In this chapter
16
I am not going to review the general theory of hyperspherical functions and the “tree” method but
rather focus on the application of the “tree” method to the problem of constructing hyperspherical
harmonics for the N -body problem.
This chapter is organized as follows, in Section 2.1 I give a detailed description of the hyperspherical coordinates in the N -body problem. The Laplace operator that correspond to this
coordinate system is given in Section 2.2. The hyperspherical functions are given in Section 2.3.
In Section 2.4 I briefly review the Raynal-Revai and T-coefficients, needed for the transformation
from one set of Jacobi coordinates to another set.
2.1
The hyperspherical coordinates
Several variants of hyperspherical coordinates have been used by different authors. They differ by
both the choice of the underlying set of “single-particle” coordinates, the number (one or three) of
spatial components specifying a “single-particle”, and certain details concerning the definition of
the angular and/or hyper-angular coordinates [Nik91, Ave89].
We shall be using a version of the hyperspherical coordinates in which the center of mass motion
of the system is removed, thereby obtaining a set of non-spurious basis functions, see Appendix B
for detailed description of the Jacobi coordinates.
To introduce these hyperspherical coordinates we start from the center-of-mass coordinate
Rcm =
1
MN
PN
η k−1 =
Mk =
Pk
i=1
i=1
s
mi ri and the normalized Jacobi coordinates Eq. (B.1)
´
1
Mk−1 mk ³
(m1 r1 + m2 r2 + · · · + mk−1 rk−1 ) ;
rk −
Mk
Mk−1
k = 2, 3, · · · , N
(2.2)
mi , in which the k’th particle is specified relative to the center of mass of particles 1
to k − 1. η k consists of a radial coordinate ηk and a pair of angular coordinates Ωk ≡ (θk , φk ).
A two-particle system is specified by the Jacobi coordinate η 1 that consists of the radial coordinate ρ1 ≡ η1 and of the angular coordinates Ω1 = (θ1 , φ1 ). For a three-particle system we
transform the two radial coordinates η1 and η2 into a hyper-radial coordinate ρ2 and a hyper17
angular coordinate ϕ2 defined via
η1 = ρ1 = ρ2 cos ϕ2
η2 = ρ2 sin ϕ2
(2.3)
The complete set of six coordinates consists of ρ2 , ϕ2 , and the solid angles Ω1 , Ω2 associated with
the unit vectors η̂ 1 and η̂ 2 respectively. The six dimensional volume element is given by
dV6 = ρ52 dρ2 dS5 ≡ ρ52 dρ2 sin2 (ϕ2 ) cos2 (ϕ2 ) d ϕ2 d Ω1 d Ω2
(2.4)
where dS5 is the angular volume element associated with the 5 dimensional hypersphere. The
hyperangle ϕ2 varies in the range
π
2
≥ ϕ2 ≥ 0.
Adding the fourth particle, third Jacobi coordinate, we define the hyper-radial coordinate ρ3
and the hyper-angular coordinate ϕ3 via
ρ2 = ρ3 cos ϕ3
η3 = ρ3 sin ϕ3
(2.5)
The hyperspherical coordinates for three Jacobi coordinates are the two hyper-angular coordinates
ϕ2 and ϕ3 , the six coordinates Ω1 , Ω2 and Ω3 and the hyper-radial coordinate ρ3 . The volume
element associated with the four particles system is given by
dV9 = ρ83 dρ3 dS8 ≡ ρ83 dρ3 sin2 (ϕ3 ) cos5 (ϕ3 ) d ϕ3 d Ω3 d S5
(2.6)
where dS8 is the angular volume element associated with the 8 dimensional hyper sphere and
dS5 is the angular volume element defined is Eq. (2.4). The hyper angle ϕ3 varies in the range
π
2
≥ ϕ3 ≥ 0.
In general, having defined the hyper-radial coordinate ρk−1 we define ρk and ϕk so as to satisfy
ρk−1 = ρk cos ϕk
where
ρ2k
=
ρ2k−1
+
ηk2
=
ηk = ρk sin ϕk
(2.7)
k
X
(2.8)
i=2
ηi2
k
1 X
=
mi mj (ri − rj )2
2Mk i,j=1
18
Therefore, the hyper-radial coordinate is symmetric with respect to permutations of the underlying
single particle coordinates, and is invariant to the choice of the Jacobi coordinates. The 3N internal
coordinates for the N + 1-particle system consist of the hyper-radial coordinate ρN , the N − 1
hyper-angular coordinates ϕ(N ) ≡ {ϕ2 , ϕ3 , · · · , ϕN }, and the 2N angular coordinates Ω(N ) ≡
{Ω1 , Ω2 , · · · , ΩN }. These coordinates depend on the set of Jacobi coordinates specified in Eq.
(B.1). For a different ordering of particle indices a different set of hyper-angular coordinates as
well as a permuted set of angular coordinates is obtained. On the other hand, the hyper-radial
coordinate is independent of the order of the particle indices, cf. Eq. (2.8). The volume element
associated with the N Jacobi coordinates η 1 , . . . , η N is
−1
−1
dV3N = ρ3N
dρN dS3N −1 ≡ ρ3N
dρN sin2 (ϕN ) cos3N −4 (ϕN )dϕN dΩN dS3N −4
N
N
(2.9)
where dS3N −1 is the volume element associated with the 3N − 1 dimensional hyper sphere, dS3N −4
is the volume element associated with the 3N − 4 dimensional hyper sphere and dΩN is the volume
element associated with the angular part of the N th Jacobi coordinate η N . The hyper angle ϕN
varies in the range
π
2
≥ ϕN ≥ 0.
In order to obtain the rotational hyperspherical harmonics we have to evaluate the matrix
elements of the generators, X̂ij , of the group O(N ), defined in equation (A.1). In particular, we
are interested to study the action of the generator X̂N,N −1 = i(η N ·∇N −1 −η N −1 ·∇N ), on the “tree”
functions. This is conveniently done by constructing the N + 1-particle hyperspherical coordinates
in terms of the set of N − 2 Jacobi coordinates corresponding to the first N − 1 particles and the
set consisting of the two Jacobi coordinates η N −1 and η N . This construction only involves the
radial parts of the various Jacobi coordinates; the angular coordinates Ω(N ) are not affected. The
hyper-angular coordinates that are related to the Jacobi coordinates of the first N − 1 particles,
ϕ(N −2) , also remain unchanged. However, instead of the two hyper-angular coordinates ϕN −1 and
ϕN we have one hyper-angular coordinate ϕN −1,N which is obtained from the transformation
ηN −1 = ρN −1,N cos ϕN −1,N
ηN = ρN −1,N sin ϕN −1,N
(2.10)
and a second hyper-angular coordinate ϕ′N which is obtained from the coordinates ρN −2 and ρN −1,N
by the relation
ρN −2 = ρN cos ϕ′N
ρN −1,N = ρN sin ϕ′N
19
(2.11)
Note that ρN −1,N =
q
2
2
ηN
−1 + ηN is not the distance between the particles N and N + 1. Note also
that the hyper-angular coordinate ϕ′N , defined in Eq. (2.11), is different from the hyper-angular
coordinate ϕN defined in Eq. (2.7).
2.2
The Laplace operator in hyperspherical coordinates
The internal kinetic energy operator for a two-particle system is given by the three-dimensional
Laplace operator, expressed in terms of the relative motion Jacobi coordinate η 1 and the corresponding angular coordinates Ω1 ,
∆(1) = ∆η 1 = ∆η1 −
where the radial part is
∆η1 =
1 2
ℓ̂
η12 1
∂2
2 ∂
+
2
∂η1 η1 ∂η1
(2.12)
(2.13)
and ℓ̂1 is the angular momentum operator of the relative motion.
The internal kinetic energy of a three-particle system is described by the six-dimensional Laplace
operator which is a sum over the three dimensional Laplace operators that act on the coordinates
η 1 and η 2 separately.
∆(2) = ∆η 1 + ∆η 2 = ∆η1 + ∆η2 −
1 2
1 2
ℓ̂
−
ℓ̂
1
η12
η22 2
(2.14)
Using Eq. (2.3) we can transform the two radial coordinates η1 and η2 into the hyper-radial
coordinate ρ2 and the hyper-angular coordinate ϕ2 , in terms of which
∆(2) = ∆ρ2 −
1 2
K̂
ρ22 2
(2.15)
The radial part in Eq. (2.15) depends only on the hyper-radial coordinate ρ2 , i.e.,
∆ρ 2 =
∂2
5 ∂
+
2
∂ρ2 ρ2 ∂ρ2
(2.16)
The hyperspherical angular momentum operator K̂22 is expressed in terms of the hyper-angular
coordinate ϕ2 and the two angular momentum operators ℓ̂21 and ℓ̂22 as follows
K̂22
∂
1
1
∂2
2
+
ℓ̂
+
ℓ̂22
= − 2 − 4 cot(2ϕ2 )
1
2
2
∂ϕ2
∂ϕ2 cos ϕ2
sin ϕ2
20
(2.17)
The internal angular momentum operator of the three-particle system is L̂2 = ℓ̂1 + ℓ̂2 . Note that
L̂22 and L̂2z commute with ∆(2) , ℓ̂21 , ℓ̂22 and K̂22 .
The 3N -dimensional Laplace operator, describing the internal kinetic energy of the N + 1particle system, is a sum over the three-dimensional Laplace operators that act on the coordinates
η1, η2, . . . , ηN
∆(N ) =
N
X
i=1
∆η i =
N
X
i=1
Ã
1
∆ηi − 2 ℓ̂2i
ηi
!
(2.18)
This N + 1-particle Laplace operator can be expressed by means of the recurrence relation
∆(N ) = ∆(N −1) + ∆η N = ∆ρN −1 + ∆ηN −
1
ρ2N −1
K̂N2 −1 −
1 2
ℓ̂
2 N
ηN
(2.19)
We can apply Eq. (2.7) and transform the coordinates ρN −1 and ηN into the hyper-radial coordinate
ρN and the hyper-angular coordinate ϕN . In this case the 3N dimensional Laplace operator, Eq.
(2.18), can be written in the form
∆(N ) = ∆ρN −
where the radial part is
∆ρ N
1 2
K̂
ρ2N N
3N − 1 ∂
∂2
= 2 +
∂ρN
ρN ∂ρN
(2.20)
(2.21)
K̂N2 , the N + 1-particle hyperspherical angular momentum operator, can be expressed in terms of
K̂N2 −1 and ℓ̂2N as follows [Efr72]
K̂N2 = −
∂2
1
1
3N − 6 − (3N − 2) cos(2ϕN ) ∂
2
K̂
+
+
+
ℓ̂2N
N
−1
2
2
∂ϕN
sin(2ϕN )
∂ϕN
cos2 ϕN
sin ϕN
(2.22)
where we define K̂12 ≡ ℓ̂21 . The internal N + 1-particle angular momentum operator is L̂N = L̂N −1 +
ℓ̂N . The operators K̂N2 −1 , ℓ̂2N , K̂N2 , L̂2N and L̂Nz commute with each other, thus each hyperspherical
state is labeled by the complete set of quantum numbers KN , KN −1 , . . . , K2 corresponding to the
hyperspherical angular momentum, LN , LN −1 , . . . , L2 , MNz corresponding to the spatial angular
momentum, and ℓN , ℓN −1 , . . . , ℓ2 , ℓ1 corresponding to the angular part of the Jacobi coordinates.
In the final paragraph of the previous section we constructed the set of N -particle hyperspherical
coordinates in terms of the hyperspherical coordinates corresponding to the first N − 2 Jacobi
coordinates, and the hyperspherical representation of the Jacobi coordinates η N −1 and η N . In
this scheme the N -particle Laplace operator has to be expressed in terms of the Laplace operators
corresponding to the two subsets of coordinates specified.
21
Applying Eq. (2.19) twice and using Eqs. (2.10) and (2.11) we obtain the following expression
for the N + 1-particle Laplace operator
∆(N ) =
(
=
(
∆ρN −2
∆ρN −2
)
(Ã
!
Ã
!)
1
+ ∆ηN − 2 ℓ̂2N
+
∆ηN −1 − 2
− 2
ρN −2
ηN −1
ηN
) (
)
1
1
1
− 2 K̂N2 −2 + ∆ρN −1,N − 2
K̂N2 −1,N = ∆ρN − 2 K̂N′2 .
ρN −2
ρN −1,N
ρN
1
K̂N2 −2
1
ℓ̂2N −1
(2.23)
The structure of the operator ∆ρN −1,N is analogous to that of ∆ρ2 (Eq. (2.16)), substituting ρN −1,N
for ρ2 . The expression for the hyperspherical angular momentum operator K̂N2 −1,N is similar to the
expression for K̂22 (Eq. (2.17)), where the hyper-angular coordinate is ϕN −1,N and the two angular
momentum operators are referred to the Jacobi coordinates η N −1 and η N , i.e.,
K̂N2 −1,N = −
∂2
∂ϕ2N −1,N
− 4 cot(2ϕN −1,N )
∂
∂ϕN −1,N
+
1
cos2 (ϕ
N −1,N )
ℓ̂2N −1 +
1
ℓ̂2
sin (ϕN −1,N ) N
2
(2.24)
The expression finally obtained for the N + 1-particle hyperspherical angular momentum operator
is
K̂N′2 = −
∂2
1
3N − 9 − (3N − 5) cos(2ϕ′N ) ∂
1
2
K̂N2 −2 +
+
+
2 ′ K̂N −1,N
′2
′
′
′
2
∂ϕN
sin(2ϕN )
∂ϕN
cos ϕN
sin ϕN
(2.25)
where ϕ′N was defined in Eq. (2.11). Note that this expression is similar to Eq. (2.22) where ϕN ,
K̂N −1 and ℓ̂N are replaced by ϕ′N , K̂N −2 and K̂N −1,N , respectively. Actually, comparing Eq. (2.20)
and Eq. (2.23) we note that K̂N′2 and K̂N2 are expressions for the same operator in terms of different
sets of coordinates.
2.3
The hyperspherical functions
The angular part of the internal state for two particles is described by the spherical harmonic
Yℓ1 , m1 (Ω1 ). While permutational symmetry adaptation will only concern us in Chapter 4, we point
out in passing that the two-particle state belongs to the irrep [2] of the symmetric group S2 if
ℓ1 is even and to [11] if ℓ1 is odd. Adding one more particle belonging to the state Yℓ2 , m2 (Ω2 )
we form the three-particle state ΦL2 M2 ;ℓ1 ℓ2 (Ω(2) ), which is an eigenstate of the operators ℓ̂21 , ℓ̂22 , L̂22
and L̂2z . This three-particle state is obtained by conventional angular momentum coupling of the
22
states Yℓ1 , m1 (Ω1 ) and Yℓ2 , m2 (Ω2 ),
ΦL2 M2 ;ℓ1 ℓ2 (Ω(2) ) =
X
m1 m2
(ℓ1 m1 ℓ2 m2 |L2 M2 )Yℓ1 , m1 (Ω1 )Yℓ2 , m2 (Ω2 )
(2.26)
in which only the single-Jacobi coordinate angles Ω1 and Ω2 are involved. Recall that Ω(2) ≡
{Ω1 , Ω2 }.
The eigenfunctions of K̂22 , Eq. (2.17), are functions of the hyper-angular coordinate ϕ2 (Eq.
(2.3)) and depend on the value of the quantum number K2 as well as on the values of ℓ2 and
K1 (≡ ℓ1 ), as follows [Efr72]
(ℓ2 + 12 ,ℓ1 + 21 )
ΨK2 ;ℓ2 ℓ1 (ϕ2 ) = N2 (K2 ; ℓ2 ℓ1 ) (sin ϕ2 )ℓ2 (cos ϕ2 )ℓ1 Pn2
(ℓ2 + 12 ,ℓ1 + 21 )
where Pn2
(cos 2ϕ2 ) ,
(2.27)
is the Jacobi polynomial, n2 is a non-negative integer and
K2 = 2n2 + ℓ1 + ℓ2 .
(2.28)
The normalization constant is [Efr72]
"
(2K2 + 4)n2 !Γ(n2 + ℓ2 + ℓ1 + 2)
N2 (K2 ; ℓ2 ℓ1 ) =
Γ(n2 + ℓ2 + 23 )Γ(n2 + ℓ1 + 32 )
#1
2
.
(2.29)
The eigenvalues of K̂22 corresponding to the eigenfunctions (2.27) are K2 (K2 + 4), where K2 ≥
ℓ1 + ℓ2 ≥ L2 ≥ 0 and has the same parity as ℓ1 + ℓ2 (cf. Eq. (2.28)).
The hyperspherical function for three particles, which is an eigenfunction of K̂22 as well as of
L̂22 , is obtained by multiplying the function (2.27) by the function (2.26),
Y[K2 ] (Ω(2) ϕ2 ) = ΨK2 ;ℓ2 ℓ1 (ϕ2 )ΦL2 M2 ;ℓ1 ℓ2 (Ω(2) )
(2.30)
The symbol [K2 ] stands for the aggregate of five good quantum numbers K2 , L2 , M2 , ℓ1 and ℓ2 ,
which completely label the state since there are five (internal) coordinates, i.e., Ω1 = (θ1 , φ1 ),
Ω2 = (θ2 , φ2 ), and ϕ2 .
The construction of basis functions that belong to well-defined irreps of the orthogonal group
O(2) will require the formation of linear combinations of the functions (2.30) with common values
of K2 , L2 and M2 . For ℓ1 and ℓ2 we will have to allow all values that are consistent with the values
of K2 and L2 specified. Consequently, K2 , L2 and M2 remain good quantum numbers after the
23
rotational symmetry adaptation, but instead of ℓ1 and ℓ2 we have the Gel’fand-Zetlin, GZ, symbol
Λ2 (see Appendix A as well as an additional multiplicity label α2 . This step will be discussed
in detail in Chapter 3, but it is convenient to think about [K2 ] as standing for K2 , L2 , M2 ; ℓ1 , ℓ2
before the rotational symmetry adaptation and for K2 , L2 , M2 ; Λ2 , α2 following it. From the point
of view of the present section, Y[K2 ] in either sense is suitable as a parent state for constructing the
four-particle hyperspherical functions.
In view of the fact that for two-particle states the angular momentum ℓ1 and the hyperspherical
angular momentum K1 quantum numbers coincide, the discussion presented above does not exhibit
the generic characteristics of the recursive construction of hyperspherical functions in general. We
therefore proceed to describe the formation of the four-particle hyperspherical functions.
Starting from any three-particle function corresponding to a given set of quantum numbers
K2 and L2 we construct a four-particle hyperspherical function by coupling that function to the
function Yℓ3 , m3 (Ω3 ). This is done in the following three steps. First, by using the O(3) ClebschGordan coefficients we couple the angular momenta L2 and ℓ3 and obtain
ΦL3 M3 ;[K2 ]ℓ3 (Ω(3) ϕ2 ) =
X
M2 m3
(L2 M2 ℓ3 m3 |L3 M3 )Y[K2 ] (Ω(2) ϕ2 )Yℓ3 , m3 (Ω3 )
(2.31)
This function has seven (internal) coordinates, i.e., Ω1 = (θ1 , φ1 ), Ω2 = (θ2 , φ2 ), ϕ2 and Ω3 =
(θ3 , φ3 ).
As a second step we use the transformation (2.5) and construct the eigenfunctions of the hyperangular momentum operator K̂32 , (Eq. (2.22) for N=3) [Efr72]
(ℓ3 + 12 ,K2 +2)
ΨK3 ;ℓ3 K2 (ϕ3 ) = N3 (K3 ; ℓ3 K2 )(sin ϕ3 )ℓ3 (cos ϕ3 )K2 Pn3
(ℓ3 + 12 ,K2 +2)
where Pn3
(cos 2ϕ3 )
(2.32)
is the Jacobi polynomial, n3 is a non-negative integer, and
K3 = 2n3 + K2 + ℓ3 .
The normalization constant N3 (K3 ; ℓ3 K2 ) is [Efr72]
(2K3 + 7)n3 !Γ(n3 + ℓ3 + K2 + 27 )
N3 (K3 ; ℓ3 K2 ) =
Γ(n3 + ℓ3 + 23 )Γ(n3 + K2 + 3)
"
(2.33)
#1
2
.
(2.34)
The eigenvalues of K̂32 corresponding to the eigenfunctions (2.32) are K3 (K3 + 7) where K3 ≥
K2 + ℓ3 ≥ 0 and has the same parity as K2 + ℓ3 (cf. Eq. (2.33)).
24
Having obtained the functions ΦL3 M3 ;[K2 ]ℓ3 (2.31) and ΨK3 ;ℓ3 K2 (2.32) we can construct the
functions Y[K3 ] as the products of these two functions
Y[K3 ] (Ω(3) ϕ(3) ) = ΨK3 ;ℓ3 K2 (ϕ3 )ΦL3 M3 ;[K2 ]ℓ3 (Ω(3) ϕ2 )
(2.35)
where [K3 ] stands for K3 , L3 , M3 , K2 , L2 , ℓ3 , as well as either ℓ1 and ℓ2 or Λ2 and α2 , depending
on the three-particle function we started from.
Assuming that rotational symmetry adapted Y[K2 ] ’s were used, we can now symmetry-adapt
to O(3). This is achieved by forming appropriate linear combinations of Y[K3 ] ’s with common
values of K3 , L3 , M3 and Λ2 but summing over the remaining quantum numbers. When this is
completed [K3 ] should be interpreted to stand for K3 , L3 , M3 , Λ3 and α3 , where Λ3 is the O(3)
Gel’fand-Zetlin symbol and and α3 is the further multiplicity label.
To formulate the general recursive procedure for the construction of the N -Jacobi coordinates
hyperspherical function let us assume that the (N − 1)-Jacobi coordinates functions Y[KN −1 ] are
already available. These functions posses well defined hyperspherical angular momentum KN −1 ,
total angular momentum LN −1 and z-component MN −1 . Assuming that they have been symmetry adapted to O(N − 1), they have as additional quantum numbers the Gel’fand-Zetlin symbol
ΛN−1 and the multiplicity label αN −1 , but these are for the time being irrelevant. The functions
Y[KN −1 ] depend on the set of single coordinate angels Ω(N −1) as well as on the set of hyper-angular
coordinates ϕ(N −1) .
The N + 1-particle hyperspherical function Y[KN ] is now obtained in the following three
steps. First, we couple the function Y[KN −1 ] and the function YℓN , mN (ΩN ) to obtain the function
ΦLN MN ;[KN −1 ]ℓN
ΦLN MN ;[KN −1 ]ℓN (Ω(N ) ϕ(N −1) )
=
X
MN −1 mN
(2.36)
(LN −1 MN −1 ℓN mN |LN MN )Y[KN −1 ] (Ω(N −1) ϕ(N −1) )YℓN , mN (ΩN ) .
Second, using the transformation (2.7) in the subspace of the coordinates (ρN −1 , ηN ) ≡ (ρN , ϕN ),
we construct the orthonormalized eigenfunctions of the hyperspherical angular momentum operator
K̂N2 (2.22)
ΨKN ;ℓN KN −1 (ϕN ) =
(2.37)
(ℓ + 12 ,KN −1 + 3N2−8 )
NN (KN ; ℓN KN −1 )(sin ϕN )ℓN (cos ϕN )KN −1 PnNN
25
(cos 2ϕN )
where nN is a non-negative integer,
KN = 2nN + KN −1 + ℓN ,
(2.38)
and the normalization coefficient is [Efr72]
(2KN + 3N − 5)nN !Γ(nN + KN −1 + ℓN + 3N2−5 )
NN (KN ; ℓN KN −1 ) =
Γ(nN + ℓN + 23 )Γ(nN + KN −1 + 3N2−6 )
"
#1
2
.
(2.39)
The appropriate eigenvalues of the operator K̂N2 (2.22) for the eigenfunctions (2.37) are
KN (KN + 3N − 2)
(2.40)
where KN ≥ KN −1 + ℓN ≥ 0 and has the same parity as KN −1 + ℓN .
Finally, we construct the functions Y[KN ] . These are the N Jacobi coordinates, N + 1-particle,
hyperspherical functions which are coupled to a total angular momentum LN . They are products
of the two functions ΦLN MN ;[KN −1 ]ℓN , Eq. (2.36), and ΨKN ;ℓN KN −1 , Eq. (2.37),
Y[KN ] (Ω(N ) ϕ(N ) ) = ΨKN ;ℓN KN −1 (ϕN )ΦLN MN ;[KN −1 ]ℓN (Ω(N ) ϕ(N −1) )
(2.41)
where [KN ] stands for KN , LN , MN , KN −1 , LN −1 , ℓN and ΛN−1, αN −1 . The hyperspherical
functions Y[KN ] defined in Eq. (2.41) form a complete and orthonormal set of functions that satisfy
< Y[KN ] |Y[KN′ ] >= δ[KN ],[KN′ ] =
(2.42)
δKN ,KN′ δLN ,L′N δMN ,MN′ δKN −1 ,KN′ −1 δLN −1 ,L′N −1 δℓN ,ℓ′N δΛ
δα ,α′
′
N−1 ,ΛN−1 N −1 N −1
.
In the construction of the many-particle hyperspherical functions I used the “tree” method
introduced in Ref. [Vil65]. The “tree” diagram corresponding to the sequential coupling scheme
formulated above is presented in Fig. 2.2. Each segment in this “tree” connects two nodes, except the top segments which connect single-coordinate nodes with Cartesian components of the
corresponding Jacobi coordinate. As I explained in the beginning of the chapter each segment is
called a propagator, and with each node we associate an angle. A propagator that extends to the
right (left) upward from a node corresponds to the cosine (sine) of the angle. The components of
the Jacobi coordinates in the “tree” are obtained by taking the product of ρN , Eq. (2.8), with all
the appropriate cosine and sine functions, starting from the vertex at the bottom of the diagram.
Further, the quantum number associated with each node is presented.
26
Figure 2.2: The “tree” structure that represents the scheme for constructing the eigenfunctions of
the hyperspherical angular momentum operator K̂N2 . The particles are added sequentially.
y
ηN
x z
φr ¡
mN@¡
θr¡
ℓN@@
@
@
ηN −1
y x z
@φ¡
r θ ¡
mN −1
@¡
r
ℓN −1
@
@
@
@
@
@
@
@
@
ηN −2
y x z
@φ¡
r θ ¡
mN −2
¡
r
ℓN @
−2
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
y
φr ¡
m@2 ¡
θr¡
@
ℓ2 @
@
@
η2
x z
@
@
@
y
η1
x z
φr ¡
m@¡
1 θ
@r¡
ℓ1¡
@ ϕ2 ¡
¡
@
@s¡K
2
@
@
@
¡
@
@
¡
@
@
ϕN −2 ¡
@¡
sK
N −2
¡
@ ϕN −1 ¡
¡
@
@
sK
@¡
N −1
¡
@
@ ϕN ¡
¡
@
@s¡K
N
The sequential scheme described above is one of many available routes for constructing the
eigenfunctions of K̂N2 . In Section 2.1 we described an alternative route, in which the last two
Jacobi coordinates are grouped together, separately from the first N − 2 Jacobi coordinates. The
eigenfunctions of the hyperspherical angular momentum operator K̂N2 −1,N (defined in Eq. (2.24))
are
ΨKN −1,N ;ℓN ℓN −1 (ϕN −1,N ) = NN −1,N (KN −1,N ; ℓN ℓN −1 )
(ℓN + 21 ,ℓN −1 + 21 )
nN −1,N
× (sin ϕN −1,N )ℓN (cos ϕN −1,N )ℓN −1 P
(2.43)
(cos 2ϕN −1,N )
where KN −1,N = 2nN −1,N + ℓN + ℓN −1 and nN −1,N is a non-negative integer. These functions
are similar to the functions (2.27) where ϕ2 , K2 , n2 , ℓ2 and ℓ1 , are replaced by ϕN −1,N , KN −1,N ,
nN −1,N , ℓN and ℓN −1 , respectively. Using these substitutions we obtain the normalization constant
NN −1,N (KN −1,N ; ℓN ℓN −1 ) from the expression for N2 (K2 ; ℓ2 ℓ1 ) (Eq. (2.29)).
The hyperspherical functions for N Jacobi coordinates, N + 1 particles, that are the eigenfunctions of the hyperspherical angular momentum operator K̂N′2 (=K̂N2 ) (cf.
Eq.
(2.25)),
are constructed from the hyperspherical functions of the first N − 2 Jacobi coordinates,
ΨKN −2 ;ℓN −2 KN −3 (ϕN −2 ), and the functions of the last two coordinates ΨKN −1,N ;ℓN ℓN −1 (ϕN −1,N ) (Eq.
27
Figure 2.3: The “tree” structure that represents the scheme for constructing the eigenfunctions
of the hyperspherical angular momentum operator K̂N′2 . The last two particles are coupled to one
another before they are coupled to the rest of the system.
y
ηN
x z
φ¡
r
¡
m@
N θ
@
¡
r
ℓN @
ηN −1
y x z
@φ¡
r
¡
mN −1
θ¡
@
r
ℓN −1¡
@ ϕN −1,N ¡
@
¡
@¡
sK
@ N −1,N
@
@
@
@
@
@
ηN −2
y x z
@φ¡
r
¡
mN −2
θ¡
@
r
ℓN −2 @
@
@
@
@
@
@
@
@
y
φr ¡
m@2 ¡
θr¡
@
ℓ2 @
@
@
η2
x z
@
@
y
η1
x z
φr ¡
m@¡
1 θ
@r¡
ℓ1¡
@ ϕ2 ¡
¡
@
@s¡
K
2
@
@
ϕ′N ¡
¡
@
¡
@
¡
¡
@
¡
@
¡
ϕN −2 ¡
sK
@¡
¡ N −2
¡
sK
@¡
N
(2.43)) as follows,
ΨKN ;KN −1,N KN −2 (ϕ′N ) = NN (KN ; KN −1,N KN −2 )
(KN −1,N +2,KN −2 + 3N2−8 )
× (sin ϕ′N )KN −1,N (cos ϕ′N )KN −2 Pn′
N
(2.44)
(cos 2ϕ′N )
where KN = 2n′N + KN −2 + KN −1,N and n′N is a non-negative integer. This expression is similar to
Eq. (2.37), replacing ϕN , nN , ℓN and KN −1 by ϕ′N , n′N , KN −1,N and KN −2 , respectively. Using these
substitutions we obtain the normalization constant NN (KN ; KN −1,N KN −2 ) from the expression for
NN (KN ; ℓN KN −1 ), (Eq. (2.39)). In addition to these substitutions, the correct form of the Jacobi
Polynomial and of the normalization constant in Eq. (2.44) require the addition of the constant
3
2
to KN −1,N and its subtraction from from KN −2 [Efr72]. The “tree” structure corresponding to
this hyperspherical coupling scheme is presented in Fig. 2.3.
28
Figure 2.4: A schematic representation of the Raynal-Revai coefficients , RR, between two sets of
Jacobi coordinates coupled to the same node. Only the relevant part of the “tree” is plotted.
ηN
rℓ
@N
@
@
ϕN −1¡
η N −1
rℓ
¡ N −1
¡
@s¡K
@ N −1,N
¡
@
¡
@ ϕN ¡
@s¡K
¡
¡
¡ ¾
¡
sK
N −2
η ′N
rℓ
@N
@
@
¡
¡
- @¡
¡
s K
@ N −1,N
¡
@
¡
@ ϕN ¡
@s¡K
RR
N
2.4
ϕ′N −1,N ¡
η ′N −1
rℓ
¡ N −1
¡
¡
sK
N −2
N
The Raynal-Revai and the T-coefficients
The hyperspherical functions, as constructed in the previous section, depend on the choice of the
Jacobi coordinates, Eq. (B.1), as well as on the “tree” diagram chosen for coupling the various
single-Jacobi coordinate functions. I first consider the transformation of the hyperspherical harmonics which results from transforming between different sets of two Jacobi coordinates connected
to the same node in the “tree” diagram. This transformation is effected by means of the RaynalRevai coefficients presented in Subsection 2.4.1. In Subsection 2.4.2 I discuss the transformation
between the two sets of hyperspherical functions corresponding to two different coupling schemes,
i.e. two different “tree” diagrams, that is effected by means of the T-coefficients. It can be easily
verified that the transformation of the HH induced by any transformation of the Jacobi coordinates
can be expressed in terms of these two sets of coefficients.
2.4.1
The Raynal-Revai coefficients
To enable the transformation between sets of hyperspherical functions obtained for the same physical system using different choices of Jacobi coordinates we use the Raynal-Revai coefficients. More
specifically these coefficients are needed in order to apply transformations between different sets of
two Jacobi coordinates connected to the same node in the “tree” diagram, as can be seen in Fig.
2.4.
As was pointed out at the bottom of Section 2.2 the N + 1-particle hyperspherical angular mo29
mentum operator K̂N2 is independent of the set of the angular coordinates Ω(N ) and of the structure
of the “tree” or of the choice of the Jacobi coordinates. Let us consider the two Jacobi coordinates η N −1 and η N . The hyperspherical functions constructed from these two Jacobi coordinates
are uniquely specified by the quantum numbers ℓN −1 , ℓN , KN −1,N , LN −1,N and MN −1,N , (as was
explained in the previous section). These hyperspherical functions can be expressed in terms of
the hyperspherical functions depending on two different Jacobi coordinates η ′N −1 and η ′N , that are
related to the first two by means of a rotation by an angle γ, using the Raynal-Revai coefficients
as follows:
|(ℓN −1 ; ℓN )KN −1,N LN −1,N MN −1,N >
=
X
ℓ′N −1 ℓ′N
< ℓN −1 ℓN |γ|ℓ′N −1 ℓ′N >KN −1,N LN −1,N |(ℓ′N −1 ; ℓ′N )KN −1,N LN −1,N MN −1,N > ,
(2.45)
here ℓ′N −1 ,ℓ′N are the values of the single-coordinate angular momentum operators corresponding
to the rotated coordinates η ′N −1 and η ′N . The sum in Eq. (2.45) is restricted by the relation
|ℓ′N −1 − ℓ′N | ≤ LN −1,N ≤ ℓ′N −1 + ℓ′N ≤ KN −1,N such that ℓ′N −1 + ℓ′N has the same parity as KN −1,N .
Note that the quantum numbers KN −1,N and LN −1,N (as well as MN −1,N ) are common to both sets
of hyperspherical functions, as was explained above.
The Raynal-Revai coefficients defined by Eq. (2.45) can be expressed in terms of the harmonicoscillator Talmi-Moshinsky brackets [Ray73]. Using this relation, an analytic expression for the
transformation brackets for hyperspherical functions was obtained by Raynal and Revai for three
particles [Ray70], [Ray73]. A generalization to four particles was studied in Ref. [Jib77].
In the context of this thesis we are not going to use the Raynal-Revai coefficients defined in Eq.
(2.45), however we are interested in the action of the infinitesimal generator of the Raynal-Revai
rotation, Eq. (A.1),
X̂N −1,N = i(η N −1 ·∇N − η N ·∇N −1 ) ,
(2.46)
on the hyperspherical harmonics. Analytic expressions for the matrix elements of the operator
X̂N −1,N in the hyperspherical basis were obtained by Raynal [Ray73]. This operator can change
the values of ℓN −1 and ℓN by ±1 and does not change the other quantum numbers K = KN −1,N ,
L = LN −1,N and MN −1,N . The expressions obtained by Raynal are (Eqs. (23) and (24) in Ref.
30
[Ray73])
1
< ℓ′N −1 ℓ′N |X̂N −1,N |ℓN −1 ℓN >K,L = [n(n + ℓN −1 + ℓN + 2)(L + ℓN −1 + ℓN + 2)] 2
"
(L + ℓN −1 + ℓN + 3)(ℓN −1 + ℓN − L + 1)(ℓN −1 + ℓN − L + 2)
×
(2ℓN −1 + 1)(2ℓN + 1)(2ℓ′N −1 + 1)(2ℓ′N + 1)
#1
2
(2.47)
where ℓ′N −1 = ℓN −1 + 1 and ℓ′N = ℓN + 1, and
<
ℓ′N −1 ℓ′N |X̂N −1,N |ℓN −1 ℓN
¸1
1
3
>K,L = (n + ℓN + )(n + ℓN −1 + )
2
2
·
"
2
(L + ℓN − ℓN −1 − 1)(L + ℓN − ℓN −1 )(L + ℓN −1 − ℓN + 1)(L + ℓN −1 − ℓN + 2)
×
(2ℓN −1 + 1)(2ℓN + 1)(2ℓ′N −1 + 1)(2ℓ′N + 1)
#1
2
(2.48)
where ℓ′N −1 = ℓN −1 + 1 and ℓ′N = ℓN − 1. The variable n is given by
n=
K − ℓN −1 − ℓN
.
2
(2.49)
The other two non-zero matrix elements i.e., ℓ′N −1 = ℓN −1 − 1, ℓ′N = ℓN − 1 and ℓ′N −1 = ℓN −1 − 1,
ℓ′N = ℓN + 1 have the same expressions as in Eqs. (2.47) and (2.48), respectively, but with opposite
signs.
2.4.2
The hyperspherical recoupling coefficients - the T-coefficients
The hyperspherical recoupling coefficients are the recoupling coefficients that enable the expression
of a hyperspherical function obtained by coupling of three subsystems in a particular order in
terms of the set of hyperspherical functions describing the same composite system, obtained using
a different coupling order.
According to the sequential scheme specified in the previous section the functions of the N ′ th
Jacobi coordinate are coupled to appropriate (N − 1)-Jacobi coordinates hyperspherical functions,
Eq. (2.37). The alternative scheme, in which we couple the hyperspherical functions of the last two
coordinates to suitable (N − 2)-coordinate sequentially coupled hyperspherical functions, results
in a different set of N -Jacobi coordinates hyperspherical functions, Eqs. (2.43) and (2.44). The
transformation between these two coupling schemes is the case which is relevant to our present
31
Figure 2.5: A schematic representation of the T-coefficient between the “tree” of Fig. 2.2 and the
“tree” of Fig. 2.3 Ȯnly the relevant parts of the two “trees” are plotted. The two-headed arrow
points to the two different intermediate quantum numbers.
rℓ
@N
@
sK
rℓ
@ N −1
¡ N −2
@ ϕN −1 ¡
@
@
¡
T
¾
@
@
¡
s
K
N −1
@
¡
@
¡
@ ϕN ¡
@s¡K
sK
rℓ
rℓ
@N
¡ N −1
¡ N −2
@ ϕN −1,N ¡
¡
@
¡
¡
- @¡
¡
s K
@ N −1,N
¡
@
¡
@ ϕ′N ¡
@s¡K
N
N
problem. In other words, we are interested in the transformation from the “tree” diagram given in
Fig. 2.2 to the “tree” diagram given in Fig. 2.3. This transformation is effected by means of the
hyperspherical recoupling coefficients (of type F) introduced in Ref. [Kil72]. The transformation
is presented in Fig. 2.5 which is the figure of case F in Ref. [Kil72] ); Only the relevant parts of the
two “trees” are plotted. This hyperspherical recoupling coefficients are known as the T-coefficients,
where ”T” stands for “tree” recoupling coefficients.
More specifically, the T-coefficients presented many years ago by Kil’dyushov [Kil73] are the
hyperspherical analogs of the celebrated angular momentum 6j symbols, and indeed they could
as justly be called 6K symbols. Let us denote by Ki , i = 1, 2, 3 an harmonic polynomial over
a coordinate space of dimension νi . We can couple three polynomials to create an harmonic
polynomial of order K over ν1 + ν2 + ν3 dimensional space in, essentially, two different ways. We
can couple K1 and K2 to obtain K12 and then couple K12 with K3 , or couple K1 and K3 to
obtain K13 and then couple K13 with K2 . The T-coefficients, like the 6j coefficients, are the matrix
elements of the transformation between these two sets of functions,
|((K1 ; K2 )K12 ; K3 )K >=
X
K13
1 K2 K3
TKK;K
|((K1 ; K3 )K13 ; K2 )K >
12 K13
(2.50)
We introduce the following definitions
3
αi ≡ Ki + νi − 1
2
K12 − K1 − K2
K13 − K1 − K3
n12 ≡
; n13 ≡
2
2
K − K1 − K2 − K3
n≡
2
32
(2.51)
and follow Raynal derivation in Ref. [Ray73b] to obtain
1 K2 K3
TKK;K
= (−)n−n12 −n13 Bn (n12 , α1 , α2 , α3 )Bn (n13 , α1 , α3 , α2 )
12 K13
4 F3 (n12
− n, n13 − n, −n − n12 − α1 − α2 − 1, −n − n13 − α1 − α3 − 1;
−2n − α1 − α2 − α3 − 1, −n − α1 , −n; 1)
(2.52)
where
Bn (n12 , α1 , α2 , α3 ) =
h
h
(2n12 +α1 +α2 +1)Γ(n12 +α1 +α2 +1)
n!
n12 !(n−n12 )! Γ(n+n12 +α1 +α2 +2)Γ(n−n12 +α3 +1)
Γ(2n+α1 +α2 +α3 +2)Γ(n12 +α2 +1)Γ(n+α1 +1)
Γ(n12 +α1 +1)Γ(n+n12 +α1 +α2 +α3 +2)
i1
2
i1
2
(2.53)
and Bn (n13 , α1 , α3 , α2 ) has the same expression as (2.53) with the appropriate substitutions. The
function 4 F3 is the generalized hypergeometric function defined by the relation
4 F3 (a1 , a2 , a3 , a4 ; b1 , b2 , b3 ; x)
=
∞
X
(a1 )k (a2 )k (a3 )k (a4 )k xk
k=0
where
ck =
and c0 = 1.
(b1 )k (b2 )k (b3 )k
k!
=
∞
X
c k xk
(2.54)
k=0
(a1 + k − 1)(a2 + k − 1)(a3 + k − 1)(a4 + k − 1)
ck−1
(b1 + k − 1)(b2 + k − 1)(b3 + k − 1)k
(2.55)
It should be noted that expression (2.52) is symmetric with respect to permutation of the indices
2 and 3. In addition, since some of the variables in the function 4 F3 are negative integers (see Eq.
(2.52)) the sum in Eq. (2.54) reduces to a polynomial of the order of the lowest absolute value of
these negative integers.
33
Chapter 3
Construction of Hyperspherical
functions Symmetrized with respect to
the Orthogonal Group
34
The construction of N + 1-body states that belong to a well defined irreducible representations of the orthogonal group is a long standing problem in physics. In this chapter I present
the algorithm, we developed [Bar97a], for constructing such states, in terms of the hyperspherical
harmonics basis, which are a standard basis to study few-body problems in nuclear, atomic and
molecular physics. However, this is a general algorithm, appropriate for any quantum mechanical
basis functions as long as they are expressed in terms of arbitrary orthogonal coordinates and form
an invariant subspace under the action of the orthogonal group O(N ).
The method described in this chapter and in Chapter 4 consists of two independent but similar
steps that are combined to yield N-particle states, expressed in terms of hyperspherical functions
which belong to well defined irreps of the orthogonal group and the symmetric group. The first
step is the construction of hyperspherical functions which belong to irreps of the orthogonal group
O(N ). The second step is the reduction of the irreps of O(N ) into irreps of the symmetric group
SN +1 .
The hyperspherical functions that are constructed in the first step belong to O(N ) irreps and
are labeled according to the group-subgroup chain O(N ) ⊃ O(N − 1) ⊃ · ⊃ O(3) ⊃ O(2). We call
these states the hyperspherical GZ states after Gel’fand and Zetlin who first introduced this O(N )
labeling scheme [Gel50]. The hyperspherical GZ states are constructed by using a variant of the
recursive approach introduced in Ref. [Nov94]. The main idea is to start from hyperspherical GZ
states of the O(N − 1) group which are expressed in terms of the first (N −1) Jacobi coordinates and
have well defined angular momentum and hyperspherical angular momentum quantum numbers.
The N ’th Jacobi coordinate state is coupled to these states, obtaining N +1-particle hyperspherical
functions with well defined angular momentum and hyperspherical angular momentum quantum
numbers. The states which have the same value of these quantum numbers and belong to the same
O(N − 1) irrep are diagonalized with respect to the second order Casimir operator of the O(N )
group. As the eigenvalues of the second Casimir operator of O(N ) are degenerated with respect
to the O(N ) irreps, after diagonalization we can only identify the O(N ) irreps that belong to the
non-degenerated eigenvalues of this second Casimir operator. The rest of the irreps are identified
using the Grahm-Schmidt orthogonalization procedure. The eigenvectors obtained in this process
are the hyperspherical orthogonal group parentage coefficients (hsopcs). Note that in order to
determine a consistent phase between states that belong to the same O(N ) irrep but originate
35
from two different O(N − 1) irreps we must introduce a phase consistent algorithm. In reference
[Bar97], I use a different approach which avoids the Grahm-Schmidt orthogonalization procedure
by simultaneous diagonalization of all the Casimir operators. Although this approach is more
elegant from mathematical point of view, the implementation of the Grahm-Schmidt procedure
presented here is simpler.
The recursive algorithm for reduction of the orthogonal group irreps into irreps of the symmetric
group will be presented in the next chapter but follows very similar lines. In this case we start
from O(N − 2) irrep states that belong to an irrep of the symmetric group SN −1 . We embed
these O(N − 2) states into an O(N − 1) irrep to obtain O(N − 1) states. Then we diagonalize
the transposition class-sum operator of the symmetric group SN within an invariant subspace that
consists of states with the same SN −1 and O(N − 1) irreps. In this case the eigenvalues uniquely
identify the SN irreps and the eigenvectors consist of the orthogonal group coefficients of fractional
parentage (ocfps).
We start this chapter by reviewing the concept of parentage coefficients in view of its role in
our algorithm. We also define the orthogonal parentage parentage coefficients, opcs, and their
properties. Section 3.2 is devoted to the 3-body case. In Subsection 3.2.1 we give an illustrative
example with two Jacobi coordinates and describe the algorithm for constructing hyperspherical
functions that belong to well defined irreps of the kinematic SO(2) group. The explicit analytic
expressions of Whitten & Smith [Whi68, Whi68] for L = 0 and ours [Bar90, Bar91] for L = 1 are
given in Subsection 3.2.2. The new algorithm for constructing N -Jacobi coordinate hyperspherical
functions that belong to well defined irreps of the SO(N ) group, is given in Section 3.3. In Section
3.4 I present in details the method of implementing the recursive algorithm of Subsection 3.2.1
and Section 3.3 into an efficient computational algorithm. For large number of particles, large
N , a considerable improvement of the computational algorithm can be achieved as for practical
calculations one needs only the O(N − 1) → O(N ) opcs (for more details see Chapter 5). The
improved algorithm is presented in Section 3.5. A review of the orthogonal group, O(N ), and of
the special orthogonal group SO(N ) is given in Appendix A.
36
3.1
The Orthogonal Parentage Coefficients
The rank, r, of the Lie algebra associated with the Lie group O(N ), is given by r = [ N2 ]. Therefore
the irreducible representations, irreps, of the orthogonal group O(N ), λN , are labeled by λN =
(λN,1 , λN,2 , . . . λN,r ). The values of λN,j are positive and either all integers or half integers. In
the case N = 2r, the O(N ) representations (λN,1 , λN,2 , . . . λN,r ) splits into the two SO(N ) irreps
(λN,1 , λN,2 , . . . λN,r ) and (λN,1 , λN,2 , . . . − λN,r ) if λN,r 6= 0. While for λN,r = 0 or odd N the
irreducible representations of O(N ) are irreducible under SO(N ). In what follows we shall be
working with the group SO(N ) remembering the connection between its irreps and the irreps of
O(N ). The basis vectors of an irrep λN are completely specified by the Gel’fand-Zetlin [Gel50]
states, ΛN . This states, ΛN , are labeled by the canonical chain of subgroups SO(N ) ⊃ SO(N −1) ⊃
·· ⊃ SO(3) ⊃ SO(2), or ΛN = [λN λN −1 ...λ2 ], and are restricted according to the following rules,
given by Gel’fand and Zetlin [Gel50], λk,j ≥ λk−1,j ≥ |λk,j+1 | and λk,j ≥ |λk−1,j |,k = 3, . . . , N .
The purpose of this chapter is to present a recursive method to transform the hyperspherical
harmonic basis, {|a1 >≡ |Y[KN ] > a1 = 1, .., dim(V )}, of an SO(N ) invariant vector space V ,
defined by the quantum numbers KN ,LN , into the Gel’fand-Zetlin basis. Here we use the notation
|a1 > for the “tree” hyperspherical harmonic basis, |Y[KN ] > ( [KN ] stands for the quantum numbers
KN , KN −1 , . . . , K2 , LN , . . . , L2 , ℓN , . . . , ℓ1 ). In other words we seek the transformation
dim(V )
|ΛN aN >=
X
a1 =1
(N )
UΛN aN ,a1 |a1 > .
(3.1)
Here aN is the degeneracy of the SO(N ) irrep λN in V , i.e. in the invariant vector space KN ,LN .
The main idea in the recursive method is to present the unitary transformation matrix U(N ) whose
matrix elements are < Y[KN ] |KN LN MN ΛN aN > as a product of a sequence of block diagonal
matrices {C(m) , m = 2, ..., N }, where C(m) , the step transformation matrix, is used to transform
the O(m − 1) GZ states into O(m) states,
|Λm am >=
X
(m)Λ
am−1
|Λm−1 am−1 > .
Cλ a m−1
m m ,am−1
(3.2)
The step transformation matrix C(m) is block diagonal since Λm = [λm , Λm−1 ] and it doesn’t alter
the value of Λm−1 . Therefore, we can see that
(k)
UΛk ak ,a1 =
X
ak−1 ,ak−2 ,...,a2
(k)Λ
(k−1)Λ
(2)
Cλ a k−2,a · ·Cλ a ,a .
Cλ a k−1
2 2 1
k−2 k−1 k−2
k k ,ak−1
37
(3.3)
The orthogonal parentage coefficients, opcs, are the matrix elements of the step transformation
(m)Λ
. The opcs, defined in equation (3.2) satisfy, the orthogonality relation
matrix Cλ a m−1
m m ,am−1
X
(m)Λ
∗
(m)Λ
= δλ′ ,λ δa′m ,am
C ′ m−1
Cλ a m−1
m
m
λ m a′m ,am−1
m m ,am−1
am−1
(3.4)
which follows from the orthogonality of the m’th step basis states. The opcs also satisfy the
completeness relation
X
(m)Λ
(m)Λ
= δa′m−1 ,am−1
)∗ Cλ a m−1
(Cλ a m−1
′
m m ,am−1
m m ,am−1
(3.5)
λm ,am
As an example, let V be the invariant subspace created by the outer product of two irreps of
O(N ), i.e. V = λ1N ⊗ λ2N . In this case the matrix elements of the transformation matrix U(N )
which transforms the outer product subspace λ1N ⊗ λ2N into the GZ states λN are just the O(N )
Clebsh-Gordon coefficients, and the opcs are the O(N ) isoscalar factors.
In what follows we will use different notation for the hyperspherical orthogonal coefficient of
fractional parentage, hsopcs. In accordance with the usual notation of coefficient of fractional
parentage, the hsopcs will be denoted by
(N )Λ
Cλ a N,a−1 ≡ [(KN −1 LN −1 λN −1 αN −1 ; ℓN )KN LN |}KN LN λN αN ] ,
N N N −1
(3.6)
with the following interpretation: the O(N ) invariant subspace V is the subspace of hyperspherical
functions with common values of KN , LN and MN . The degeneracy, aN , of the O(N ) irrep λN
in V is replaced by αN in order to emphasize the fact that in the recursive construction of the
GZ hyperspherical functions αN stands for the degeneracy of the irreps of O(N ) in an invariant
subspace of N Jacobi coordinates with definite values of KN and LN , as a result aN −1 in the left
hand side of Eq. (3.6) should be identified with (KN −1 LN −1 αN −1 ℓN ).
3.2
3.2.1
Construction of 3-Body Functions
The 3-Body Algorithm
The three body HH functions are usually expressed in terms of the Jacobi coordinates when the
center of mass is removed. For three particle systems the two normalized Jacobi coordinates are
38
(see Appendix B)
η1 =
s
m1 m2
(r2 − r1 ) ; η 2 =
m1 + m2
s
(m1 + m2 )m3 ³
m 1 r1 + m 2 r2 ´
r3 −
m1 + m2 + m3
m1 + m2
(3.7)
As we have seen in Chapter 2 the hyperspherical coordinates are obtained from these two Jacobi coordinates ρ1 ≡ η1 , and η2 , by transforming them into a hyper-radial coordinate ρ2 and a
hyperangular coordinate ϕ2 via
η1 = ρ1 = ρ2 cos ϕ2
η2 = ρ2 sin ϕ2
.
(3.8)
This transformation does not change the spherical angular coordinates Ω1 ≡ {θ1 , φ1 } and Ω2 ≡
{θ2 , φ2 } of the coordinates η 1 and η 2 , respectively. The angular part of the HH functions for three
particles, Eq. (2.26), is obtained by angular momentum coupling of the two spherical harmonic
functions Yℓ1 ,m1 (Ω1 ) and Yℓ2 ,m2 (Ω2 )
ΦL2 M2 ;ℓ1 ℓ2 (Ω(2) ) =
X
m1 m2
(ℓ1 m1 ℓ2 m2 |L2 M2 )Yℓ1 , m1 (Ω1 )Yℓ2 , m2 (Ω2 )
(3.9)
where Ω(2) ≡ {Ω1 , Ω2 }, and (ℓ1 m1 ℓ2 m2 |L2 M2 ) are the Clebsch-Gordan coefficients. The hyperan-
gular part of the three particles HH functions is given by Eq. (2.27). These functions are functions
of the hyperangular coordinate ϕ2 and depend on the hyperspherical angular momentum K2 and
the angular momenta ℓ1 (≡ K1 ≡ L1 ) and ℓ2 [Nov94]
(ℓ2 + 21 ,ℓ1 + 21 )
ΨK2 ;ℓ2 ℓ1 (ϕ2 ) = N2 (K2 ; ℓ2 ℓ1 ) (sin ϕ2 )ℓ2 (cos ϕ2 )ℓ1 Pn2
(ℓ2 + 12 ,ℓ1 + 21 )
where Pn2
(cos(2ϕ2 )) ,
(3.10)
is the Jacobi polynomial, n2 is a non-negative integer and
K2 = 2n2 + ℓ1 + ℓ2 .
(3.11)
The normalization constant N2 (K2 ; ℓ2 ℓ1 ) is given in Eq. (2.29). The eigenvalues of K̂22 correspond-
ing to the eigenfunctions (3.10) are K2 (K2 + 4), where K2 ≥ ℓ1 + ℓ2 ≥ 0, and has the same parity
as ℓ1 + ℓ2 (cf. Eq. (3.11)).
The hyperspherical state, expressed in terms of the two Jacobi coordinates η 1 and η 2 is constructed as a product of the function (3.9) by the function (3.10),
|(ℓ1 ; ℓ2 )K2 L2 M2 >= ΦL2 M2 ;ℓ1 ℓ2 (Ω(2) )ΨK2 ;ℓ2 ℓ1 (ϕ2 ) .
39
(3.12)
The product of the two functions on the right hand side of Eq. (3.12) by the function (of the
2
hyper-radial coordinate) ρK
2 is a polynomial of the order K2 in the first two Jacobi coordinates η 1
and η 2 . This result is obtained since the function in Eq. (3.9) is a polynomial of the order ℓ1 + ℓ2
(a product of two spherical harmonics) and the Jacobi polynomial in Eq. (3.10) is of the order
K2 − ℓ1 − ℓ2 .
The states in Eq. (3.12) are eigenfunctions of K̂22 as well as of L̂22 . However, they do not belong
to a well defined irrep of the kinematic SO(2) group. On the other hand, the states (3.12) that
have the same values of L2 and K2 , span an invariant subspace with respect to the kinematic
SO(2) group since its generator commutes with K̂2 and it is a scalar operator with respect to space
rotations. Our task is to determine the linear combinations of these states which yield the states
with well defined orthogonal symmetry. In other words, we have to find the following hsopcs
|K2 L2 M2 λ2 α2 >=
X
ℓ1 ,ℓ2
[(ℓ1 ; ℓ2 )K2 L2 |}K2 L2 λ2 α2 ] |(ℓ1 ; ℓ2 )K2 L2 M2 >
(3.13)
where the values of ℓ1 and ℓ2 are restricted by the rules that result from Eqs. (3.9) and (3.10).
We need the additional label α2 in Eq. (3.13) to remove possible degeneracy of the three-particle
hyperspherical states that belong to the same SO(2) irrep and have the same quantum numbers.
To evaluate the hsopcs it is sufficient to diagonalize the quadratic Casimir operator of SO(2),
2
C2 (2) = X̂1,2
defined in Eq. (A.16). Since our functions are polynomials of the order K2 the SO(2)
irreps that we expect to obtain are restricted by this number, i.e., |λ2,1 | ≤ K2 . This results from the
fact that we realize a polynomial irrep of SO(2) of order λ2,1 , in the space of order K2 polynomials.
2
Our task is actually to calculate the matrix elements of the operator C2 (2) = X̂1,2
within the
invariant subspace consisting of the states (3.13) with the same quantum numbers K2 , L2 and M2 ,
as we explained above. Analytic expressions, obtained by Raynal [Ray73], for the matrix elements
of the operator X̂1,2 in the hyperspherical basis were presented in Subsection 2.4.1. Substituting
ℓ1 for ℓN −1 , ℓ2 for ℓN , K2 for K and L2 for L, we can see that this operator can change the values
of ℓ1 and ℓ2 by ±1 and does not change the other quantum numbers K2 , L2 and M2 . The non zero
matrix elements, Eqs. (2.47),(2.48), are
1
< ℓ′1 ℓ′2 |X̂1,2 |ℓ1 ℓ2 >K2 ,L2 = [n2 (n2 + ℓ1 + ℓ2 + 2)(L2 + ℓ1 + ℓ2 + 2)] 2
"
(L2 + ℓ1 + ℓ2 + 3)(ℓ1 + ℓ2 − L2 + 1)(ℓ1 + ℓ2 − L2 + 2)
×
(2ℓ1 + 1)(2ℓ2 + 1)(2ℓ′1 + 1)(2ℓ′2 + 1)
40
#1
2
(3.14)
where ℓ′1 = ℓ1 + 1 and ℓ′2 = ℓ2 + 1, and
<
ℓ′1 ℓ′2 |X̂1,2 |ℓ1 ℓ2
¸1
3
1
>K2 ,L2 = (n2 + ℓ2 + )(n2 + ℓ1 + )
2
2
·
2
"
(L2 + ℓ2 − ℓ1 − 1)(L2 + ℓ2 − ℓ1 )(L2 + ℓ1 − ℓ2 + 1)(L2 + ℓ1 − ℓ2 + 2)
×
(2ℓ1 + 1)(2ℓ2 + 1)(2ℓ′1 + 1)(2ℓ′2 + 1)
#1
2
(3.15)
where ℓ′1 = ℓ1 + 1 and ℓ′2 = ℓ2 − 1. The variable n2 is defined in Eq. (3.11). The other two non-zero
matrix elements i.e., ℓ′1 = ℓ1 − 1, ℓ′2 = ℓ2 − 1 and ℓ′1 = ℓ1 − 1, ℓ′2 = ℓ2 + 1 have the same expressions
as in Eqs. (3.14) and (3.15), respectively, but with opposite signs. Using Eqs. (3.14)-(3.15) we can
2
calculate the matrix elements of the operator X̂1,2
by introducing a complete intermediate set of
hyperspherical states associated with the first two hyperspherical coordinates, i.e.,
2
|ℓ1 ℓ2 >K2 ,L2 =
< ℓ′1 ℓ′2 |X̂1,2
X
′′
ℓ′′
1 ℓ2
< ℓ′1 ℓ′2 |X̂1,2 |ℓ′′1 ℓ′′2 >K2 ,L2 < ℓ′′1 ℓ′′2 |X̂1,2 |ℓ1 ℓ2 >K2 ,L2
(3.16)
In the method described in this subsection we can construct hyperspherical functions, resulting
from the first two Jacobi coordinates, (i.e., 3-particles) which belong to well defined irreps of the
orthogonal group O(2). In order to split those O(2) irreps into its SO(2) components we have,
in addition, to diagonalize the X̂1,2 generator, as I explained in Sect. 3. In the next section I
present the general recursive procedure for constructing hyperspherical functions in terms of N
Jacobi coordinates that belong to well defined irreps of SO(N ).
3.2.2
Explicit Construction of 3-Body Functions for L = 0, 1
We will reconsider now the three body problem, and present analytic expressions for the rotational
hyperspherical harmonics using a different parameterization of the six dimensional sphere. For
three nonrelativistic particles the mass weighted Jacobi coordinates (B.1) are given by
η1 =
s
m1 m2
(r2 − r1 ) ; η 2 =
m1 + m2
s
m 1 r1 + m 2 r2 ´
(m1 + m2 )m3 ³
r3 −
m1 + m2 + m3
m1 + m2
(3.17)
Consider the coordinate system whose origin is located in the center of mass motion and whose axes
1, 2 are laying in the plane defined by a triangle where vertices are determined by the positions
of the three particles. Since in this body-fixed coordinate system η13 = η23 = 0, the diagonal
41
components of the inertia tensor are given by [Lan60]
I11 = (η12 )2 + (η22 )2 , I22 = (η11 )2 + (η21 )2
I33 =
X
i = 12[(η1i )2 + (η2i )2 ] = I11 + I22 ≡ ρ22
(3.18)
while the only non diagonal component is
I12 = I21 = −[η11 η12 + η21 η22 ]
(3.19)
Choosing the direction of axes 1, 2 along the principal axes force the condition I12 = 0. We can
use the following parameterization [Whi68, Whi68] for the components of the Jacobi vectors η 1
and η 2
Θ
φ
Θ
φ
cos , η21 = ρ2 cos sin
2
2
2
2
φ
Θ
φ
Θ
= −ρ2 sin sin , η22 = ρ2 sin cos .
2
2
2
2
η11 = ρ2 cos
η12
(3.20)
One can see, that since
q
I1 = ρ2 sin
Θ q
Θ q
, I2 = ρ2 cos , I3 = ρ2
2
2
(3.21)
(where we use the notation Ia instead of Iaa ), the variables ρ2 ,Θ determine the values of the
components of the inertia tensor. The additional variable φ fixes the scalar product of the Jacobi
vectors:
1
η 1 · η 2 = (I2 − I1 ) sin φ
2
From Eq. (3.21) one can assume that the variables are changing in the intervals
0 ≤ ρ2 < ∞ , 0 ≤
π
φ
Θ
≤ , 0 ≤ ≤ 2π
2
2
2
(3.22)
(3.23)
However,in view of (3.22), the change of Θ to π − Θ results only in the interchange of I1 and I2 .
This reflection of the axes 1, 2 , however, is properly described by 180o rotation of the plane of the
triangle around the axis in the plane which bisects the angle between the axes 1 and 2. Since such
a rotation is contained among the Euler rotations of the plane, it should not be included among
transformations describing triangle deformations. Therefore the proper interval should be
0 ≤ ρ2 < ∞ , 0 ≤ Θ ≤
π
, 0 ≤ φ ≤ 4π
2
(3.24)
which base a choice of directions of the axes 1 and 2 on the condition that the difference I2 − I1
is always positive. In order to be able to use an algebra of irreducible tensors one has to deal with
42
the Jacobi coordinates in the spherical representation. Following [Edm60], one can introduce the
spherical components
1
ηk± = −[± √ (ηk1 ± iηk2 )] , ηk0 = ηk3 , k = 1, 2
2
(3.25)
M
M
M
η±
1 = −[±(η1 ± iη2 )] , M = ±, 0
(3.26)
and
2
Simple calculation gives [Whi68, Whi68]
a
φ
exp i
2
2
a
φ
= −ρ2 sin exp −i
2
2
a
φ
= −ρ2 sin exp i
2
2
a
φ
= ρ2 cos exp −i
2
2
+
= ρ2 cos
η+
1
2
+
η−
1
2
−
η+
1
2
−
η−
1
2
(3.27)
where instead of Θ we introduce a new variable a = Θ + π2 . Eq. (3.27) could be more elegantly
written with the help of the Wigner D functions [Whi68, Whi68]
1
1
1
M
ησM = (−1)( 2 +σ)( 2 − 2 ) ρ2 Dσ,2 M (φ, a, 0).
(3.28)
2
A rotation through the Euler angles finally gives the presentation of the Jacobi vectors in the
space-fixed coordinate system [Whi68, Whi68]:
ησM =
X
M =±,0
1
1
1
M
1
DM,N
(ϕ, θ, ψ)(−1)( 2 +σ)( 2 − 2 ) ρ2 Dσ,2 M (φ, a, 0).
(3.29)
2
Here ϕ, θ, ψ are the Euler angles defining an orientation of a triangle the vertices of which are determined by positions of the three particles and ρ2 , a, φ are so called kinematic variables characterizing
the triangle itself. Introducing complex vector notations [Bad66, Sim66],
z = −η 1 , z∗ = (−η 1 )∗ = η − 1
2
2
2
(3.30)
one can write [Nyi69]
φ
z = ρ2 e−i 2 [l− cos (a/2) − l+ sin (a/2)]
φ
z∗ = ρ2 ei 2 [l+ cos (a/2) − l− sin (a/2)]
(3.31)
In derivation of Eq. (3.31) in order to comply with the notations used in the previous papers
[Bar90, Man80, Haf87, Haf88, Haf89, Haf89b, Man90, Haf90b] we have change the notations a
43
to −a and φ to −φ, φ = λ in these references. The angles φ,a are changing in intervals (0, 4π)
and (0, π/2), respectively. The components M of the complex vectors l± and l0 = i(l− × l+ ) are
expressed through the Wigner D-function [Edm60] in the following fashion
M
1
lN
= DM,N
(ϕ, θ, ψ), M, N = ±, 0.
(3.32)
In view of (3.26) z, z∗ are simply a linear combinations of the Jacobi coordinates:
z = η 1 + iη 2 , z∗ = η 1 − iη 2
(3.33)
(ρ2 )2 = z · z∗ , z2 = (ρ2 )2 sin (a)e−iφ , z∗ 2 = (ρ2 )2 sin (a)e−iφ .
(3.34)
Clearly
In correspondence with Eq. (B.8) the kinetic energy operator in the center of mass system of the
three particles is equivalent to the Laplace operator ∆(2) , given by
∆(2)
∂2
∂2
∂2
=
+
=4
.
∂η 21 ∂η 22
∂z∂z∗
(3.35)
In terms of the kinematic variables, ρ2 , a, φ, and the Euler angels ,ϕ, θ, ψ, the Laplace operator
could be written as,
∆(2) =
1 ∂ 5 ∂
K̂22
ρ
−
ρ52 ∂ρ2 2 ∂ρ2
ρ22
(3.36)
with K̂22 given by [Bar90]
K̂22 = −[[
∂2
∂
1
1
1
+ 2 cot 2a −
(Ŝ02 − cos aL̂0 Ŝ0 + L̂20 )] −
[(L̂2 − L̂20 ) − sin a(L̂2+ + L̂2− )]]
2
2
∂a
∂a sin a
4
2 cos a2
(3.37)
Here L̂M are the familiar angular-momentum operators in the body-fixed coordinate system, the
axes of which coincide with the principal axes of the moment of inertia of the three particles, and
Ŝ0 ≡ 12 X̂1,2 is the generator of kinematic rotations in the plane of the Jacobi coordinates η 1 and
η2,
Ŝ0 = −i
∂
.
∂φ
(3.38)
The expression (3.37) for the angular part of the Laplace operator ∆(2) was originally derived in
a different way by Zickendraht [Zic65] and later by Whitten and Smith [Whi68, Whi68]. In our
derivation we have followed mostly work of Nyiri and Smorodinskii [Nyi69], who, however, used
a slightly different parameterization of the vectors z, z∗ and therefore have obtained somewhat
different form of the equation. It should be noted that Eq. (3.37) is equivalent to Eq. (2.17).
44
However, although all the eigenfunctions of (2.17) are easily obtained through the ”tree” method,
only a limited class of eigenfunctions for (3.37) are known so far. Inspection of Eq. (3.37) suggest
that the 3-body rotational hyperspherical harmonics has the following form [Nyi69, Zic65]
e−iνφ
L2
X
L2
DM,m
Fm (a)
(3.39)
m=−L2
The summation involves m even or odd for parities P = +1 or P = −1, respectively, and the
functions Fm (a) are determined by solving the eigenfunction equation. It is evident that these
functions have definite SO(2) symmetry as they are eigenfunctions of X̂1,2 with eigenvalue ν ≡
1
λ .
2 2,1
The scalar solutions of the Laplace Eq. (3.37) were found about thirty years ago by Whitten and
Smith [Whi68, Whi68]. These functions along with the pseudovector (1+ ) states, which we have
recently found [Bar90], could be easily obtained due to the fact that they contain no summation
on m and in both cases only m = 0 is allowed. The expression for the the scalar functions is
00
Yµν
where µ =
K2
2
1
=
4
s
(µ + 1) µ/2
Dν/2,−ν/2 (φ, 2a, −φ)
π3
(3.40)
and ν = − µ2 , − µ2 + 1, . . . , µ2 . The pseudovector states have similar form
1+ M
Yµν
1
=
4
s
3(µ + 1) µ/2
1
D(1+ν)/2,(1−ν)/2 (φ, 2a, −φ)DM,0
(ϕ, θ, ψ),
π3
(3.41)
where here ν = − µ−1
, − µ−1
+ 1, . . . , µ−1
.
2
2
2
However, as was already pointed out by several authors [Bad66, Kor61, Lev65, Nyi69, Sim66,
Whi68, Zic65], it is very difficult to find a direct general solution of Eq. (3.37). Though some
success in this direction was achieved by Zickendraht [Zic65] and by Mayer [May75], who found the
hyperspherical functions for the total angular momenta L2 = 0, 1, 2 for arbitrary K2 , the resulting
expressions turned out to be too cumbersome and thus unfit for subsequent calculations of matrix
elements of two body potentials. To circumvent the difficulties, Pustovalov and Simonov [Pus66]
suggested finding the L > 0 hyperspherical functions by calculating multiple derivatives
∂
, ∂
∂zi ∂zj∗
of a scalar solution of the Laplace equation, since the gradient operators, in view of Eq. (3.35),
commute with the Laplace operator. We utilized this idea for generating vector hyperspherical
harmonic functions [Bar91], and got the following expression:
1− M
Yµν
=
s
3(µ) i(ν+1/2)φ X X µ ν µ − 1 µ − α 1 α
e
, |
,
,
16π 3
2
2 2 2
α=±1 β=±1 2 2
µ
45
¶
(µ−1)/2
1/2
1
d(ν−α)/2,−(ν+1)/2 (2a)dα/2,β/2 (a)DM,β
(ϕ, θ, ψ).
(3.42)
In view of the fact that the vector solution (3.42) was obtained by differentiation of the scalar
solution (3.40) one can expect that ν takes on the same values as in Eq. (3.40). However, from
the properties of the d functions it follows [Edm60] that the values ν = µ are forbidden which
means that in Eq. (3.42) ν only takes on [2(µ/2) + 1] − 1 = µ values ν = − µ2 , − µ2 + 1, . . . , µ2 − 1.
We have thus µ =
K2 +1
2
different vector functions with 3µ different components. However from
the work of Pustovalov and Simonov [Pus66] one knows that there should be 3(K2 + 1) = 6µ
different components. The additional 3µ components could be obtained from ( 3.42) by the complex
conjugation.
Summing up, in this subsection we have shown that three-body rotational hyperspherical harmonic functions, realizing irreducible representations of the orthogonal group O(2), could be expressed through the Wigner d-functions for L2 = 0, 1. However, considering the amount of work
needed for the construction of these functions and the limited success up to date, one could realize
that the new algorithm presented in the next section is the only practical way to construct these
functions.
3.3
The Hyperspherical Gel’fand-Zetlin states
The N + 1 particle HH functions are usually expressed in terms of N Jacobi coordinates (see
Appendix B) transformed into an hyper-radius and 3N − 1 hyperangles. As we have seen in
Chapter 2 the hyperspherical coordinates can be obtained recursively from the normalized Jacobi
coordinates using the “tree” method. Having defined the hyper-radial coordinate ρk−1 we get the
hyper-radial coordinate ρk and the hyperangular coordinate ϕk from the hyper-radial coordinate
ρk−1 and the Jacobi coordinate ηk by the following transformation, Eq. (2.7),
ρk−1 = ρk cos ϕk
ηk = ρk sin ϕk
where
ρ2k = ρ2k−1 + ηk2 =
k
X
i=1
46
(3.43)
ηi2
(3.44)
Therefore, the hyper-radial coordinate, ρk , is a function of the quadratic form and is invariant with
respect to the orthogonal group O(k). Note that the transformation (3.43) does not change the
spherical angular coordinates Ωk ≡ {θk , φk } of η k where k = 1, 2, . . . , N .
To construct hyperspherical basis functions that are expressed in terms of the N Jacobi coordinates, Eq. (B.1), and belong to a well defined irrep of the orthogonal group SO(N ) we first assume
that a complete set of hyperspherical functions in (N − 1) Jacobi coordinates have been obtained.
These basis functions are characterized by the GZ state ΛN −1 , the quantum numbers KN −1 , LN −1 ,
MN −1 , and the additional label αN −1 . Note that these basis functions describe N -particle systems.
We want to add the N th Jacobi coordinate. The function associated with this coordinate is
the spherical harmonic function YℓN mN (ΩN ) and its corresponding state is denoted by |ℓN mN >.
We first couple the angular momenta LN −1 and ℓN to LN , obtaining the state
|(KN −1 LN −1 ΛN −1 αN −1 ; ℓN )LN MN >=
X
MN −1 mN
(LN −1 MN −1 ℓN mN |LN MN )
×|KN −1 LN −1 MN ΛN −1 αN −1 > |ℓN mN > .
(3.45)
Then we use the transformation (3.43) in the subspace of the coordinates (ρN −1 , ηN ) ≡ (ρN , ϕN ),
and construct the orthonormalized eigenfunctions of the hyperspherical angular momentum operator K̂N2 , Eq. (2.37),
(ℓ + 12 ,KN −1 + 3N2−5 )
ΨKN ;ℓN KN −1 (ϕN ) = NN (KN ; ℓN KN −1 )(sin ϕN )ℓN (cos ϕN )KN −1 PnNN
(cos(2ϕN ))
(3.46)
where
KN = 2nN + KN −1 + ℓN ,
(3.47)
and nN is a non-negative integer. The normalization coefficient, NN (KN ; ℓN KN −1 ) is given in Eq.
(2.39). The eigenvalues of the operator K̂N2 are
KN (KN + 3N − 2)
(3.48)
where KN ≥ KN −1 + ℓN ≥ 0 and has the same parity as KN −1 + ℓN .
The product of the N −1 Jacobi coordinates basis function denoted by the state vector (3.45) and
the function (3.46) yields hyperspherical function of N Jacobi coordinates that is a simultaneously
47
eigenvector of the operators K̂N2 and L̂2N . We denote this state by
|(KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN >
= ΨKN ;ℓN KN −1 (ϕN )|(KN −1 LN −1 ΛN −1 αN −1 ; ℓN )LN MN > .
(3.49)
This state does not belong to an irrep of the orthogonal group SO(N ).
In order to obtain the states that belong to a well defined irrep of the SO(N ) group we have
to diagonalize the quadratic Casimir operator Ĉ2 (N ) of SO(N ), defined in Eq. (A.16), within an
invariant subspace. This operator does not change the total angular momentum and the hyperspherical angular momentum of the states (3.49) and it commutes with all the Casimir operators
of SO(i), i = 2, 3, . . . , N − 1. Therefore, the vectors (3.49) with the same good quantum numbers
KN ,LN and MN which belong to the same GZ state ΛN −1 form a finite dimensional invariant
subspace with respect to the operator Ĉ2 (N ) of SO(N ).
As we noted in the previous subsection for the SO(2) group, the polynomial irreps of SO(N )
of the order λN,1 + λN,2 + λN,3 are embedded in the polynomial space of order KN . Therefore, the
number of the various irreps that we expect to get after diagonalization of the quadratic Casimir
operator Ĉ2 (N ) in the basis states (3.49) is restricted by the KN value, i.e.,
λN,1 + λN,2 + λN,3 ≤ KN .
(3.50)
as well as by the GZ rules (A.5)-(A.6). For example, for the orthogonal group SO(3) we obtain
that |λ2,1 | ≤ λ3,1 ≤ K3 . The eigenvalues (A.17) of the Ĉ2 (N ) operator are sufficient to determine
uniquely the SO(N ) irreps in most of the cases. For degenerate cases the Grahm-Schmidt algorithm
is used, as we will explicitly describe in the next subsection. According to Eq. (A.16) the quadratic
Casimir operator Ĉ2 (N ) of SO(N ) has two terms. The first term Ĉ2 (N − 1) is diagonal in the
invariant subspace of the basis states (3.49) mentioned above. The matrix element of this term
between two states of the same invariant subspace is
′
′
< (KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN |Ĉ2 (N − 1)|(KN′ −1 L′N −1 ΛN −1 αN
−1 ; ℓN )KN LN MN >
= δKN −1 ,KN′ −1 δLN −1 ,L′N −1 δαN −1 ,α′N −1 δℓN ,ℓ′N
3
X
i=1
(λN −1,i + N − 1 − 2i)λN −1,i
(3.51)
The second term in Eq. (A.16) requires the calculation of all the matrix elements of the
2
for i = 1, 2, . . . , N − 1. From the theory of the group algebra (see Ref. [Ham62])
operators X̂i,N
48
we recall that an element gi always exists in the group SO(N − 1) whose corresponding operator,
Ô(gi ), yields the relation X̂i,N = Ô(gi−1 )X̂N −1,N Ô(gi ). Therefore, the matrix element of the second
term in Eq. (A.16) is
< (KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN |
=
N
−1
X
i=1
N
−1
X
i=1
2
′
′
X̂i,N
|(KN′ −1 L′N −1 ΛN −1 αN
−1 ; ℓN )KN LN MN >
< (KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN |Ô(gi−1 )X̂N2 −1,N Ô(gi )|
′
′
(KN′ −1 L′N −1 ΛN −1 αN
−1 ; ℓN )KN LN MN > .
(3.52)
Replacing the Ô(gi ) operators by their corresponding representation matrices D
λ N −1
Λ′N −2 ΛN −2
(gi )
we obtain the following expression for the matrix element (3.52)
< (KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN |
=
N
−1 |λX
N −1 | µ
N −1 | |λX
X
i=1 Λ′N −2 Λ′′
N −2
D
λ N −1
Λ′N −2 ΛN −2
(gi )
¶∗ µ
N
−1
X
i=1
D
2
′
′
X̂i,N
|(KN′ −1 L′N −1 ΛN −1 αN
−1 ; ℓN )KN LN MN >
λ N −1
Λ′′
N −2 ΛN −2
(gi )
¶
< (KN −1 LN −1 λN −1 Λ′N −2 αN −1 ; ℓN )KN LN MN |X̂N2 −1,N
′
′
|(KN′ −1 L′N −1 λN −1 Λ′′N −2 αN
−1 ; ℓN )KN LN MN >
(3.53)
where |λN −1 | is the dimension of the SO(N − 1) irrep λN −1 . Noting that the matrix element
on the left hand side of Eq. (3.53) is independent of the irreps of SO(N − 2), and using the
orthogonality theorem for the matrix representations [Ham62] we obtain for the matrix element
(3.53) the expression
< (KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN |
=
|λ N −1 |
N
−1
X
i=1
2
′
′
X̂i,N
|(KN′ −1 L′N −1 ΛN −1 αN
−1 ; ℓN )KN LN MN >
N −1 X
< (KN −1 LN −1 λN −1 Λ′N −2 αN −1 ; ℓN )KN LN MN |X̂N2 −1,N |
|λN −1 | Λ′
N −2
′
′
(KN′ −1 L′N −1 λN −1 Λ′N −2 αN
−1 ; ℓN )KN LN MN > .
(3.54)
Since the operator X̂N −1,N commutes with all the group algebra operators of SO(N − 2), the
sum in the last equation is independent of the GZ basis for SO(N − 3) and depends only on the
irreps of SO(N −2). Therefore, we sum over these irreps and multiply each term by the appropriate
dimension, obtaining
< (KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN |
N
−1
X
i=1
49
2
′
′
X̂i,N
|(KN′ −1 L′N −1 ΛN −1 αN
−1 ; ℓN )KN LN MN >
=
N −1 X ′
|λN −2 | < (KN −1 LN −1 λN −1 λ′N −2 ΛN −3 αN −1 ; ℓN )KN LN MN |X̂N2 −1,N |
|λN −1 | λ ′
N −2
′
′
(KN′ −1 L′N −1 λN −1 λ′N −2 ΛN −3 αN
−1 ; ℓN )KN LN MN
> .
(3.55)
where we choose an arbitrary GZ basis ΛN −3 . From Eq. (3.55) we conclude that instead of
2
calculating the matrix elements of all the (N − 1) operators X̂i,N
, i = 1, 2, . . . , N − 1 we have to
evaluate only the matrix elements of X̂N2 −1,N .
The calculation of the matrix element on the right hand side of Eq. (3.55) requires the separation of the hyperspherical function associated with the last two Jacobi coordinates from the
N -coordinate hyperspherical functions in the bra and the ket states. Therefore, the first step is to
use the hsopcs for the (N − 1)-coordinate states in order to write these states as a linear combi-
nation of (N − 2)-coordinate states (which belong to the GZ basis ΛN −2 ) that are coupled with
the (N − 1)th-coordinate state to an angular momentum LN −1 (L′N −1 ) and to an hyperspherical
angular momentum KN −1 (KN′ −1 ). The second step is to study the effect of the rotation in the
plane of the last two Jacobi coordinates on the hyperspherical functions, i.e.:
|(KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN >ξ
=
X
KN −2 LN −2 αN −2 ℓN −1
×
X
′
′
′
′
KN
−1 LN −1 ℓN −1 ℓN
[(KN −2 LN −2 λN −2 αN −2 ; ℓN −1 )KN −1 LN −1 |}KN −1 LN −1 λN −1 αN −1 ]
K L K
L
N −2
UKNN−1NLNN−1−2ℓN −1
ℓN ,K ′
′
′
′
N −1 LN −1 ℓN −1 ℓN
(φ)
|((KN −2 LN −2 ΛN −2 αN −2 ; ℓ′N −1 )KN′ −1 L′N −1 ; ℓ′N )KN LN MN > .
(3.56)
where the operator Û (φ) = eiφX̂N −1,N rotates the last two Jacobi coordinates η N −1 and η N by an
angle φ into the coordinates ξ N −1 and ξ N . This rotation is done by separation of the hyperspherical
functions associated with the last two Jacobi coordinates from the hyperspherical functions of the
other coordinates. For this purpose we have to change the coupling order of the angular momentum
and the hyperspherical angular momentum of the original state in Eq. (3.56). In Fig. 3.1 we
demonstrate the action of this operator.
In order to calculate the matrix element of the operator X̂N2 −1,N on the right hand side of Eq.
(3.55) we apply the expansion in Eq. (3.56) on the bra and the ket states, obtaining
′
′
< (KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN |X̂N2 −1,N |(KN′ −1 L′N −1 ΛN −1 αN
−1 ; ℓN )KN LN MN >
50
Figure 3.1: A schematic representation of the action of the rotation operator Û on the “tree”
functions. The coupling coefficients needed to apply the transformation are the 6j coefficients, ’6j’,
the T-coefficient, ’T’, and the Raynal-Revai coefficients ’RR’. Only the relevant parts of the two
“trees” are plotted.
ℓq N
@
@
ℓq N −1
qKN −2
¡
@
@
¡
T, 6j
@¡
q
@
KN −1
¡
@
@q¡K
N
X
=
KN −2 LN −2 αN −2 ℓN −1 ℓ′N −1
×
h
′
′
qKN −2
qℓN −1
q ℓN
qKN −2
qℓN −1
q ℓN
¡
@
¡
¡
@
¡
¡
¡
¡
@
¡
@
RR - ¡
T, 6j
- @¡
q KN −1,N ¡
@q KN −1,N ¡
¡
@
¡
@
@q¡K
@q¡K
N
N
′
qKN −2
q ℓN −1
¡
@
@
¡
q ′
@¡
KN −1
¡
@
@q¡K
N
[(KN −2 LN −2 λN −2 αN −2 ; ℓN −1 )KN −1 LN −1 |}KN −1 LN −1 λN −1 αN −1 ]∗
′
(KN −2 LN −2 λN −2 αN −2 ; ℓ′N −1 )KN′ −1 L′N −1 |}KN′ −1 L′N −1 λN −1 αN
−1
×(−)
′
q ℓN
@
@
-@
d2 KN LN KN −2 LN −2
U
′
′
′
′ (φ) |φ=0 .
dφ2 KN −1 LN −1 ℓN −1 ℓN ,KN −1 LN −1 ℓN −1 ℓN
i
(3.57)
where we take the second derivative of Û (φ) and substitute φ = 0. Using the angular momentum
coupling coefficients and the hyperspherical angular momentum T-coefficients [Kil72] for the bra
and the ket states in Eq. (3.57) we get
d2 KN LN KN −2 LN −2
U
′
′
′
′ (φ) |φ=0
dφ2 KN −1 LN −1 ℓN −1 ℓN ,KN −1 LN −1 ℓN −1 ℓN
′
q
′
= (−1)ℓN +ℓN −1 +ℓN +ℓN −1 (2LN −1 + 1)(2L′N −1 + 1)
×
X
X
K
L



(2L + 1) 
K ;ℓ ℓ
ℓN
ℓN −1
 LN −2
N N N −1
TKK
N −1
KN −2
LN
L
N −1
ℓ′N
 LN −2

LN −1 
KN ;ℓ′N ℓ′N −1 KN −2
TKK ′




ℓ′N −1
LN
L




L′N −1 
< ℓN ℓN −1 |X̂N2 −1,N |ℓ′N ℓ′N −1 >K,L
(3.58)
where K is the hyperspherical angular momentum associated with the last two Jacobi coordinates,
i.e., η N −1 η N and L is the angular momentum related to these coordinates. The hyperspherical
recoupling coefficients, that are usually called the T-coefficients, are given in analytic form in
Subsection 2.4.2.
The matrix element at the end of Eq. (3.58) is evaluated in the same way that we calculated
2
the matrix element of the operator X̂1,2
in Eq. (3.16), i.e., by introducing a complete intermediate
51
set of hyperspherical states
< ℓN ℓN −1 |X̂N2 −1,N |ℓ′N ℓ′N −1 >K,L
=
X
′′
ℓ′′
N ℓN −1
< ℓN ℓN −1 |X̂N −1,N |ℓ′′N ℓ′′N −1 >K,L < ℓ′′N ℓ′′N −1 |X̂N −1,N |ℓ′N ℓ′N −1 >K,L .
(3.59)
The expression for the matrix elements of the operator X̂N −1,N is given by Eqs. (2.47)-(2.48).
From Eq. (3.51) and Eqs. (3.55) - (3.59) we can calculate the matrix elements of the second
Casimir operator Ĉ2 (N ) in the N Jacobi coordinates basis states (3.49) which have the same
quantum number KN and LN and belong to the same GZ state ΛN −1 . The eigenvalues of this
operator (A.17) uniquely identify the various SO(3) irreps, since only one number describe these
irreps. In general we might obtain degenerate SO(N ) irreps, i.e., irreps with the same eigenvalue.
One kind of degeneracy results from the fact that for N = 2k, reflection of the N th Jacobi
coordinate takes the SO(N ) irrep (λN,1 , λN,2 , . . . λN,k ) into (λN,1 , λN,2 , . . . − λN,k ). This kind of
degeneracy can be removed by diagonalization of X̂N −1,N in the highest weight states. In the next
section I will present an algorithm for separating the SO(N ) irreps which have the same eigenvalue,
using the Grahm-Schmidt process. The appropriate eigenvectors are the hsopcs.
Let us assume that states with well defined SO(N ) irreps were obtained in this algorithm.
Then, each state is expressed in terms of the basis states (3.49) by using the hsopcs
|KN LN MN ΛN αN >=
X
KN −1 LN −1 αN −1 ℓN
[(KN −1 LN −1 λN −1 αN −1 ; ℓN )KN LN |}KN LN λN αN ]
|(KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN > .
(3.60)
Note that the hyperspherical coefficients in this equation depend on the given irreps λN −1 and
λN but not on the complete sequence of irreps λ2 , λ3 , . . . , λN −2 . This independence results
directly from the derivation of the matrix element of the second Casimir operator. Another direct
consequence of the derivation is that these hsopcs are independent of the z projection of the angular
momentum, i.e., of the M quantum number.
Since the hsopcs defined in Eq. (3.60), are the orthogonal eigenvectors of an hermitian matrix
they satisfy the orthogonality relation, Eq. (3.4),
X
KN −1 LN −1 αN −1 ℓN
[(KN −1 LN −1 λN −1 αN −1 ; ℓN )KN LN |}KN LN λN αN ]∗
′
]= δ
×[(KN −1 LN −1 λN −1 αN −1 ; ℓN )KN LN |}KN LN λ′N αN
52
λ N ,λ ′N
δαN ,α′N (3.61)
as well as the completeness relation, Eq. (3.5),
X
αN λ N
[(KN −1 LN −1 λN −1 αN −1 ; ℓN )KN LN |}KN LN λN αN ]∗
′
′
× [(KN′ −1 L′N −1 λN −1 αN
−1 ; ℓN )KN LN |}KN LN λN αN ]
= δKN −1 ,KN′ −1 δLN −1 ,L′N −1 δαN −1 ,α′N −1 δℓN ,ℓ′N .
(3.62)
When states with a given SO(N ) irrep can be obtained from more than one SO(N − 1) irrep
our procedure leaves the relative phases between these states undetermined. However, from the
discussion in Sect. 3 we obtain that the operator X̂N −1,N does connect these states, i.e.,
′
< KN LN MN ΛN αN |X̂N −1,N |KN LN MN Λ′N αN
>
= δ ΛN −2 ,Λ′N −2 δ
λ N ,λ ′N
δαN ,α′N F (λN , λN −1 , λ′N −1 , λN −2 )
(3.63)
where the F function is given, in general, by Eqs. (A.7), (A.8) and (A.9), and in simple forms for
the orthogonal groups SO(2), SO(3) and SO(4) in Eqs. (A.10) - (A.15).
The relation (3.63) yields a connection between hsopcs resulting from two different irreps of the
SO(N − 1) group (and may be used as well to separate the two SO(N ) irreps that belong to the
same O(N ) irrep). This connection is obtained if we use the expansion (3.60), and write the bra
and the ket states in Eq. (3.63) in terms of the appropriate hsopcs. Then, we multiply both sides
of Eq. (3.63) by the hsopc
h
′
′
′
′
(KN′ −1 L′N −1 λ′N −1 αN
−1 ; ℓN )KN LN |KN LN λN αN
i
′
and sum over λ′N and αN
. From the completeness relation (3.62) we can express every hsopc
originating from the irrep λ′N −1 as a sum over all the hsopcs originating from the irrep λN −1
multiplied by the appropriate matrix elements of the operator X̂N −1,N :
′
′
[(KN′ −1 L′N −1 λ′N −1 αN
−1 ; ℓN )KN LN |}KN LN λN αN ]
X
1
=
[(KN −1 LN −1 λN −1 αN −1 ; ℓN )KN LN |}KN LN λN αN ]
F(λN , λN −1 , λ′N −1 , λN −2 ) KN −1 LN −1 αN −1 ℓN
′
′
× < (KN′ −1 L′N −1 λ′N −1 αN
−1 ; ℓN )KN LN |X̂N −1,N |(KN −1 LN −1 λN −1 αN −1 ; ℓN )KN LN >
(3.64)
In conclusion, whenever the SO(N ) irrep λN originates from more than one SO(N − 1) irrep
we keep, after the diagonalization of the second order Casimir operator, only the set of hsopcs that
53
originate from one particular irrep λN −1 of SO(N − 1). The hsopcs that originate from the other
SO(N − 1) irreps are constructed from the relation (3.64).
3.4
The computational Algorithm
In the previous sections I described the method of diagonalizing the second Casimir operator
Ĉ2 (N ) in the N Jacobi coordinates basis states (3.49), in order to obtain states which belong to
well defined irreps of the group SO(N ). In this recursive method we calculate the matrix elements
of the operator X̂N2 −1,N using the hsopcs of the group SO(N − 1), that by assumption have already
been calculated.
To carry out our algorithm systematically we choose some maximum value Kmax for the hyperspherical angular momentum. This value restricts also the total angular momentum LN since
0 ≤ LN ≤ KN ≤ Kmax [Nov94]. (Note that for 2 Jacobi coordinate states the minimum value of
L2 is 1 for odd K2 and 0 for even K2 [Nov94].) In addition the possible SO(N ) irreps are restricted
according to the relation (3.50).
The one Jacobi coordinate states have no special orthogonal symmetry. Starting from the two
Jacobi coordinate states we construct the desired SO(2) states (3.13) according to the detailed
explanation given in Subsection 3.2.1. Note that from the GZ basis ( Appendix A) the SO(2)
irreps are described by only one number λ2,1 and the eigenvalues of the operator Ĉ2 (N ) are λ22,1
(A.17). Consequently, by diagonalization of this operator we uniquely identify the O(2) irreps and
by diagonalization of X̂1,2 in the space of the degenerate eigenvalues we can split these irreps into
irreps of SO(2).
In general, starting from N −1 Jacobi coordinate states which belong to well defined SO(N −1)
irreps we construct the N Jacobi coordinate states by the following steps:
1. Consider all the SO(N − 1) irreps and arrange these irreps in decreasing order according to
the values of λN −1,1 . Irreps with the same value of λN −1,1 order according to the values of
λN −1,2 , and for irreps with the same values of λN −1,1 and λN −1,2 use the value of λN,3 for
54
ordering. The first SO(N −1) irrep in this order has λN −1,1 = Kmax , and λN −1,2 = λN −1,3 = 0.
2. Start from the first SO(N − 1) irrep and consider all the possible values for KN within the
range 0 ≤ KN ≤ Kmax .
3. For every given value of KN consider all the values of LN , where 0 ≤ LN ≤ KN for N 6= 2,
and for N = 2 the minimum value of L2 is 0 for even K2 and 1 for odd K2 [Nov94].
4. Construct the basis states (3.49) where the possible values of ℓN are restricted by the relation
|LN − LN −1 | ≤ ℓN ≤ min (LN + LN −1 , KN − KN −1 ), and they should also have the same
parity as KN − KN −1 [Nov94].
5. Calculate the matrix elements of the second Casimir operator Ĉ2 (N ) in this basis and diagonalize the matrix. For the first irrep in the ordered list mentioned above all the eigenvalues
are the same since we can obtain only one SO(N ) irrep - identical , in labels, to the SO(N −1)
irrep that we have started from (following the rules on Appendix A).
6. Use the phase relation (3.64) and Eq. (A.7) or Eq. (A.9), to calculate the N coordinate
states (with their corresponding hsopcs) originating from the SO(N − 1) irrep where the
value of λN −1,1 is less by one from the value of λN,1 , and repeat this step until you cover all
the possible SO(N − 1) states in the given SO(N ) irrep.
7. Repeat the steps 2-6 for the next SO(N −1) irrep, i.e., the irrep λN −1,1 = Kmax −1, λN −1,2 = 1
and λN −1,3 = 0. (Note that this irrep does not exist for SO(2) and SO(3), as I present in
Appendix A.) In this case we can also obtain only the SO(N ) irrep that is identical to the
SO(N − 1) irrep. As before we have to calculate the N Jacobi coordinate states originating
from other SO(N − 1) states. This is done with the help of the phase relation (3.64). We
cover all the possible SO(N − 1) states step by step, where in each step we change the value
of λN −1,j ; j = 1, 2, 3 by one.
8. Repeat steps 2-6 for the next SO(N − 1) irrep, i.e., λN −1,1 = Kmax − 1, and λN −1,2 =
λN −1,3 = 0. In this case we expect to obtain three SO(N ) irreps : One with λN,1 = Kmax − 1,
λN,2 = λN,3 = 0 the second with λN,1 = Kmax , λN,2 = λN,3 = 0, and the third with
λN,1 = Kmax − 1, λN,2 = 1 and λN,3 = 0. Note that the states of the second SO(N ) irrep
are already known from steps 5 and 6, and the states of the third SO(N ) irrep are already
known from step 7, unless N = 4. Therefore we should ignore the states of these irreps that
55
where obtained by diagonalization of the second Casimir operator Ĉ2 (N ). Moreover, if the
eigenvalues of this operator for the first (and third if N = 4) SO(N ) irrep are degenerate with
the second SO(N ) irrep we can obtain the states uniquely using the Grahm-Schmidt process,
and of course, not from diagonalization. If N = 4 we must diagonalize the generator X̂N,N −1
in the highest weight SO(3) irrep to remove the two-fold degeneracy of the eigenvalues of the
third irrep, and use the phase relations to calculate the rest of the states.
9. Continue to use steps 2-6 for the other SO(N − 1) irreps. Ignore the states obtained by
diagonalization for SO(N ) irreps that were already calculated and keep only the states for
the new SO(N ) irreps. For each new SO(N ) irrep use the phase process repeatedly (step
6) to calculate the N Jacobi coordinate states originating from the other SO(N − 1) states
contained in the SO(N ) irrep. Since the SO(N − 1) irreps are taken in the order defined
in step 1 we may expect to obtain for N > 6, and for N = 3, 5, one new SO(N ) irrep for
any new SO(N − 1) irrep, that is the identical irrep. Therefore, if this new SO(N ) irrep
is degenerate we can obtain its states uniquely by using the Grahm-Schmidt process, and
otherwise, by diagonalization. For N = 4 and N = 6 we might obtain more than one SO(N )
irrep. However, since these new SO(N ) irreps are different only in the value of λ4,2 and λ6,3 ,
respectively, they are degenerate only with respect to reflection. In order to remove this twofold degeneracy we must diagonalize the generator X̂N,N −1 in the highest weight SO(N − 1)
irrep.
A computer code implementing the formalism described above for one row states was developed.
The code starts from one coordinate states with a given value for Kmax . Then invariant sets of
2 hyperspherical coordinates are constructed and the operator X̂1,2 is diagonalized within these
invariant subspaces. In this way, we obtain the SO(2) hyperspherical states with their appropriate
hsopcs. Then we apply our algorithm to hyperspherical states with 3 Jacobi coordinates and obtain
the SO(3) states, and so on.
Comparison of the algorithm described above with the state of the art symmetrization algorithm, Ref. [Nov94], proves the computational advantage of the present algorithm. In Table 3.1
we compare the total number of hscfps and of hsopcs for 4-body hyperspherical functions i.e., hyperspherical functions of 3 Jacobi coordinates, where the values of K3 - the hyperspherical angular
momentum are 4, 6, 8, 10. From this table we see that the number of hsopcs are reduced drastically
56
Table 3.1: The total number of hscfps and hsopcs for 4-particle systems where K3 = 4, 6, 8, 10.
K3
hscfps hsopcs ratio
4
471
295
1.59
6
5805
2608
2.22
8
49675
17179
2.86
10
313127
88115
3.55
in comparison with the number of hscfps as the value of K3 is increasing. This occurs because the
GZ basis includes more irreps where K3 increases (see Subsection 3.3) and therefore the number
of states in the invariant subspaces are reduced significantly in our method as we can see in Table
3.2. Moreover, because of the relatively small size of the matrices that we have to diagonalize in
our method we also reduce significantly the calculation time as we can see in Table 3.3. (Both
calculations were carried out on the same machines, namely Silicon Graphics SGI Indy SC4600 for
K3 = 4, 6, 8 and Digital Alfa for K3 = 10.)
The comparison presented above is actually not complete since in our method we constructed
hyperspherical states with well defined SO(N ) irreps whereas in Ref. [Nov94] the states have
well defined SN irreps. However, in the next chapter I will present the method for constructing
SO(N − 1) irrep states which belong tow well defined SN irreps, and I will show that this is an
almost trivial computer calculation.
3.5
Shortcut for large N
In the previous section I described the recursive algorithm to construct GZ hyperspherical states.
In this recursive method we calculate the the opcs for the group O(N ), i.e., the O(N − 1) → O(N )
opcs, using the opcs of the group O(N − 1), that by assumption have already been calculated.
However, as we will see in Chapter 5, in order to calculate the matrix elements of two-body
57
Table 3.2: The Maximal size of the Invariant Subspace for the calculation of the hscfps (MIS-hscfps)
and the hsopcs (MIS-hsopcs) for 4-particle systems where K3 = 4, 6, 8, 10.
K3
MIS-hscfps MIS-hsopcs ratio
4
10
6
1.67
6
30
13
2.33
8
73
27
2.70
10
165
51
3.24
Table 3.3: The CPU time (in seconds) required for the calculation of the hscfps (CPU-hscfps) and
the hsopcs (CPU-hsopcs) for K3 = 4, 6, 8, 10.
K3
CPU-hscfps CPU-hsopcs ratio
4
0.5
0.26
1.92
6
13.5
7.59
1.79
8
479
226
2.13
10
4078
997
4.09
58
interaction we only need the O(N − 1) → O(N ) hsopcs. In this section I will take advantage of
the fact that the hsopcs of O(2), O(3), . . . O(N − 1) are redundant and present a shortcut, i.e., a
version of the recursive algorithm that avoids most of the work associated with the calculation of
the hsopcs. Inspecting Eq. (3.55) we see that in order to calculate the O(N ) hsopcs starting from
the O(N − 1) irrep λN−1, we must have all the hsopcs associated with λN−1, i.e., all the hsopcs
[(KN −2 LN −2 λN −2 αN −2 ; ℓN −1 )KN −1 LN −1 |}KN −1 LN −1 λN −1 αN −1 ] ,
however we need only one of hsopcs associated with λN −2 . Using this fact we may propagate the
scalar representation λn = (0, 0, 0) starting from the scalar representation of O(2), then calculating
the hsopcs corresponding to the scalar representation of O(3) and so on all the way up to the scalar
representation of O(N − 6). Then using the fact that one-row O(n) irreps, i.e., irreps of the form
(λn,1 , 0, 0) contains the O(n − 1) scalar representation, we may use the ’scalar’ hsopcs to build the
’one-row’ states. Similarly, as the two-rows irreps of O(n), i.e., irreps of the form (λn,1 , λn,2 , 0),
contains at least one O(n − 1) one-row irrep, we may use the O(n − 1) ’one-row’ hsopcs to construct
the O(n) two-rows states. And of course we may use the two-rows irreps to build the three-rows
states.
To carry out this algorithm systematically we choose some maximum value Kmax for the hyperspherical angular momentum. This value restricts also the total angular momentum LN since
0 ≤ LN ≤ KN ≤ Kmax [Nov94]. (Note that for 2 Jacobi coordinate states the minimum value of
L2 is 1 for odd K2 and 0 for even K2 [Nov94].) In addition, the possible O(N ) irreps are restricted
according to the relation (3.50). In what follows we will assume that N ≥ 9, thus the O(N ) irreps
coincide with the SO(N ) irreps. As the scalar representation plays a major rule in the current
algorithm, we arrange the irreps of the orthogonal group in increasing order according to the values
of λN,3 . Irreps with the same value of λN,3 are ordered according to the values of λN,2 , and for
irreps with the same values of λN,3 and λN,2 we use the value of λN,1 for ordering. The first O(N )
irrep in this order is of course the scalar representation (λN,1 , λN,2 , λN,3 ) = (0, 0, 0). An ordered
list of the irreps of the orthogonal groups O(2), O(3), . . . , O(12) up to the maximum value of
λN,1 + λN,2 + λN,3 = Kmax ,
(3.65)
for Kmax = 4, is presented in Table 3.4. The table starts with the one coordinate state, N = 1,
and for each irrep presents its dimension and the eigenvalue of the quadratic Casimir operator in
this irrep.
59
Table 3.4: An ordered list of the irreps of the orthogonal groups O(2) . . . O(12), limited by the
restriction λN,1 + λN,2 + λN,3 ≤ Kmax = 4. The dimension (DIM) and the eigenvalue of the
quadratic Casimir operator (CASIMIR) are also presented.
N
#
λN
DIM
CASIMIR
#
λN
DIM
CASIMIR
1
1
(000)
1
0
55
(300)
112
27
2
2
(000)
1
0
56
(400)
294
40
3
(100)
1
1
57
(110)
28
12
4
(200)
1
4
58
(210)
160
21
5
(300)
1
9
59
(310)
567
32
3
4
5
6
7
N
6
(400)
1
16
60
(220)
300
28
7
(000)
1
0
61
(111)
56
15
8
(100)
3
2
62
(211)
350
24
9
(200)
5
6
63
(000)
1
0
10
(300)
7
12
9
64
(100)
9
8
11
(400)
9
20
65
(200)
44
18
12
(000)
1
0
66
(300)
156
30
13
(100)
4
3
67
(400)
450
44
14
(200)
9
8
68
(110)
36
14
15
(300)
16
15
69
(210)
231
24
16
(400)
25
24
70
(310)
910
36
17
(110)
3
4
71
(220)
495
32
18
(210)
8
9
72
(111)
84
18
19
(310)
15
16
20
(220)
5
12
21
(000)
1
0
22
(100)
5
4
76
(200)
54
20
23
(200)
14
10
77
(300)
210
33
24
(300)
30
18
78
(400)
660
48
25
(400)
55
28
79
(110)
45
16
26
(110)
10
6
80
(210)
320
27
27
(210)
35
12
81
(310)
1386
40
28
(310)
81
20
82
(220)
770
36
10
73
(211)
594
28
74
(000)
1
0
75
(100)
10
9
29
(220)
35
16
83
(111)
120
21
30
(000)
1
0
84
(211)
945
32
31
(100)
6
5
85
(000)
1
0
32
(200)
20
12
86
(100)
11
10
33
(300)
50
21
87
(200)
65
22
34
(400)
105
32
88
(300)
275
36
35
(110)
15
8
89
(400)
935
52
36
(210)
64
15
90
(110)
55
18
37
(310)
175
24
91
(210)
429
30
38
(220)
84
20
92
(310)
2025
44
39
(111)
10
9
93
(220)
1144
40
40
(211)
45
16
94
(111)
165
24
41
(000)
1
0
36
42
(100)
7
6
43
(200)
27
14
44
(300)
77
24
45
(400)
182
36
46
(110)
21
10
11
12
60
95
(211)
1430
96
(000)
1
0
97
(100)
12
11
98
(200)
77
24
99
(300)
352
39
100
(400)
1287
56
The one Jacobi coordinate states have no special orthogonal symmetry. Thus, like in the basic
algorithm, Sec. 3.4, we start from the two Jacobi coordinate states and construct the desired O(2)
states (3.13) according to the detailed explanation given in Subsection 3.2.1. However, we keep
only those two Jacobi coordinate states that correspond to the scalar representation λ2,1 = 0, irrep
number 2 in Table 3.4, and discard all the rest.
In general, for n = 3, . . . , N − 6, we start from n − 1 Jacobi coordinate states which belong to
the scalar representation of O(n − 1) and construct the n Jacobi coordinate scalar states by the
following steps:
1. Start from the scalar SO(n − 1) irrep, (λn−1,1 , λn−1,2 , λn−1,3 ) = (0, 0, 0), and consider all the
possible values for Kn within the range 0 ≤ Kn ≤ Kmax .
2. For every given value of Kn consider all the values of Ln , where 0 ≤ Ln ≤ Kn for n 6= 2, and
for n = 2 the minimum value of L2 is 0 for even K2 and 1 for odd K2 [Nov94].
3. Construct the basis states (3.49) where the possible values of ℓn are restricted by the relation
|Ln − Ln−1 | ≤ ℓn ≤ min (Ln + Ln−1 , Kn − Kn−1 ), and they should also have the same parity
as Kn − Kn−1 [Nov94].
4. Calculate the matrix elements of the second Casimir operator Ĉ2 (n) in this basis and diagonalize the matrix. According to the GZ rules (A.5)-(A.6), the expected O(n) irreps are
the one-row irreps. As can be seen in Table 3.4 these irreps are uniquely identified by the
eigenvalues of the second Casimir operator. The eigenvectors are the (0, 0, 0) −→ (λn,1 , 0, 0)
hsopcs.
5. Discard all the states constructed in the previous step and keep only those states that belong
to the scalar irrep of O(n).
For n = N − 5, we skip step 5 above and keep the one-row O(N − 5) states and the hsopcs of
the form (0, 0, 0) → (λN −5,1 , 0, 0), i.e.,
[(KN −6 LN −6 (0, 0, 0)αN −6 ; ℓN −5 )KN −5 LN −5 |}KN −5 LN −5 (λN −5,1 , 0, 0)αN −5 ] .
61
Then, going from O(N − 5) to O(N − 4) we may repeat steps 1 − 4 above, calculate the one-row
O(N − 4) states and use these coefficients and the phase relation, Eq. (3.64), to calculate all the
O(N − 4) hsopcs of the form (λN −5,1 , 0, 0) → (λN −4,1 , 0, 0), i.e.
[(KN −5 LN −5 (λN −5,1 , 0, 0)αN −5 ; ℓN −4 )KN −4 LN −4 |}KN −4 LN −4 (λN −4,1 , 0, 0)αN −4 ] .
For n = N − 3, we follow steps 1 − 4 above, this time we repeat this steps for each one-row irrep
of O(N − 4) according to the order presented above. Starting from O(N − 4) one-row irreps, the
expected O(N −3) irreps are, according to the GZ rules, the tow-rows irreps. Since irreps with more
then one row may have degenerated eigenvalues of the second Casimir, see for example Table 3.4,
the O(N −3) irreps are not uniquely identified. Working in the order defined above, we may expect
to obtain new O(N − 3) irreps with labels (λN −3,1 , λN −4,1 , 0) for each O(N − 4) irrep (λN −4,1 , 0, 0).
Inspecting Eq. (A.17), one may conclude that these irreps are non-degenerated among themselves.
Therefore, if a new O(N − 3) irrep is degenerate with previously obtain irrep we can uniquely
obtain its states by using the Grahm-Schmidt process, and otherwise, by diagonalization. In this
process we ignore those states that belong to O(N − 3) irreps that were already obtained and
keep only those states that belong to new irreps. For each new O(N − 3) irrep use the phase
process repeatedly (step 6 in the previous section) to calculate the N − 3 Jacobi coordinate states
originating from the other one-row O(N − 4) states contained in the O(N − 3) irrep, i.e., to obtain
all the (λN −4,1 , 0, 0) → (λN −3,1 , λN −3,2 , 0) hsopcs,
[(KN −4 LN −4 (λN −4,1 , 0, 0)αN −4 ; ℓN −3 )KN −3 LN −3 |}KN −3 LN −3 (λN −3,1 , λN −3,2 , 0)αN −3 ] .
Going now to n = N − 2 we have to repeat the process described for N − 3, however this time
we may use the (λN −4,1 , 0, 0) → (λN −3,1 , λN −3,2 , 0) hsopcs in order to calculate the full O(N − 2)
two-rows hsopcs (λN −3,1 , λN −3,2 , 0) → (λN −2,1 , λN −2,2 , 0).
In order to obtain the O(N −1), O(N ) states, we follow the same method used for the calculation
of the O(N − 3), O(N − 2) states respectively. This time going from two-rows irreps to three-rows
irreps, thus obtaining all the possible O(N ) states.
A computer code implementing this formalism was developed. The code starts from one coordinate states with a given value of Kmax and N . Then invariant sets of 2 hyperspherical coordinates
2
is diagonalized within these invariant subspaces. In this
are constructed and the operator X̂1,2
62
way, we obtain the O(2) hyperspherical states with their appropriate hsopcs. Ignoring all but the
O(2) scalar states, we apply our algorithm to hyperspherical states with 3 Jacobi coordinates and
obtain the scalar SO(3) states, and so on up to N − 6 coordinate states. Stepping from N − 6
coordinates states to N coordinates states, we start building more and more O(n) irreps as de-
scribed above, until we get all the possible O(N ) states with the appropriate hsopcs. An output
list of the calculated GZ hyperspherical states with the appropriate hsopcs is given in Table 3.5
for N = 12. The N coordinates states are labeled in increasing order, presented in the left most
column of this table. This label is equivalent to the degeneracy label αN used through out this
chapter. It can be seen that state number 180 is the first state of 12 coordinates. The other
labels of the GZ hyperspherical states, i.e., KN , LN and λN are given in the next three columns,
the O(N ) irreps are indexed according to Table 3.4. For each state, the appropriate hsopcs are
given in the lines corresponding to that state. For each hsopc the initial N − 1 coordinate state
and its labels are given in columns 5 − 8 and the N ’th coordinate state ℓN is given in the 9’th
column. The numerical value of the hsopc is given in the last column. In order to demonstrate the
computational advantage of this shortcut algorithm, Table 3.8 presents the number of hsopcs and
the CPU time we need in order to transform the hyperspherical functions with KN ≤ Kmax = 6
into the GZ hyperspherical states. One can see that although the number of the hyperspherical
functions grows like an exponent the CPU time and the number of hsopcs remain approximately
constant.
63
Table 3.5: The calculated GZ hyperspherical states for N = 12 with the appropriate hsopcs. Each
line in the table stands for one hsopc connecting the N coordinate state with the N − 1 coordinates
GZ hyperspherical state and the N ’th coordinate state ℓN . For details see text.
#
KN
LN
λN
#
KN −1
LN −1
λ N −1
ℓN
HSOPCs
180
0
0
96
145
0
0
85
0
1.000000
181
4
0
96
145
0
0
85
0
.095238
146
4
0
85
0
.916589
147
2
2
85
2
.388321
0
0
85
2
.288675
182
2
2
96
145
147
2
2
85
0
.957427
183
4
2
96
145
0
0
85
2
.062994
147
2
2
85
0
-.027399
147
2
2
85
2
.398409
148
4
2
85
0
.914631
145
0
0
85
4
.133630
147
2
2
85
2
.361873
149
4
4
85
0
.922599
0
0
85
1
1.000000
184
4
4
96
185
1
1
97
145
150
1
1
86
0
-1.000000
186
3
1
97
145
0
0
85
1
.308606
147
2
2
85
1
.951189
150
1
1
86
0
.035169
150
1
1
86
2
-.286794
187
188
189
190
191
192
193
3
3
2
4
4
2
4
2
3
0
0
1
2
2
97
97
98
98
98
98
98
151
3
1
86
0
-.957346
147
2
2
85
1
1.000000
150
1
1
86
2
.301511
152
3
2
86
0
-.953462
145
0
0
85
3
.462910
147
2
2
85
1
.886405
150
1
1
86
2
-.267261
153
3
3
86
0
-.963624
145
0
0
85
0
1.000000
150
1
1
86
1
-1.000000
154
2
0
87
0
1.000000
145
0
0
85
0
.388321
146
4
0
85
0
-.393397
147
2
2
85
2
.833333
150
1
1
86
1
-.267261
151
3
1
86
1
-.963624
154
2
0
87
0
-.067115
155
4
0
87
0
.955071
157
2
2
87
2
.288675
147
2
2
85
2
1.000000
150
1
1
86
1
.308606
151
3
1
86
1
-.834522
152
3
2
86
1
-.456435
156
4
1
87
0
.957427
157
2
2
87
2
-.288675
145
0
0
85
2
.957427
147
2
2
85
0
-.288675
150
1
1
86
1
-1.000000
157
2
2
87
0
1.000000
145
0
0
85
2
.398862
147
2
2
85
0
.917010
147
2
2
85
2
.000000
148
4
2
85
0
.000000
150
1
1
86
1
-.113960
150
1
1
86
3
-.325669
151
3
1
86
1
-.410890
152
3
2
86
1
.502518
153
3
3
86
1
-.677935
154
2
0
87
2
.275240
157
2
2
87
0
-.028618
2
2
87
2
.000000
4
2
87
0
.938523
4
2
87
0
.206391
157
158
159
64
Table 3.6: (Continued)
#
KN
LN
λN
#
KN −1
LN −1
λ N −1
ℓN
HSOPCs
196
4
4
98
145
0
0
85
4
.586301
147
2
2
85
2
.721687
149
4
4
85
0
-.367990
150
1
1
86
3
-.433012
153
3
3
86
1
-.901387
157
2
2
87
2
.250000
161
4
4
87
0
.968245
145
0
0
85
1
.951189
147
2
2
85
1
-.308606
150
1
1
86
0
-.748392
150
1
1
86
2
-.642416
151
3
1
86
0
.164957
154
2
0
87
1
.745355
157
2
2
87
1
.666666
162
3
1
88
0
-1.000000
145
0
0
85
3
.886405
147
2
2
85
1
-.462910
150
1
1
86
2
-.963624
153
3
3
86
0
.267261
157
2
2
87
1
1.000000
197
198
199
200
201
3
3
4
4
4
1
3
0
2
4
99
99
100
100
100
163
3
3
88
0
-1.000000
145
0
0
85
0
.916589
146
4
0
85
0
.071428
147
2
2
85
2
-.393397
150
1
1
86
1
-.963624
151
3
1
86
1
.267261
154
2
0
87
0
.750375
155
4
0
87
0
-.142373
157
2
2
87
2
.645497
162
3
1
88
1
-1.000000
1.000000
164
4
0
89
0
145
0
0
85
2
.879664
147
2
2
85
0
-.382617
147
2
2
85
2
-.278174
148
4
2
85
0
.049123
150
1
1
86
1
-.806225
150
1
1
86
3
-.493710
151
3
1
86
1
.223606
152
3
2
86
1
.000000
153
3
3
86
1
.237170
154
2
0
87
2
.603807
157
2
2
87
0
.627809
157
2
2
87
2
.456435
158
4
2
87
0
-.130108
159
4
2
87
0
-.126534
162
3
1
88
1
-.836660
163
3
3
88
1
-.547722
165
4
2
89
0
1.000000
145
0
0
85
4
.798994
147
2
2
85
2
-.590096
149
4
4
85
0
.115727
150
1
1
86
3
-.901387
153
3
3
86
1
.433012
157
2
2
87
2
.968245
161
4
4
87
0
-.250000
163
3
3
88
1
-1.000000
166
4
4
89
0
1.000000
1
1
86
1
1.000000
202
2
1
101
150
167
2
1
90
0
1.000000
203
4
1
101
150
1
1
86
1
.142857
3
1
86
1
.515078
152
3
2
86
1
-.845154
167
2
1
90
0
-.035874
151
65
Table 3.7: (Continued)
#
KN
LN
λN
#
KN −1
LN −1
λ N −1
ℓN
205
4
3
101
150
1
1
86
3
.377964
152
3
2
86
1
-.487950
153
3
3
86
1
.786795
167
2
1
90
2
.267261
170
4
3
90
0
.963624
150
1
1
86
0
.662323
150
1
1
86
2
-.710669
151
3
1
86
0
.237227
154
2
0
87
1
.666666
157
2
2
87
1
-.745355
167
2
1
90
1
-1.000000
171
3
1
91
0
-1.000000
150
1
1
86
2
.953462
152
3
2
86
0
.301511
157
2
2
87
1
-1.000000
167
2
1
90
1
1.000000
172
3
2
91
0
-1.000000
150
1
1
86
1
.940400
151
3
1
86
1
.195615
152
3
2
86
1
.278174
156
4
1
87
0
-.288675
157
2
2
87
2
-.957427
162
3
1
88
1
1.000000
167
2
1
90
0
.914631
167
2
1
90
2
-.393397
168
4
1
90
0
-.093205
171
3
1
91
1
-.790569
172
3
2
91
1
-.612372
173
4
1
92
0
1.000000
150
1
1
86
1
.515078
150
1
1
86
3
-.735980
151
3
1
86
1
.357142
152
3
2
86
1
-.227128
153
3
3
86
1
-.117851
154
2
0
87
2
.677003
157
2
2
87
0
-.703914
157
2
2
87
2
.000000
158
4
2
87
0
-.209860
159
4
2
87
0
-.046150
162
3
1
88
1
-.547722
163
3
3
88
1
.836660
167
2
1
90
2
-.963624
169
4
2
90
0
-.267261
171
3
1
91
1
-.866025
172
3
2
91
1
.500000
174
4
2
92
0
1.000000
150
1
1
86
3
.879664
152
3
2
86
1
.454256
153
3
3
86
1
-.140859
157
2
2
87
2
-.957427
160
4
3
87
0
-.288675
163
3
3
88
1
1.000000
167
2
1
90
2
.963624
170
4
3
90
0
-.267261
172
3
2
91
1
-1.000000
175
4
3
92
0
1.000000
154
2
0
87
0
.657595
155
4
0
87
0
.259937
157
2
2
87
2
-.707106
3
1
91
1
-1.000000
4
0
93
0
1.000000
154
2
0
87
2
.316227
157
2
2
87
0
.328797
206
207
208
209
210
211
3
3
4
4
4
4
1
2
1
2
3
0
102
102
103
103
103
104
171
176
212
4
2
104
66
HSOPCs
Table 3.8: The total number of hsopcs and the CPU time (in seconds) required for their calculation
for N-coordinate system, and Kmax = 6. NHH is the number of hyperspherical functions with
KN ≤ Kmax . The calculations where done on workstation HP 9000/730.
N
10
12
14
16
18
20
NHH
1.9 · 106
5.2 · 106
12.1 · 106
25.5 · 106
49.6 · 106
90.2 · 106
# hsopcs
CPU
14116
4.46
14202
4.19
14288
4.03
14374
4.78
14460
4.68
14546
4.70
67
Chapter 4
Permutational Symmetry
68
In the previous chapter I introduced a recursive method for constructing hyperspherical functions which belong to well defined irreps of the orthogonal group. Actually, for physical systems
composed of identical particles, we are interested in states with well defined permutational symmetry, i.e., symmetric states for bosons and antisymmetric states for fermions. Since there are
additional quantum numbers of unitary type, like spin (color and flavor for quarks), other permutational symmetries may occur in each unitary subspace [Che89a]. Therefore we have to solve
the problem of reducing a given O(N − 1) irrep into irreps of the symmetric group SN . This
reduction is also relevant in nuclear physics in the shell-model as well as in collective models
[Row80]. For atomic systems, where we have a nuclei and N − 1 electrons, we have to consider
the O(N − 1) ↓ SN −1 reduction problem rather then the reduction of O(N − 1) into SN . Since
SN ⊃ SN −1 , this problem is solved once we have reduced the irreps of O(N − 1) into irreps of SN .
However, only small modification are needed to adapt the algorithm presented in this chapter for
dealing directly with the O(N − 1) ↓ SN −1 reduction problem.
The computational method for the reduction of O(N − 1) irreps into SN irreps is presented in
Section 4.1, later on, I use the results of this section to construct hyperspherical functions with
well defined permutational symmetry in Section 4.2.
4.1
The reduction problem O(N − 1) ↓ SN
A few years ago Novoselsky, Katriel and Gilmore, NKG, [Nov88] developed a new algorithm for the
construction of irreps of the symmetric group. In their approach they used the canonical nature
of the group subgroup chain of the symmetric groups, Sn ⊃ Sn−1 ⊃ ·· ⊃ S2 ⊃ S1 , and formulated
a recursive algorithm in which one starts with basis functions that belong to the irreps of S1 , then
transform this basis into new basis functions that belong to well defined irreps of S2 . These basis
functions are then symmetrized with respect to S3 , and so on. After n − 1 recurrence steps one gets
basis functions that belong to well defined irreps of the symmetric group Sn and are characterized
by the Yamanouchi symbols. Assuming that basis functions with well defined Sn−1 symmetry have
already been constructed, NKG have shown, that the eigenvalues of the transposition class-sum
69
operator uniquely identify the irreps of symmetric group Sn , and therefore, the eigenvectors are the
new basis functions which belong to well defined irreps of Sn . The key point in this algorithm is
that the calculation of the matrix elements of the transposition class-sum operator can be reduced
to the calculation of the matrix elements of the transposition (n, n − 1) in each recurrence step.
The algorithm has been successfully applied for a number of mathematical and physical problems,
such as the calculation of shell model coefficients of fractional parentage [Nov88] and the evaluation
of the inner product isoscalar factors and outer product isoscalar factors of the permutation group
[Che89a]. It was also applied to the long standing problems [Efr95] of constructing harmonic
oscillator [Nov89] and hyperspherical [Nov94] states with definite permutational symmetry. In all
these applications the same NKG algorithm has been employed using the appropriate realization of
the transposition (n, n − 1) and the invariant subspaces. Comparing it with other symmetrization
methods [Che89a] the NKG algorithm was proven to be very efficient in terms of cpu time and
memory.
In this section I use the NKG algorithm, and apply it to the orthogonal group. However,
this application is much more simple since the O(N ) states are defined only by the GZ basis,
introduced in Appendix A. Basically, this algorithm is similar to the algorithm presented in the
previous chapter in which we diagonalize the second Casimir operator of the O(N ) group within an
invariant subspace of hyperspherical functions. In this method we diagonalize the second Casimir
operator of the symmetric group within an invariant subspace of GZ states.
The second Casimir operator of the symmetric group is the transposition class-sum
[(2)]N =
N
X
(i, i′ ).
(4.1)
i<i′
The eigenvalue of [(2)]N corresponding to a given irrep of SN , presented by the Young diagram ΓN ,
can be written in the following equivalent ways
¯
[(2)]N ¯¯
ΓN

1X



γi (γi − 2i + 1)



2 i


=







or
X
i,j∈ΓN
(4.2)
(j − i)
where i and j are the row and column indices of the N boxes of the irrep presented by the Young
diagram ΓN and γi is the number of boxes in the i’th row. These eigenvalues are sufficient to
70
characterize the irrep to which a state of N particles belongs, given that it belongs to some definite
irrep of SN −1 [Nov94]. In fact, the eigenvalues corresponding to the Young diagrams ΓN −1 and ΓN
differ by the “content” of the box which was added to ΓN −1 in order to form ΓN ; the “content”
being the difference j − i between the column index j and the row index i of that box [Nov94].
The recursive symmetry adaptation process starts from the S2 states. The S2 states with even
and odd symmetries, i.e., states that belong to the irreps [2] and [11] of S2 respectively, can be
constructed by taking the following symmetric and anti-symmetric linear combinations of the two
SO(2) irreps λ2,1 ,and −λ2,1
−i
|λ2,1 , [2] >= √ (|λ2,1 > +| − λ2,1 >)
2
1
|λ2,1 , [11] >= √ (|λ2,1 > −| − λ2,1 >)
2
(4.3)
where for λ2,1 = 0, there might be an odd state as well as even state. Going on to S3 , we may follow
Raynal [Ray73] and notice that for identical particles, kinematic rotation of the Jacobi coordinates
η 1 , η 2 by the angle φ = 34 π results in cyclic permutation. Thus, the O(2) vectors, Eq. (4.3), with
λ2,1 which is a multiple of 3 form a symmetric S3 representation in the even S2 subspace and an
antisymmetric one in the odd subspace. The O(2) vectors with λ2,1 = 3n + m; m = 1, 2 form a
mixed representation up to a phase factor.
Assuming now that we have already constructed all the (N − 1)-particle states, each one is
characterized by the O(N − 2) irrep λN −2 and by the Yamanouchi symbol YN −1 (the symmetric
group irreps that correspond to the group-subgroup chain S1 ⊂ S2 ⊂ . . . ⊂ SN −1 , and are presented
by their appropriate Young diagrams Γ1 , Γ2 , . . . ΓN −1 , respectively). Usually, we need an additional
label βN −1 to remove possible degeneracy of the SN −1 states in O(N − 2). The N -particle O(N − 1)
states are constructed by embedding these O(N − 2) states within the O(N − 1) irreps, λN −1 ,
according to the GZ rules mentioned in Sect. 3
|λN −2 YN −1 βN −1 >−→ |(λN −2 YN −1 βN −1 )λN −1 > .
(4.4)
Our aim is to transform these basis functions into states that belong to an irrep of the symmetric
group, SN . We shall refer to the elements of this transformation matrix as the ocfps.
The actual process of symmetry adaptation for these O(N − 1) states will take place by diago-
nalizing the class sum operator [(2)]N , Eq. (4.1), within a finite invariant subspace, which consists
71
of functions that belong to a given Yamanouchi symbol YN −1 and a given O(N − 1) irrep λN −1 .
The calculation of the matrix elements of the operator [(2)]N within this set of states reduces to the
calculation of the matrix elements of the transposition (N − 1, N ). This results from the recurrence
relation
[(2)]N = [(2)]N −1 +
N
−1
X
(i, N )
(4.5)
i=1
since the N -particle basis states are eigenfunctions of [(2)]N −1
¯
< λN −1 λN −2 YN −1 |[(2)]N −1 |λN −1 λ′N −2 YN −1 >= δλ N −2 ,λ ′N −2 [(2)]N −1 ¯¯
In addition, the matrix element of the sum
PN −1
i=1
ΓN −1
.
(4.6)
(i, N ) can be expressed in terms of the matrix
elements of the transposition (N − 1, N ) using the relation obtained in Ref. [Nov88]
< ΓN −1 |
N
−1
X
i=1
(i, N )|Γ′N −1 >= δΓN −1 ,Γ′N −1
X
N −1
< YN −2 ΓN −1 |(N − 1, N )|YN −2 ΓN −1 >
|ΓN −1 | YN −2 ∈ΓN −1
(4.7)
where |ΓN −1 | is the dimension of the irrep ΓN −1 of SN −1 . The sum on the right-hand-side of Eq.
(4.7) ranges over all the Yamanouchi symbols YN −2 which can give rise to ΓN −1 by adding a single
box.
In Ref. [Nov94] it was shown that the operation of the transposition (N − 1, N ) on the Jacobi
coordinates η N −2 and η N −1 is equivalent to a rotation by an angle φ = arccos
³
1
N −1
´
, preceded by
a reflection of the coordinate η N −1 . The action of this reflection on a SO(N − 1) GZ vector yields
Pη N −1 |(λN −1,1 , λN −1,2 , . . . λN −1,k ) ΛN −2 >= (−1)qN −1 +qN −2 |(λN −1,1 , λN −1,2 , . . . λN −1,k ) ΛN −2 >
(4.8)
for N = 2k + 2, and
Pη N −1 |(λN −1,1 , λN −1,2 , . . . λN −1,k ) ΛN −2 >= (−1)qN −1 +qN −2 |(λN −1,1 , λN −1,2 , . . . − λN −1,k ) ΛN −2 >
(4.9)
for N = 2k + 1. qN −1 (qN −2 ) is the sum over the GZ basis numbers for the irrep λN −1 ( λN −2 ). For
example, qN −1 = λN −1,1 + λN −1,2 + λN −1,3 for hyperspherical basis functions since they are three
dimensional vectors, as we explained in Appendix A. The phase can be obtained by applying the
reflection operators in two steps and noting that the orthogonal states presented by the GZ basis
ΛN −1 (ΛN −2 ) are monomials of order qN −1 (qN −2 ). Therefore, the first step is to reflect all the N −1
Jacobi coordinates which yields the phase (−1)qN −1 and the second step is to reflect only the first
72
N − 2 coordinates which yields the phase (−1)qN −2 . So, only the coordinate η N −1 is reflected after
these two operations and all the other coordinates remain unchanged. For the groups O(3), O(5) one
should also consider inverse symmetry irreps, i.e., irreps with qN = λN,1 + λN,2 + λN,3 + 1. Matching
the GZ states with the rotational hyperspherical harmonics ( Chapter 3), see next section, we must
demand that the two states have the same parity, i.e. qN and KN must have the same parity.
The rotation of the coordinates η N −2 and η N −1 by the angle φ, mentioned above, can be achieved
by using the representation matrices of the rotation operator gN −2,N −1 (φ) = eiφX̂N −2,N −1 . The
application of this operator on a GZ state yields
gN −2,N −1 (φ)|λN −1 λN −2 ΛN −3 >=
X
λ ′N −2
(λ
,λ
)
Dλ ′ N −1,λ NN−2−3 (φ)|λN −1 λ′N −2 ΛN −3 > ,
N −2
(4.10)
where the GZ basis ΛN −3 as well as the irrep λN −1 remain unaltered. The representation matrices
are, actually, the rotation operator matrices between the appropriate irreps of the O(N − 1) group
(λ
,λ
)
Dλ ′ N −1,λ NN−2−3 (φ)
N −2
µ
iφMN −2,N −1 (λN −1 ,λN −3 )
= e
¶
λ ′N −2 ,λ N −2
,
(4.11)
where the matrix representation of X̂N −2,N −1 is given by
< λN −1 Λ′N −2 |X̂N −2,N −1 |λN −1 ΛN −2 > ≡
δΛ′
N −3
,ΛN −3 MN −2,N −1 (
λN −1 , λN −3 )λ ′N −2 ,λ N −2
≡ δΛ′N −3 ,ΛN −3 F(λN −1 , λ′N −2 , λN −2 , λN −3 ) (4.12)
and the F function, introduced in Eq. (3.63), is given by Eqs. (A.7)-(A.9). Since X̂N −2,N −1 is an
hermitian operator, we can diagonalize its corresponding matrix MN −2,N −1 and write it as follows
MN −2,N −1 = UDU −1
where D is a diagonal matrix. Therefore, the representation matrices in Eq. (4.11) can be rewritten
as
(λ
,λ
)
³
Dλ ′ N −1,λ NN−2−3 (φ) = UeiφD U −1
N −2
´
λ ′N −2 ,λ N −2
.
(4.13)
This form is very efficient and accurate from a computational point of view.
Using Eqs. (4.5) - (4.7) and applying the reflection and rotation operators, we obtain the
matrix representing the operator [(2)]N (Eq. (4.1)) in the basis specified above. Its eigenvalues
(4.2) uniquely identify the Young diagram ΓN that can be obtained from the Young diagram ΓN −1
by adding one box. The eigenvectors are the ocfps, (the orthogonal group coefficients of fractional
73
parentage) that are the coefficients of the linear transformation of the states (4.4) into states with
well-defined irreps of SN , i.e.,
|λN −1 YN βN >=
X
λ N −2 βN −1
[(λN −2 ΓN −1 βN −1 ) λN −1 |}λN −1 ΓN βN ]| (λN −2 YN −1 βN −1 ) λN −1 > (4.14)
Note that the ocfps depend on the given Young diagrams ΓN and ΓN −1 and not on the complete
sequence of diagrams Γ2 , Γ3 , . . . , ΓN −2 i.e., the Yamanouchi basis. This is a direct result from the
expressions for the matrix elements given in Eqs. (4.10) - (4.13).
The ocfps, introduced in Eq. (4.14), form the columns of an unitary matrix. We can invert
this matrix in order to express the original N -particle states (4.4) in terms of the symmetrized
N -particle states, i.e.,
| (λN −2 YN −1 βN −1 ) λN −1 >=
X
ΓN ,βN
[(λN −2 ΓN −1 βN −1 ) λN −1 |}λN −1 ΓN βN ]∗ |λN −1 YN βN > . (4.15)
The scalar product of two permutationally symmetrized N -particle states with a common Young
diagram ΓN −1 and the same O(N − 1) irrep λN −1 yields the orthogonality relation when we sum
over all the (N − 1)-particle states, i.e., the irreps λN −2 and the additional quantum numbers βN −1
X
λ N −2 βN −1
[(λN −2 ΓN −1 βN −1 ) λN −1 |}λN −1 ΓN βN ]
′ ∗
×[(λN −2 ΓN −1 βN −1 ) λN −1 |}λN −1 Γ′N βN
] = δΓN ,Γ′N δβN ,βN′
(4.16)
Similarly, by taking the scalar product of the states in Eq. (4.16) and summing over the SN irreps
ΓN and the additional quantum numbers βN we obtain the completeness relation
X
ΓN βN
[(λN −2 ΓN −1 βN −1 ) λN −1 |}λN −1 ΓN βN ]
³
′
×[ λ′N −2 ΓN −1 βN
−1
´
λN −1 |}λN −1 ΓN βN ]∗ = δλ N −2 λ ′N −2 δβN −1 ,βN′ −1 .
(4.17)
The different components of the set of permutationally symmetrized N -particle states spanning
any irrep of the symmetric group SN are obtained from distinct Yamanouchi symbols YN −1 , except
the one-dimensional irreps, i.e., the symmetric and the antisymmetric irreps. As we noted above,
the ocfps depend only on the last two irreps, ΓN −1 and ΓN . Therefore, states with different
Yamanouchi symbols YN −2 have the same (N − 1) to N ocfps. However, states with different irrep
74
Γ′N −1 have a different ocfps set and consequently a different set of N -particle states. In our method
the relation between these two different sets of ocfps is arbitrary. However, this is inconsistent with
the requirement that the following relation should also exist between these two set of states
√
σ2 − 1
′
δΓN ,Γ′N δβN ,βN′
< λN −1 YN −2 Γ′N −1 Γ′N βN
|(N − 1, N )|λN −1 YN −2 ΓN −1 ΓN βN >=
(4.18)
|σ|
where σ is the difference between the “contents” (see the explanation after Eq. (4.2) ) of the boxes
in which the particles N and (N − 1) are placed in the Yamanouchi symbol YN ≡ YN −2 ΓN −1 ΓN .
Note that the range of values of βN depends only on ΓN and is independent of the Yamanouchi
symbol YN −2 , for a given λN −1 irrep.
In order to calculate the matrix element in Eq. (4.18) we have to separate the last two particles.
This is done by using the (N − 1) to N ocfps and the (N − 2) to (N − 1) ocfps in the bra and in
the ket states. Then, by applying the completeness relation, Eq. (4.17), we can express any ocfp
originating from the irrep Γ′N −1 , as a sum over the ocfps originating from the irrep ΓN −1 times the
appropriate matrix element of the transposition (N − 1, N ), as follows:
³
´
′
[ λ′N −2 Γ′N −1 βN
−1 λN −1 |}λN −1 ΓN βN ]
=√
X
|σ|
[(λN −2 ΓN −1 βN −1 ) λN −1 |}λN −1 ΓN βN ]
σ 2 − 1 λ N −2 βN −1
³
´
′
× < λ′N −2 Γ′N −1 βN
−1 λN −1 |(N − 1, N )| (λN −2 ΓN −1 βN −1 ) λN −1 >
(4.19)
The calculation of the matrix elements of the transposition operator (N −1, N ) was already derived
through Eqs. (4.10) - (4.13).
From Eq. (4.19) we conclude that in order to construct states with consistent phases we should
keep after the diagonalization of [(2)]N only the set of ocfps that originate from one particular
ΓN −1 . For definiteness we choose ΓN −1 to be the diagram obtained from Γn by deleting a box from
the top row, i.e., with highest possible “content” [Nov94]. The ocfps which originate from the other
(N − 1)-particle Young diagrams are constructed with consistent phases, using Eq. (4.19).
75
4.2
Symmetrized Hyperspherical Harmonics
In this chapter and in the previous one two independent algorithms were introduced. In the
first algorithm (Chapter 3) we constructed hyperspherical functions with well defined O(N ) irreps.
(Although we mentioned the SO(N ) irreps in Chapter 3 we pointed out that we actually referred
to the O(N ) irreps that are directly related to the SO(N ) irreps.) In the second algorithm (Sect.
4.1) we constructed O(N − 1) irrep states with well defined irreps of the symmetric group SN .
Although the main ideas of these two algorithms are similar they are very distinct in their details.
In the first algorithm the GZ basis is used for identifying the O(N ) irreps and the hsopcs are the
eigenvectors of the second Casimir operator Ĉ2 (N ) of the group O(N ). In the second the Young
diagrams are used for identifying the SN irreps and the ocfps are the eigenvectors of the class sum
operator [(2)]N .
In these two algorithms the states are constructed recursively. In the first algorithm the N Jacobi coordinate state with a well defined O(N ) irrep is described as a linear combination of
hyperspherical states with O(N − 1) symmetry times the appropriate hsopcs (see Eq. (3.60) ). In
the second algorithm we express each state with a well defined SN irrep by a linear combinations
of states with SN −1 symmetry times the appropriate ocfps (see Eq. (4.14) ).
Our main goal in this section is to construct hyperspherical functions with well defined O(N −1)
irreps and well defined SN irreps. This goal is achieved by taking the “inner-product” of GZ O(N −
1) states with O(N − 1) states characterized by a well defined SN symmetry. For hyperspherical
functions with well defined quantum numbers KN −1 , LN −1 and MN −1 , the “inner-product” yields
the following combined set of hyperspherical states that belong to a given Yamanouchi basis YN
|KN −1 LN −1 MN −1 λN −1 αN −1 YN βN >
=
X
ΛN −2
|KN −1 LN −1 MN −1 λN −1 ΛN −2 αN −1 >< λN −1 ΛN −2 |λN −1 YN βN > .
(4.20)
In this equation there are two additional labels αN −1 and βN . These numbers were introduced to
remove degeneracy in the states (3.60) and the states (4.14), respectively. Therefore, we should
keep these two labels in the combined states (4.20).
In Ref. [Nov94] an algorithm for the construction of symmetrized hyperspherical functions was
76
formulated. As we noted in the introduction, these states have no special properties under O(N −1)
transformations. However, the number of hyperspherical states obtained in Ref. [Nov94] for any
invariant subspace, determined by the quantum numbers KN −1 , LN −1 and the Yamanouchi symbol
YN , is the same as the number of states obtained by the current procedure ( Eq. (4.20) ). On the
other hand, because the current approach and the approach of Ref. [Nov94] are different, each one
of the states obtained in the current approach is a linear combination of the states obtained in the
other approach, within the same invariant subspace.
The hyperspherical states, constructed in Eq. (4.20) are composed from hyperspherical states
with O(N − 1) symmetry and from states constructed from O(N − 1) irreps with SN symmetry.
From Eqs. (3.60) and (4.14) we obtain that each one of these N -particle states can be expressed
in terms of (N − 1)-particle states coupled to the N ’th particle state by using the ocfps and the
hsopcs as follows:
|KN −1 LN −1 MN −1 λN −1 αN −1 ΓN YN −1 βN >
=
X
λ N −2 βN −1
×
[(λN −2 ΓN −1 βN −1 ) λN −1 |}λN −1 ΓN βN ]
X
KN −2 LN −2 αN −2 ℓN −1
[(KN −2 LN −2 λN −2 αN −2 ; ℓN −1 )KN −1 LN −1 |}KN −1 LN −1 λN −1 αN −1 ]
|(KN −2 LN −2 λN −2 αN −2 ΓN −1 YN −2 βN −1 ; ℓN −1 )KN −1 LN −1 MN −1 λN −1 > .
(4.21)
Note that in this equation we start from the sum over the hsopcs which determines the irrep λN −2
of O(N − 2) and then we can sum over the ocfps. Since we calculate and store, in our computer
code, the hsopcs and ocfps separately, we are saving a lot of computer space and time compare to
the code based on the previous algorithm [Nov94] in which the “inner-product” states are directly
calculated. See Subsection 3.4 and Tables 3.1 - 3.3 in which we compare between the computer
achievements of the computer code based on the current algorithm and the code based on the
previous algorithm [Nov94].
The O(N −1) symmetry of the states in Eq. (4.21) is also important for physical calculations. In
order to calculate the various two-body matrix elements in the hyperspherical basis an appropriate
rotation should be applied for any relative two-body coordinates. Since the O(N − 1) irrep yields
the rotational symmetry, the desired rotation for each two-body matrix element can be easily
obtained [Nov95], as we will see in the next chapter.
77
Chapter 5
Matrix Elements of Two Body
Potentials Between symmetrized
Hyperspherical States
78
The application of the hyperspherical functions for the solution of the Schroedinger equation,
requires the calculation of the matrix elements of the potential energy operator between different hyperspherical states. The matrix elements of two-body operators between the permutational
symmetry adapted hyperspherical functions presented by Novoselsky and Katriel [Nov94], were
derived two years ago by Novoselsky and Barnea [Nov95]. Actually, for identical particles, only
the matrix element of the two-body operator between the last two particles has to be calculated.
This matrix element can be calculated using the hyperspherical cfps, the Raynal-Revai coefficients
[Ray70, Ray73], the angular momentum recoupling coefficients - the 6j coefficients, and the hyperspherical recoupling coefficients - the T-coefficients [Kil72, Kil73]. A detailed description of the
method for this calculation is given in Ref. [Nov95]. Although this method is complete, it requires
many calculation steps for each matrix element, involving the Raynal-Revai and the T-coefficients.
This results from the fact that the two-body interaction depends on the relative coordinate between
the two particles whereas the hyperspherical functions are expressed in terms of the Jacobi coordinates. The transformation from Jacobi coordinates to the relative coordinates, although requires
simple rotations, is difficult to implement.
This difficulty is greatly reduced by introducing the group sub-group chain O(3N − 3) ⊃ O(3)⊗
O(N − 1) ⊃ O(3) ⊗ SN that I have presented in the previous chapters. The difficulty in the
calculation of the matrix elements is reduced since we do not have to use the Raynal-Revai and
the T-coefficients. We just have to use the appropriate representation matrices of the orthogonal
group.
In the previous chapters I have presented a recursive algorithm for constructing hyperspherical
basis functions that belong to a given irrep of the orthogonal group and are at the same time
characterized by a well-defined permutational symmetry. The main idea was to construct independently the hyperspherical functions with well defined orthogonal symmetry, and then reduce the
irreps of the orthogonal group into the appropriate irreps of the symmetry group.
In this chapter I present the advantage of using the hyperspherical states with rotational symmetry for solving the difficulty, mentioned above. Since the states belong to well defined irreps of
the rotation group each rotation of two coordinates is associate with an appropriate representation
matrix. Any complicate rotation can be expressed in terms of several consecutive two-coordinate
79
rotations i.e., as a product of representation matrices. So, the transformation from the Jacobi
coordinates to the relative coordinate is reduced to a product of the appropriate representation
matrices. Moreover, the implementation of the representation matrices for the rotation is applicable, as well, to systems with non-identical particles, i.e., particles with different masses. In this
chapter I will present, in full detail, the rotation method described above as well as the calculation of the two-body matrix elements which requires also the various hyperspherical coefficients of
fractional parentage.
The presentation in this chapter is organized as follows: In Section 5.1 we demonstrate the rotation from the Jacobi coordinates to the relative coordinates, where the appropriate representation
matrices are given in Section 5.2. The matrix elements of a general two-body interaction between
the hyperspherical wave functions with orthogonal and permutational symmetry are derived in
Section 5.3. In Section 5.4 we express the scalar two-body interaction as a power expansion in
the relative coordinate and a simple expression for the matrix element of each term is given. An
analytic expression is derived in Section 5.5.
5.1
Kinematic rotations
The hyperspherical functions, expressed in terms of the Jacobi coordinates, are very convenient for
N -body calculations since spurious states (center-of-mass excitations) are not included. However,
two-body interactions are usually expressed in terms of the relative coordinates. Therefore, in
order to calculate the two-body matrix elements we have to introduce an efficient algorithm for
the transformation from Jacobi to the relative coordinates. More specifically, (see Section 5.3), we
have to transform the Jacobi coordinates into another set of coordinates where the last coordinate
is the desired relative coordinate.
For a general system of N -particles (not necessarily identical particles) where the mass of the
i′ th particle is mi , i = 1, 2, . . . , N the normalized Jacobi coordinates are expressed in terms of the
single particle coordinates as (B.1)
ηj =
s
j
´
1 X
Mj mj+1 ³
rj+1 −
m i ri ) ,
Mj+1
Mj i=1
80
;
j = 1, 2, . . . , N − 1
(5.1)
where Mj =
Pj
i=1
mi . Assuming that we want to calculate the matrix element of the two-body
interaction between particles n1 and n2 , where n1 > n2 . We have to rotate the set of coordinates
η j , j = 1, 2, . . . , N − 1 , into another set of coordinates η ′j where
rn1 ,n2 = κn1 ,n2 η ′N −1
and rn1 ,n2 = rn1 − rn2 . (We will obtain that the constant κn1 ,n2 is equal to
(5.2)
r
mn1 +mn2
.)
mn1 mn2
Actually,
the coordinates η l where l = 1, 2, . . . , n2 − 2 are not changed. This transformation is achieved by
an appropriate rotation gn1 ,n2 ∈ O(N − 1) and its realization matrix R(gn1 ,n2 ) i.e.,
η ′i =
N
−1
X
Rij (gn1 ,n2 )η j .
(5.3)
j=1
In particular, from the last two equations we obtain that
rn1 ,n2 = κn1 ,n2 η ′N −1 = κn1 ,n2
N
−1
X
j=1
RN −1,j (gn1 ,n2 )η j .
(5.4)
In general, the desired rotation is achieved by a sequence of N − n2 successive “rotations”,
where each “rotation” is actually a reflection or a two dimensional rotation, by the angle ϑk , in the
plane of the coordinates η k and η k+1 where k = n2 − 1, n2 , . . . , N − 2, i.e.,
N −3
n2
n2 −1
gn1 ,n2 = gnN1−2
,n2 gn1 ,n2 . . . gn1 ,n2 gn1 ,n2
(5.5)
( Note that for the relative coordinate r2,1 we need only N − n2 − 1 successive rotations, since
this coordinate is proportional to the Jacobi coordinate η 1 . ) Therefore, we can write the rotation
matrix R(gn1 ,n2 ) as a product of N − n2 two-dimension rotation matrices as follows
N −3
n2
n2 −1
R(gn1 ,n2 ) = RN −2 (gnN1−2
,n2 )RN −3 (gn1 ,n2 ) . . . Rn2 (gn1 ,n2 )Rn2 −1 (gn1 ,n2 )
(5.6)
The two-dimension rotations consist, in general, from three different “types” of rotations;
A. Rotation of two successive Jacobi coordinates η k and η k+1 to the relative coordinate rk+2,k+1
(times a constant) and its appropriate orthogonal coordinate. We will refer to this rotation
as the relative coordinate rotation.
81
B. Applying the transposition (k + 2, k + 1) (i.e., transposing the particle k + 2 by the particle
k + 1 and vice versa) on the two successive Jacobi coordinates η k and η k+1 . This application
resulted into a rotation of these two-coordinates by an appropriate angle in addition to a
reflection of the coordinate η k+1 [Nov94]. We will refer to this rotation as the transposition
operator rotation.
C. Rotation by 900 of the two successive Jacobi coordinates η k and η k+1 . We will refer to this
rotation as the 900 rotation.
In the following subsections we will present these three rotation “types” in detail, with some
examples for a four-particle system.
5.1.1
The relative coordinate rotation - “type” A
In this transformation we rotate the two successive Jacobi coordinates η k and η k+1 , defined in Eq.
(B.1), by an appropriate angle to the two orthogonal coordinates ζ k and ζ k+1 where
v
Ã
!
u
k
u Mk (mk+1 + mk+2 ) mk+2 rk+2 + mk+1 rk+1
1 X
t
ζk =
−
m i ri
Mk+2
and
ζ k+1 =
s
mk+2 + mk+1
mk+2 mk+1
(rk+2 − rk+1 ) =
mk+1 + mk+2
s
Mk
(5.7)
i=1
mk+2 mk+1
rk+2,k+1 .
mk+1 + mk+2
(5.8)
It is not difficult to show that these two new coordinates can be expressed in terms of the two
Jacobi coordinates η k and η k+1 as follows;
ζk =
ζ k+1
Since
s
Mk+2 mk+1
η +
Mk+1 (mk+1 + mk+2 ) k
s
s
Mk mk+2
= −
η +
Mk+1 (mk+1 + mk+2 ) k
Mk mk+2
Mk+1 (mk+1 +mk+2 )
+
Mk+2 mk+1
Mk+1 (mµr
k+1 +mk+2 )
angle ϑk = θ(rk+2,k+1 ) = arcsin
Mk mk+2
η
Mk+1 (mk+1 + mk+2 ) k+1
s
Mk+2 mk+1
η
.
Mk+1 (mk+1 + mk+2 ) k+1
(5.9)
= 1 this transformation is, actually, a rotation by the
Mk mk+2
Mk+1 (mk+1 +mk+2 )
¶
.
For a system of N identical particles, like the nucleons in the nuclei, only one two-body matrix
element has to be calculated, provided that the wave functions belong to a well defined irrep of the
82
symmetric group [Nov88a, Che89a]. For convenience, we usually choose to calculate the interaction
between the last two particles which is proportional to the coordinate rN,N −1 . Therefore, the
rotation from Eq. (5.9) should be apply to the coordinates η N −2 and η N −1 . In particular, for a
four-particle system we have to rotate the two coordinates η 2 and η 3 by the angle ϑ2 = θ(r4,3 ) =
arcsin
³q
M2 m4
M3 (m3 +m4 )
´
and then we obtain the coordinate
η ′3
ζ3 =
5.1.2
1
(r4 − r3 ) =
=
κ4,3
s
m4 m3
(r4 − r3 ) .
m3 + m4
(5.10)
The transposition rotation - “type” B
In this subsection we are interested to apply the transposition operator (k + 2, k + 1) on the two
successive Jacobi coordinates η k and η k+1 . Using simple algebraic techniques we can obtain that
(k + 2, k + 1)η k = ζ k =
s
mk+2 mk+1
η +
(Mk + mk+2 )Mk+1 k
s
Mk Mk+2
η
(Mk + mk+2 )Mk+1 k+1
(5.11)
and
(k + 2, k + 1)η k+1 = ζ k+1 =
Since
Mk Mk+2
(Mk +mk+2 )Mk+1
+
s
Mk Mk+2
η −
(Mk + mk+2 )Mk+1 k
mk+2 mk+1
(Mk +mk+2
µ)Mk+1
angle θ(k +2, k +1) = − arccos
q
s
mk+2 mk+1
η
.
(Mk + mk+2 )Mk+1 k+1
(5.12)
= 1 we obtain that this transformation is a rotation by the
mk+2 mk+1
(Mk +mk+2 )Mk+1
¶
associated by a reflection of the coordinate η k+1 .
In the notation of Eq. (5.5) the transformation gnk1 ,n2 is a rotation by the angle θ(k + 2, k + 1),
associated by a reflection of the coordinate η k+1 .
For example, assuming that we are interested to calculate an interaction that is related to the
relative coordinate r4,2 = r4 − r2 for a four-particle system. Starting from the Jacobi coordinates,
Eq. (B.1), we apply the transposition
(2,¶3) to the coordinates η 1 and η 2 i.e., the rotation by the
µ
angle ϑ1 = θ(3, 2) = arccos
the coordinate
ζ2 =
q
s
m3 m2
(m1 +m3 )M2
associated by an appropriate reflection and we obtain
(m1 + m3 )m2
1
r2 −
(m1 r1 + m3 r3 )
M3
m1 + m3
as well as the coordinate ζ 1 =
µ
q
m1 m3
(r
m1 +m3 3
¶
(5.13)
− r1 ) . In order to obtain the relative coordinate r4,2
we have to apply the rotation (5.9) to the coordinates ζ 2 and η 3 which
yields the
coordinate
¶
µr
q
(m1 +m3 )m4
m2
η ′3 = mm24+m
.
(r4 − r2 ) by a rotation in the angle ϑ2 = θ(r4,2 ) = arcsin
M3 (m2 +m4 )
4
83
5.1.3
The 900 rotation - “type” C
As we noted at the beginning of this section, our goal is to construct a set of coordinates where
the last coordinate η ′N −1 is proportional to the desired relative coordinate rn1 ,n2 . This relative
coordinate already exists for n1 = 2 and n2 = 1 since η 1 =
q
m1 m2
r2,1
M2
. In general, using the
relative coordinate rotation and the transposition rotation described in the previous subsections
we can always obtain that
η ′n1 −1
=
s
mn1 mn2
(rn1 − rn2 ) .
mn1 + mn2
(5.14)
However, when N > n1 the new coordinate in Eq. (5.14) is not the last coordinate. Therefore, we
have to rotate by 900 the two coordinates η ′n1 −1 and η n1 which yields that η ′n1 = η ′n1 −1 and then
η ′′n1 −1 = −η n1 . ( In the notation of Eq. (5.5) the transformation gnk1 ,n2 is a rotation by the angle
ϑn1 −1 = 900 .) If N = n1 + 1 then the last coordinate is the desired relative coordinate. If not, we
should continue to rotate by 900 the next two coordinates η ′n1 and η n1 +1 and so on till we reach
the last coordinate η ′N −1 .
As a simple example, let us assume that the interaction between the first two particles has to
be calculated, for a system with four particles. In this case the coordinate η 1 is already the desired
relative coordinate, as we noted above. In this case we have to rotate by 900 the two coordinates
η 1 and η 2 to obtain η ′2 = η 1 and η ′1 = −η 2 . Then we rotate again the two coordinates η 3 and η ′2
to obtain η ′3 = η ′2 = η 1 and η ′′2 = −η 3 .
5.1.4
The general procedure
In the preceding subsections we have given the algorithm for transforming the set of Jacobi coordinates η j where j = 1, 2, . . . (N − 1), Eq. (B.1), into another set of coordinates η ′j , Eq. (5.3),
where the last coordinate η ′N −1 is proportional to the relative coordinate rn1 ,n2 , Eq. (5.2). We
have shown that the appropriate rotation can be obtained as a product of N − n2 two dimension
successive rotations where the rotation angles are uniquely determined by the relative coordinate
rn1 ,n2 . Actually we introduced three “types” of two dimensional rotations; the relative coordinate
rotation, the transposition rotation and the 900 rotation.
84
The procedure for applying these three “types” of rotations depends actually on the numbers
N ≥ n1 > n2 where the relative coordinate is rn1 ,n2 . If n1 > n2 + 1 we should first apply the
“type” B rotation n1 − n2 − 1 times. Then the indices n1 and n2 are “close by” in the coordinate
η n1 −1 , and we can apply the relative coordinate rotation ( “type” A ). After this rotation the new
coordinate η ′n1 −1 is proportional to the relative coordinate rn1 ,n2 . If N = n1 − 1 the procedure is
completed. Otherwise, we have to apply the “type” C rotation N − n1 times in order to replace the
coordinate η N −1 by the coordinate η’n1 −1 which we will designate now by η’N −1 . For the special
case where N = n1 = n2 + 1 the relative coordinate rotation is sufficient.
5.2
The representation matrices of the rotation (and reflection) operators
In this section we calculate the matrix elements of the A, B, and C “types” operators, introduced
in the previous section, between states with well defined orthogonal symmetry, i.e., GZ states.
Actually the operators A and C are rotations by a given angle, and the operator B involves,
in addition, a reflection of one coordinate. Therefore, we present here a general algorithm for
calculating the matrix elements of the rotation (and reflection) operators between GZ states. The
calculation is independent of the basis states that we are using, i.e., for example, the hyperspherical
states that are constructed recursively in Eq. (3.60). Therefore, the states will be denoted only by
their GZ basis.
Our aim is to calculate the following matrix element
< λN −1 ΛN −2 |gn1 ,n2 |λN −1 Λ′N −2 > ,
(5.15)
where gn1 ,n2 is the general operator that yields the transformation from the set of the Jacobi
coordinates η j where j = 1, 2, . . . N − 1, Eq. (B.1), into another set of coordinates η ′j , Eq. (5.3).
This operator is a product of two-dimension rotation operators (see Eq. (5.5)), where each operator,
gnk1 ,n2 , is a rotation by the angle ϑk of the appropriate two Jacobi coordinates, designated by the
indices k and k + 1. For “type” B, the operator gnk1 ,n2 also reflects the coordinate η k+1 .
Based on Group theory [Ham62], the application of the operator gnk1 ,n2 on a GZ state can be
85
obtained by using the D representation matrices for the rotation associated with a phase σ for the
reflection
gnk1 ,n2 |λN −1 ΛN −2 >
= σλ k+1 ,λ k
X
λ′
k
(λ
Dλ ′ k+1
,λ k
,λ k−1 )
k
(ϑk )|λN −1 . . . λk+1 λ′k λk−1 . . . λ2 > .
In Section 4.1 we have shown that the reflection phase
σλ
k+1 ,λ k
(5.16)
depends on the sum over the GZ
labels of the irreps λk+1 and λk as follows
σλ k+1 ,λ k = (−1)qk+1 +qk ,
where in general qk =
states where qk =
Pmin(3,[ k2 ])
i=1
Pmin(3,[ k2 ])
i=1
(5.17)
λk,i , and for k = 3, 5 one should also consider inverse parity
λk,i + 1 . Starting from Eq. (5.5), and using Eqs. (5.16) - (5.17), we
can rewrite the matrix element in Eq. (5.15) in the form
′
′
N −3
n2
n2 −1
< λN −1 λN −2 . . . λ2 |gnN1−2
,n2 gn1 ,n2 . . . gn1 ,n2 gn1 ,n2 |λN −1 λN −2 . . . λ2 >
σλ
= δλ ,λ ′ . . . δλ
σ ′
. . . σλ ′
′
′ σ ′
′
′
,λ ′
2
n2 −2 ,λ n −2 λ n +1 ,λ n2 λ n +2 ,λ n +1
N −1 ,λ N −2
2
N −2
N −3
2
2
2
2
<
X
λN −1 λN −2 . . . λ2 |
λ ′′
(λ ′n ,λ ′
)
n2 −2
D ′′ 2
(ϑn2 −1 )
λ
,λ ′
n2 −1
n2 −1
n2 −1
X
λ ′′
N −3
(λ ′
,λ ′′
)
Dλ ′′N −2,λ ′ N −4 (ϑN −3 )
N −3
N −3
X
λ ′′
n
(λ ′
,λ ′′
)
n +1
n2 −1
D ′′ 2 ′
(ϑn2 ) . . .
λ n ,λ n
2
2
2
X
λ ′′
N −2
(λ N −1 ,λ ′′
N −3 )
(ϑN −2 )|λN −1 λ′′N −2
′
N −2 ,λ N −2
Dλ ′′
. . . λ′′n2 −1 . . . λ′2 >
(λ ′
,λ n −1 )
(λ ′n ,λ n −2 )
n2 +1
2
2
2
σλ
D
(ϑn2 −1 )D
. . . σλ ′
(ϑn2 ) . . .
′
,λ ′
,λ ′
,λ ′n σλ ′
λ n2 ,λ ′n
N −1 ,λ N −2 λ n −1 ,λ ′
n2 +2
n2 +1
N −2
N −3
2
2
n2 −1
2
(λ ′
,λ N −3 )
(λ ′
,λ N −4 )
λ N −1
(gn1 ,n2 ) .
(ϑN −2 ) ≡ σλ
(ϑN −3 )D N −1 ′
D N −2 ′
′
Λ
Λ′
λ N −2 ,λ
λ N −3 ,λ
N −1 ,ΛN −2
N −2 N −2
N −2
N −3
= σλ ′
n2 +1
D
(5.18)
The D matrices, used in the above equations, are the representation matrices of the rotation
operator eiϑk X̂k,k+1 , where the hermitian operator X̂k,k+1 is the infinitesimal generator of the O(N )
group defined in Eq. (A.1). Since X̂k,k+1 ∈ O(k + 1) and it commutes with the generators of
O(k − 1) the D matrix alters only the O(k) irrep λk and depends only on the values of the O(k + 1)
irrep λk+1 and the O(k − 1) irrep λk−1 . The representation matrix of X̂k,k+1 is given by
Mk,k+1 (λk+1 , λk−1 )λ ′k ,λ k = F(λk+1 , λk , λ′k , λk−1 ) ,
(5.19)
where the F function was introduced in Section 3.3 (Eq. (3.63)). The analytic expressions for
this function was first derived by Gel’fand and Zetlin [Gel50] and also by Pang and Hecht [Pan67].
86
Since X̂k,k+1 is an hermitian operator, we can diagonalize its corresponding matrix Mk,k+1 and
write it as follows
Mk,k+1 = UDU −1
where D is a diagonal matrix. Therefore, the D representation matrices, between the appropriate
irreps of the orthogonal group, can be rewritten as
(λ
Dλ ′ k+1
,λ k
k
,λ k−1 )
³
(ϑk ) = eiϑk Mk,k+1 (λ k+1 ,λ k−1 )
´
λ ′ ,λ k
k
=
³
UeiφD U −1
´
λ ′ ,λ k
k
.
(5.20)
This form is very efficient from a computational point of view.
5.3
Matrix elements of arbitrary two-body operators
In this section we present the method for calculating the matrix elements of a general two-body
interaction Vn1 ,n2 between N -particle states expressed in terms of the symmetrized hyperspherical
functions
′
′
< KN −1 LN −1 MN −1 λN −1 αN −1 YN βN |Vn1 ,n2 |KN′ −1 L′N −1 MN′ −1 λ′N −1 αN
−1 YN βN > .
(5.21)
The states belong to well defined irreps of the O(N − 1) group and to a given Yamanouchi symbol
YN , as in Eq. (4.20).
The states in Eq. (5.21) are expressed in terms of the Jacobi coordinates η j where j =
1, 2, . . . N − 1 (Eq. (B.1)). However, a general two-body interaction between particles n1 and
n2 depends on the relative coordinate rn1 ,n2 = rn1 − rn2 , or its conjugate momenta. Therefore,
(except for the first two particles), we have to rotate the Jacobi coordinates in Eq. (B.1) to a new
set of coordinates where the last coordinate is the desired relative coordinate. This rotation was
fully explained in the previous section. We have explicitly shown the application of this rotation
on a state with a well defined GZ basis. However, the states in Eq. (5.21) belong only to the irreps
λN −1 and λ′N −1 of the O(N − 1) group. Therefore, we have to express these states in terms of
states with a complete GZ basis. Practically, if n2 > 4 it is sufficient to determine the orthogonal
group irreps down to the O(n2 − 2) group. In other words, we have to apply the expression appeared in Eq. (4.14) several times, and to use the appropriate ocfps in order to obtain the following
87
hyperspherical states, identified by the desired orthogonal group irreps
|KN −1 LN −1 MN −1 λN −1 αN −1 ΓN YN −1 βN >
=
X
λ n βn+1
×
×
[(λn Γn+1 βn+1 ) λn+1 |}λn+1 Γn+2 βn+2 ]
X
X
X
... ×
X
[(λN −2 ΓN −1 βN −1 ) λN −1 |}λN −1 ΓN βN ]
λ n+1 βn+2
λ N −2 βN −1
... ×
λ n+2 βn+3
λ N −3 βN −2
[(λN −3 ΓN −2 βN −2 ) λN −2 |}λN −2 ΓN −1 βN −1 ]
×|KN −1 LN −1 MN −1 λN −1 αN −1 λN −2 . . . λn Yn+1 βn+1 ) > .
(5.22)
where O(n) is the minimal group, i.e., O(2) for n2 ≤ 4 and O(n2 − 2) for n2 > 4. Using this
derivation in the bra and the ket states in Eq. (5.21) we obtain that, actually, the following matrix
element has to be calculated
< KN −1 LN −1 MN −1 λN −1 αN −1 λN −2 . . . λn Yn+1 βn+1 )|(Vn1 ,n2 )|
′
′
′
′
KN′ −1 L′N −1 MN′ −1 λ′N −1 αN
−1 λN −2 . . . λn Yn+1 βn+1 ) > .
(5.23)
The states in Eq. (5.23) are functions of the Jacobi coordinates (B.1). Now, after obtaining
all the desired orthogonal irreps in these states we can apply the operator gn1 ,n2 (as we explained
in the previous section) to transform the set of the Jacobi coordinates η j into η ′j , where j =
1, 2, . . . , N − 1 . The idea is to apply the operator (gn1 ,n2 )−1 gn1 ,n2 before and after the two-body
interaction Vn1 ,n2 in Eq. (5.23) i.e.,
< KN −1 LN −1 MN −1 λN −1 αN −1 λN −2 . . . λn Yn+1 βn+1 )
|(gn1 ,n2 )−1 gn1 ,n2 (Vn1 ,n2 )(gn1 ,n2 )−1 gn1 ,n2 |
′
′
′
′
KN′ −1 L′N −1 MN′ −1 λ′N −1 αN
−1 λN −2 . . . λn Yn+1 βn+1 ) > .
(5.24)
The application of the g operator on the GZ states is given in terms of the D functions, as we
have seen in Eq. (5.18), whereas the application of g and g −1 on the two-body interaction, i.e.,
gn1 ,n2 (Vn1 ,n2 )(gn1 ,n2 )−1 yields the two-body interaction as a function of the coordinate η N −1 times
the coefficient κn1 ,n2 (as it is obvious from Eq. (5.2)). The two-body interaction is independent
of the other coordinates η j , j = 1, . . . , N − 2 , since it is a scalar with respect to the action of
the groups O(N − 2) ⊃ O(N − 3) ⊃ · ⊃ O(3) ⊃ O(2). This property and Eq. (5.18) leads to the
following expression for the matrix element (5.24)
σλ
σ ′
N −1 ,ΛN −2 λ
N −1
,Λ′
N −2
X
D
λ N −1 ∗
Λ′′
Λ
N −2 N −2
Λ′′
N −2
88
(gn1 ,n2 )
D
λ ′N −1
(gn1 ,n2 )
Λ′′
Λ′
N −2 N −2
< KN −1 LN −1 MN −1 λN −1 αN −1 Λ′′N −2 Yn+1 βn+1 )|V (κn1 ,n2 η N −1 )|
′
′′
′
KN′ −1 L′N −1 MN′ −1 λ′N −1 αN
−1 ΛN −2 Yn+1 βn+1 ) > .
(5.25)
Here, for simplicity, we are using the symbol ΛN −2 to denote the irreps λN −2 λN −3 . . . λn although,
usually, this symbol stands for a complete GZ basis . Note that the sum in Eq. (5.25) over the
irreps λ′′N −2 is restricted to those irreps which belong to the irrep λN −1 and to the irrep λ′N −1 as
well.
Since the two-body interaction in Eq. (5.25) depends only on the coordinate η N −1 we should
separate the hyperspherical functions related to this coordinate in the bra and the ket states. So,
the last step in the derivation consists of using the hsopcs introduced in Eq. (3.60). Then, the
matrix element in Eq. (5.25) is
X
KN −2 LN −2 αN −2 ℓN −1 ,ℓ′N −1
[(KN −2 LN −2 λN −2 αN −2 ; ℓN −1 )KN −1 LN −1 |}KN −1 LN −1 λN −1 αN −1 ]
h
′
× (KN −2 LN −2 λN −2 αN −2 ; ℓ′N −1 )KN′ −1 L′N −1 |}KN′ −1 L′N −1 λ′N −1 αN
−1
i
× < (KN −2 LN −2 λN −2 αN −2 ΛN −3 Yn+1 βn+1 ; ℓN −1 )KN −1 LN −1 MN −2 |V (κn1 ,n2 η N −1 )|
(KN −2 LN −2 λN −2 αN −2 ΛN −3 Yn+1 βn+1 ; ℓ′N −1 )KN′ −1 L′N −1 MN′ −2 > .
(5.26)
This matrix element can be calculated for every given two-body interaction expressed in terms of
the hyperspherical functions. In the next section I will present a detailed calculation for a general
scalar interaction.
5.4
Explicit derivation of the matrix elements of a central
two-body interaction
A scalar two-body interaction between particles n1 and n2 is usually expressed in terms of the
absolute value of the relative coordinate rn1 ,n2 = |(rn1 − rn2 )| and is independent of the appropriate
angular coordinates. In this case the two-body operator is a sum over N (N − 1)/2 two-body
interactions depending on the norm of the relative coordinate ri,j i.e., V =
PN
i<j=2
Vi,j . Actually,
when the interaction between any two particles is equal, and the bra and the ket states belong
89
to well defined arbitrary irreps of the symmetric group it is sufficient to calculate only one twobody interaction [Nov88a, Che89a]. For symmetric or antisymmetric Yamanouchi basis this result
is straightforward, since each pair of particles is symmetric or antisymmetric for exchange. In
general, this result is a direct consequence of the sum over path overlaps method, introduced in
Refs. [Nov88a, Che89a].
Usually, the derivation of the matrix elements of the two-body operator between the last two
particles, i.e., the (N − 1)th and the N th particles is straightforward, whereas the matrix elements
of the other two-body operators are difficult to obtain. Nevertheless, I have shown in the previous
section, that in our method we can achieve, in a relatively straightforward manner, the expression for the two-body matrix element for any given two particles. We have calculated the matrix
elements of two-body operators between N -particle states expressed in terms of the symmetrized
hyperspherical functions. We have reached the conclusion that, finally, we can express the interaction in terms of the last Jacobi coordinate and the basis functions of this coordinate can be
separated in the bra and the ket states. In other words, the calculation reduced to the derivation
of the matrix element in Eq. (5.26). In this section we will explicitly evaluate this matrix element
for any scalar operator.
For every realistic scalar potential we can express the two-body interaction as a power expansion
in the relative coordinate as follows
Vn1 ,n2 =
∞
X
p=−2
p
Vp (rn1 ,n2 ) ≡
∞
X
p=−2
Vp (ρN −1 sin ϕn1 ,n2 )p ,
(5.27)
where ρN −1 is the hyper-radial coordinate (see Eq. (2.7)). Therefore, the matrix elements that we
have to calculate (Eq. (5.21)) are, practically, of the form
′
′
< KN −1 LN −1 MN −1 λN −1 αN −1 YN βN |(sin ϕn1 ,n2 )p |KN′ −1 LN −1 MN −1 λ′N −1 αN
−1 YN βN > .
(5.28)
Note that the total angular momentum LN −1 and its projection MN −1 are the same in the bra and
the ket states because of the scalar interaction.
By using the algorithm developed in the previous section we obtained that, actually, only the
matrix element from Eq. (5.26) has to be calculated. From the relation (5.27) and since the
last Jacobi coordinate is ηN −1 = ρN −1 sin ϕN −1 we obtain the following expression for the matrix
90
element in Eq. (5.26)
∞
X
p=−2
Vp (κn1 ,n2 ρN −1 )p
< (KN −2 LN −2 λN −2 αN −2 ΛN −3 Yn+1 βn+1 ; ℓN −1 )KN −1 LN −1 MN −1 |(sin ϕN −1 )p |
(KN −2 LN −2 λN −2 αN −2 ΛN −3 Yn+1 βn+1 ; ℓN −1 )KN′ −1 LN −1 MN −1 >
≡
∞
X
p=−2
Vp (κn1 ,n2 ρN −1 )p < KN −1 |(sin ϕN −1 )p |KN′ −1 >KN −2 ,ℓN −1 .
(5.29)
The short notation of the matrix element, in this equation, can be used since the angle ϕN −1
is invariant with respect to the action of the group O(3(N − 2)) associated with the first N − 2
Jacobi coordinates. Therefore, the matrix element in Eq. (5.29) is diagonal in KN −2 , and since
ρN −1 sin ϕN −1 = ηN −1 the matrix element is diagonal in ℓN −1 as well. In addition, the matrix
element is independent on the values of LN −2 , αN −2 , and λN −2 .
In Appendix A of [Nov95] we derived an analytic expression for the matrix elements in Eq.
(5.29). In the next section, I will use a somewhat different approach to obtain a more simple
analytic expression for the same matrix elements.
It should be noted that so far we have studied the matrix elements of the hyperspherical states.
Expanding the wave function in terms of the symmetrized states, each hyperspherical state is
coupled to an appropriate hyper radial function. After we integrate over the hypersphere and use
the results of this section for evaluating the matrix elements we obtain a set of coupled differential
equations for the hyper radial components of the wave function which we have to solve. A method
to solve this set of equations will be presented in the next chapter.
5.5
Analytic expression for the matrix element
In Section 3.3 it was shown that the N + 1-body (N Jacobi coordinates) hyperspherical functions
that belong to well defined irrep of the orthogonal group O(N − 1) corresponding to N particles
(N − 1 Jacobi coordinates) can be written as in Eq. (3.49),
|(KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN >
= ΨKN ;ℓN KN −1 (ϕN )|(KN −1 LN −1 ΛN −1 αN −1 ; ℓN )LN MN > ,
91
(5.30)
where, Eq. (3.45),
|(KN −1 LN −1 ΛN −1 αN −1 ; ℓN )LN MN >=
X
MN −1 mN
(LN −1 MN −1 ℓN mN |LN MN )
×|KN −1 LN −1 MN −1 ΛN −1 αN −1 > |ℓN mN > ,
(5.31)
and, Eq.(3.46),
ΨKN ;ℓN KN −1 (ϕN ) = NN (KN ; ℓN KN −1 )
(ℓ + 12 ,KN −1 + 3N2−5 )
(sin ϕN )ℓN (cos ϕN )KN −1 PnNN
(cos 2ϕN ) .
(5.32)
The explicit dependence of the N particles state |KN −1 LN −1 MN −1 ΛN −1 αN −1 > on the Jacobi
coordinates η 1 , . . . η n−1 is not of our concern right know. However, the dependence of the state
|(KN −1 LN −1 ΛN −1 αN −1 ; ℓN )KN LN MN > on the N th Jacobi coordinate is written explicitly. The
function associated with the angular part of this coordinate is the spherical harmonic function
YℓN mN (ΩN ) and its corresponding state is denoted by |ℓN mN >. This state is coupled, Eq. (3.45),
with the angular momentum LN −1 to LN , obtaining the state |(KN −1 LN −1 ΛN −1 αN −1 ; ℓN )LN MN >.
The radial part is associated with the function ΨKN ;ℓN KN −1 (ϕN ) through the transformation to hyperspherical coordinates (ρN −1 , ηN ) −→ (ρN , ϕN ).
The normalization coefficient,
NN (KN ; ℓN KN −1 ) is given in Eq. (2.39). The hyperspherical angular number KN has the same
parity as KN −1 + ℓN , and is restricted by KN ≥ KN −1 + ℓN ≥ 0. The non-negative integer
nN =
KN −KN −1 −ℓN
2
(ℓ + 12 ,KN −1 + 3N2−5 )
is the order of the Jacobi polynomial PnNN
(cos 2ϕN ). The angular
volume element associated with the 3N − 1 dimensional hyper sphere S3N −1 is, Eq. (2.9),
dS3N −1 = sin2 (ϕN ) cos3N −4 (ϕN )dϕN dΩN dS3N −4
(5.33)
where dS3N −4 is the volume element associated with the 3N − 4 dimensional hyper sphere and
dΩN is the volume element associated with the angular part of the N th Jacobi coordinate η N . The
hyper angle ϕN varies in the range
matrix element in Eq. (5.29) as
π
2
≥ ϕN ≥ 0. Using the Eqs. (5.30)-(5.33), we can rewrite the
< KN′ |(sin ϕN )p |KN >KN −1 ℓN = NN (KN′ ; ℓN KN −1 )NN (KN ; ℓN KN −1 )
×
Z
π
2
dϕN sin2 (ϕN ) cos3N −4 (ϕN ) sin2ℓN (ϕN ) cos2KN −1 (ϕN )
0
+ 3N −5 )
+ 3N −5 )
(ℓ + 1 ,K
(ℓ + 1 ,K
×Pn′ NN 2 N −1 2 (cos 2ϕN )PnNN 2 N −1 2 (cos 2ϕN ) sinp (ϕN )
. (5.34)
Substituting x = cos(2ϕN ), in Eq. (5.34) we get the following expression for the matrix element:
(α,β)
< KN′ |(sin ϕN )p |KN >KN −1 ℓN = NN (KN′ ; ℓN KN −1 )NN (KN ; ℓN KN −1 ) In′ N nN (p) ,
92
(5.35)
where
1
(α,β)
In′ n (p) ≡
2α+β+p/2+2
Z
1
−1
and
α = ℓN +
(α,β)
dx(1 − x)α+p/2 (1 + x)β Pn′
(x)Pn(α,β) (x) ,
(5.36)
3N − 5
1
; β = KN −1 +
.
2
2
(5.37)
(α,β)
In order to evaluate the integral In′ n (p) in Eq. (5.36), we express the Jacobi polynomials in Eq.
(5.36) by the explicit expressions [Mag66]
Pn(α,β) (x)
n
X
n Γ(α + β + n + r + 1) x − 1
Γ(α + n + 1)
=
n!Γ(α + β + n + 1) r=0 r
Γ(α + r + 1)
2
à !
and
(α,β)
Pn′ (x)
µ
¶r
,
′
′
i
(−1)n
dn h
1
α+n′
β+n′
= n′ ′
(1
−
x)
(1
+
x)
.
2 n ! (1 − x)α (1 + x)β dxn′
(5.38)
(5.39)
(α,β)
Doing so we see that In′ n (p), Eq. (5.36), is reduced to a sum of integrals of the following type,
Z
1
−1
(1 − x)s (1 + x)t = 2s+t+1 B(s + 1, t + 1) = 2s+t+1
Γ(s + 1)Γ(t + 1)
Γ(s + t + 2)
(5.40)
where B(a, b) is the Beta function. These integrals are convergent for s, t > −1, which is equivalent
to demanding p ≥ −2.
Using Eqs. (5.38), (5.39) and (5.40) we get after some algebra
′
(α,β)
In′ n (p)
where
1 (−)n Γ(α + n + 1)Γ(β + n′ + 1)
=
2 n′ !
Γ(α + β + n + 1)
n
X
(−)r Γ(α + β + n + r + 1) Γ(1 + r + p/2)
×
Γ(α + r + 1)
Γ(1 + r + p/2 − n′ )
r=rmin r!(n − r)!
Γ(α + r + p/2 + 1)
,
×
Γ(α + β + n′ + r + p/2 + 2)



0
for odd p
rmin = 
 max(0, n′ − p/2) for even p
,
(5.41)
(5.42)
and n = nN , n′ = n′ N . One could check that for odd p Eq. (5.41) coincides with Eq. (26) in
Ref. [Kri90] where the matrix elements of 3 and 4-body systems where considered. For even p, the
(α,β)
(α,β)
matrix element Eq. (5.41) is equal zero unless n ≥ n′ − p/2. However since In′ n (p) = Inn′ (p)
(see Eq. (5.35)) we conclude that for even p the matrix element < KN′ | sinp ϕN |KN >KN −1 ℓN = 0
unless |KN − KN′ | ≤ p. In particular this demand recovers the δ function for the p = 0 case.
93
Chapter 6
Application of the Method for Physical
Systems
94
So far I’ve presented the cumbersome mathematical apparatus of the hyperspherical harmonic
method, the symmetrization algorithm and the way to apply this algorithm to physical problems.
As I mentioned in the introduction, the hyperspherical functions are very useful basis functions
for the study of few-body systems and were extensively used for the study of such systems. Mandelzweig et al. have applied this method to the atomic three-body system [Haf87, Haf89], Wang
et al. studied the He & Li atoms [Wan95]. Viviani and his collaborators studied nuclear systems
like the triton and the α particle [Viv92, Viv95], and there are many other researchers who have
applied this method to various physical problems of interest.
In this chapter I will present some numerical and analytic applications of the hyperspherical
harmonic method to physical problems. In Section 6.1 the Faddeev equations are considered, the
problems of identical and non-identical spinless particles in three dimensional space, interacting
via harmonic oscillator two-body potentials are solved analytically for any excitation and for an
arbitrary value of the total angular momentum. These solutions are the only examples of general
analytic solution of the Faddeev equation in three dimensional space, and it is a unique example
of a solution of the Faddeev equations for non-identical particles.
The full treatment of the N -body problem in our formalism will be given in Section 6.2, for
the case of N particles interacting via two-body spin-isospin independent potentials. The generalization of the derivation for spin-isospin dependent potentials and for three-body forces is not
difficult, however, it is beyond the scope of this work. The usefulness of the method for few
body calculation is demonstrated through the application of the method to the four-body problem.
Few examples of four-body calculations are presented. The applicability of the method to nuclear
physics is demonstrated in Subsection 6.2.1, where I use the symmetrized hyperspherical harmonics
to calculate the ground state of four nucleons interacting with the Malfliet-Tjon effective two-body
s-wave potential. The atomic four-body system is considered in Subsections 6.2.2 and 6.2.3, where
I use the symmetrized hyperspherical harmonics to study the three electron system. In Subsection
6.2.2 the ground state and the first excited state of hydrogen like Li, i.e. no interaction between
the electrons, are calculated. This problem is a good test case for the computational method since
it as a well known spectrum. The complete Li atom is considered in Subsection 6.2.3.
95
6.1
Analytic Solution of Faddeev Equation
The Faddeev equations [Fad61] were extensively used during the last 15 years to obtain wave
functions of different three body systems. In order to develop a better understanding of these
equations and their solutions, analytically solvable examples are needed. Very few such solutions
were found, mostly in one dimension [Dod70, Him74, Him77, Maj78]. Indeed, due to the complexity
of the Faddeev equations, to find their analytic solutions in three dimensional space even for the
simplest known pair potentials is not an easy task. In the general case of three non-identical spinless
particles with an arbitrary total angular momentum a solution of three coupled inhomogeneous
partial differential equations in six variables should be found. Even for the case of three identical
spinless particles in the state of zero total angular momentum, an inhomogeneous partial differential
equation in three variables must be solved. This explains why up to now only a single analytically
solvable example exists for pair potentials, interacting in all partial waves. This is a case of three
identical zero spin bosons in their ground state, connected via harmonic oscillator two-body forces
which was considered by Friar, Gibson and Payne [Fri80]. Since the harmonic oscillator forces
are associated with small deviations from the equilibrium state and are of immense significance in
different branches of physics, it is very interesting to find an exact solution of the Faddeev equations
in the general case. The purpose of this section is to generate the harmonic oscillator Faddeev wave
functions for any excitation and for an arbitrary value of the total angular momentum of the three
particles. In Subsection 6.1.1 I consider the case of three identical particles [Bar92], the case of
non-identical particles is discussed in Subsection 6.1.2 [Bar94].
6.1.1
Identical Oscillators
The ground state of the harmonic oscillator Faddeev wave function could be obtained [Fri80] by
direct solution of the Faddeev equations. However, for higher excitations and nonzero angular
momenta it is very difficult to find a direct solution of the Faddeev equation, even for the simple
case of harmonic oscillator pair potentials. In order to obtain the general solution I will use here the
powerful hyperspherical harmonic method, presented in Chapter 2 of this thesis. For the sake of
brevity I will consider in this subsection three-identical but distinguishable particles and set their
96
masses to unity. In this method the rest frame solutions Ψ and Φ of the three body Schroedinger
1
(− ∆(2) + W )Ψ = EΨ
2
(6.1)
1
( ∆(2) + E)Φ = V Ψ
2
(6.2)
and Faddeev equation
corresponding to the total angular momentum L2 and its projection M2 , are represented by the
expansions
Ψ=
X
ψ(ρ2 )YK2 L2 M2 ℓ1 ,ℓ2 (Ω(2) , ϕ2 ) ,
(6.3)
X
φ(ρ2 )YK2 L2 M2 ℓ1 ,ℓ2 (Ω(2) , ϕ2 ) .
(6.4)
K2 ,ℓ1 ,ℓ2
Φ=
K2 ,ℓ1 ,ℓ2
Here ∆(2) is the six dimensional Laplace operator defined in Eq. (2.14), ρ2 is the hyperspherical
radius and Ω(2) , ϕ2 are the hyperspherical angels, where Ω(2) stands for angles (Ω1 , Ω2 ) defined in
Section 2.1. YK2 L2 M2 ℓ1 ,ℓ2 (Ω(2) , ϕ2 ) are the hyperspherical harmonic functions presented in Section
2.3. The harmonic oscillator pair potential V between particles 1, 2 and the sum W of all three
pair potentials are given in terms of the Jacobi coordinates η 1 , η 2 by the expressions
1
1
2
V = ω 2 r12
= ω 2 η 21 ,
6
3
(6.5)
1
1
1
2
2
2
W = ω 2 (r12
+ r23
+ r31
) = ω 2 ρ22 = ω 2 (η 21 + η 22 ).
6
2
2
(6.6)
Using Eqs. (2.15),(2.16), and (6.6), we see that in hyperspherical coordinates the Schroedinger
equation (6.1) has the form
"
#
1
1 ∂ 5 ∂
ρ2
− 2 K̂22 + 2E − ω 2 ρ22 Ψ = 0 .
5
ρ2 ∂ρ2 ∂ρ2 ρ2
(6.7)
Since in this case the Hamiltonian depends only on the generalized angular momentum operator
K̂22 which commutes with the operators of total and partial angular momenta L̂2 , ℓ̂1 , ℓ̂2 , there is no
coupling of angular and radial quantum numbers and therefore no summation in Eq. (6.3). The
radial part of the wave function depends only on the quantum number K2 :
Ψ = ψ(ρ2 )YK2 L2 M2 ℓ1 ,ℓ2 (Ω(2) , ϕ2 ) .
(6.8)
Its substitution into Eq. (6.7) gives the following equation for the radial function:
"
#
1 d 5 d
K2 (K2 + 4)
+ 2E − ω 2 ρ22 ψ(ρ2 ) = 0 ,
ρ2
−
5
ρ2 dρ2 dρ2
ρ22
97
(6.9)
since, as we have seen in Section 2.3, the hyperspherical functions YK2 L2 M2 ℓ1 ,ℓ2 (Ω(2) , ϕ2 ) are eigen-
functions of the generalized angular momentum operator K̂22 with eigenvalues K2 (K2 + 4). It can
2
be seen that at very small and at asymptotically large ρ2 the proper solution behaves as ρK
2 and
exp (−ωρ22 /2) respectively. Thus, one has to look for a solution in the form
2
ψK2 (ρ2 ) = ρK
2 exp (−
ωρ22
)χ(ρ2 ) .
2
(6.10)
Substitution of Eq. (6.10) into Eq. (6.9) shows that the function χ is the confluent hypergeometric
function F (α, γ, z) satisfying the Kummer equation [Gra80]
"
d2
d
z 2 + (γ − z) − α F (α, γ, z) = 0 .
dz
dz
#
(6.11)
E − (K2 + 3)ω
, γ = K2 + 3 .
2ω
(6.12)
Here the variable z = ωρ22 and
α=−
The restriction of F being a polynomial, necessary to insure the proper asymptotic behavior at
infinity, demands α = −nρ , nρ being a positive integer radial quantum number. This results in the
following equation for the energy eigenvalues:
E = ω(n + 3) , n = 2nρ + K2 .
(6.13)
On the other hand, Eq. (6.1), in view of Eqs.(2.14) and (6.6), is the Schroedinger equation for two
uncoupled spherical oscillators, therefore the principal quantum number n could also be written as
[Flu71]
n = 2(ν1 + ν2 ) + ℓ1 + ℓ2 ,
(6.14)
ν1 , ν2 being the corresponding radial quantum numbers. Since in view of (6.13) and (6.14), ℓ1 + ℓ2
and K2 have the same parity as n, for a given even (odd) excitation n the quantum number K2
takes one of the even (odd) values between n and ℓ1 + ℓ2 :
K2 = n, n − 2, ..., ℓ1 + ℓ2 .
(6.15)
We finally have, up to a normalization constant, the following solution corresponding to excitation
n, global momentum K2 and partial momenta ℓ1 , ℓ2 coupled to the total momentum L2 :
Ψ ≡ ΨnK2 L2 M2 ℓ1 ℓ2 (ρ2 , Ω(2) , ϕ2 )
¶
µ
ωρ22
n − K2
K2
2
= ρ2 exp (−
)F −
, K2 + 3, ωρ2 YK2 L2 M2 ℓ1 ℓ2 (Ω(2) , ϕ2 ).
2
2
98
(6.16)
As one can expect, in case of the six dimensional oscillator there is a very strong degeneracy, with
many functions (6.16) with different quantum numbers K2 , L2 , M2 , ℓ1 , ℓ2 corresponding to the same
energy eigenvalue (6.13).
n
Denoting the radial part of this solution as ψK
(ρ2 ), one obtains the following equation for the
2
radial part φnK2 ,ℓ1 ,ℓ2 (ρ2 ) of the Faddeev wave function:
"
1 d 5 d
K2′ (K2′ + 4)
+ 2En φnK2′ ,ℓ′1 ,ℓ′2 (ρ2 )
ρ
−
ρ52 dρ2 2 dρ2
ρ22
2
K′ K
n
= δℓ1 ℓ′1 δℓ2 ℓ′2 ω 2 ρ22 γℓ1 ℓ22 2 ψK
(ρ2 ) .
2
3
#
(6.17)
Here we used the orthonormality of the hyperspherical harmonic basis (2.42) and the expressions
for the matrix elements of powers of sin ϕ2 calculated in Section 5.5. For ℓ1 6= ℓ′1 or ℓ2 6= ℓ′2 as well
K ′ K2
for K2′ for which γℓ1 ℓ22
= 0, the right hand site of Eq. (6.17) vanishes. In this case the functions
K ′ K2
φnK ′ ,ℓ′ ,ℓ′ are set to zero. The quantities γℓ1 ℓ22
2
1
2
K ′ K2
γℓ1 ℓ22
are given by
≡ < K2′ | cos2 (ϕ2 )|K2 >ℓ2 ,ℓ1 = δK2′ K2 − < K2′ | sin2 (ϕ2 )|K2 >ℓ2 ,ℓ1
(ℓ2 + 21 ,ℓ1 + 12 )
= δK2′ K2 − N2 (K2′ ; ℓ2 ℓ1 )N2 (K2 ; ℓ2 ℓ1 ) Is′ s
(2) ,
(6.18)
where s = (K2 − ℓ1 − ℓ2 )/2, s′ = (K2′ − ℓ1 − ℓ2 )/2. The normalization coefficient, N2 (K2 ; ℓ2 ℓ1 ) is
(ℓ2 + 12 ,ℓ1 + 21 )
given in Eq. (2.29), and the matrix element Is′ s
(2) is given in Eq. (5.41). According to the
K ′ K2
discussion at the end of Section 5.5 we conclude that γℓ1 ℓ22
= 0 unless K2′ = K2 or K2′ = K2 ± 2.
With the help of a new function unK ′ ,ℓ1 ,ℓ2 (ρ2 ) = ρ22 φnK ′ ,ℓ1 ,ℓ2 (ρ2 ), Eq. (6.17) could be written as
2
"
2
d
1 d
+
−
2
dρ2 ρ2 dρ2
(K2′
2
2
#
2
+ 2)
K′ K
n
+ 2En unK2′ ,ℓ1 ,ℓ2 (ρ2 ) = ω 2 ρ42 γℓ1 ℓ22 2 ψK
(ρ2 ) .
2
2
ρ2
3
(6.19)
In order for φnK ′ ,ℓ1 ,ℓ2 (ρ2 ) to be finite at small and very large ρ2 one has to chose a solution of this
2
equation vanishing at least as ρ22 at zero and increasing slower as ρ22 at infinity.
Let us consider a Green’s function equation
"
1 d
(K2′ + 2)2
d2
+
+ 2En GK2′ (ρ2 , ρ′2 ) = δ(ρ2 − ρ′2 ) ,
−
2
2
dρ2 ρ2 dρ2
ρ2
#
(6.20)
where GK2′ (ρ2 , ρ′2 ) satisfies the above-mentioned boundary conditions. A substitution of x =
√
√
2En ρ2 , gK2 = 2En GK2 transforms this equation to
"
d2
1 d
(K2′ + 2)2
+
−
+ 1 gK2′ (x, x′ ) = δ(x − x′ ) ,
dx2 x dx
x2
#
99
(6.21)
whose general solution with a proper behavior at small and large x is expressed through the Bessel
and Neumann functions [Gra80]
α
gK
(x, x′ ) = αJK2 +2 (x)
2
πx′
+
[JK2 +2 (x)NK2 +2 (x′ )θ(x′ − x) + JK2 +2 (x′ )NK2 +2 (x)θ(x − x′ )],
2
(6.22)
where α is an arbitrary constant. The solution of Eq. (6.19) is therefore given by
φnK2′ ,ℓ1 ,ℓ2 (ρ2 )
2 ω 2 K2′ K2 Z ∞ α
4
n
γ
=
GK2′ (ρ2 , ρ′2 )ψK
(ρ′2 )ρ′2 dρ′2 ,
2
2 ℓ1 ℓ2
3 ρ2
0
(6.23)
where
q
q
q
GαK2 ( 2En ρ2 , 2En ρ′2 ) = αJK2 +2 ( 2En ρ2 )
q
q
q
q
πρ′2
+
[JK2 +2 ( 2En ρ2 ) NK2 +2 ( 2En ρ′2 ) θ( 2En ρ′2 − 2En ρ2 )
2
q
q
q
q
JK2 +2 ( 2En ρ′2 ) NK2 +2 ( 2En ρ2 ) θ( 2En ρ2 − 2En ρ′2 )] .
(6.24)
A similar form of the Green function was obtained for negative energy solutions in work [Bad66].
n
In view of Eq. (6.9), the function ρ22 ψK
(ρ2 ) satisfies the following equation
2
"
d2
1 d
(K2 + 2)2
n
n
+
+ 2E ρ22 ψK
(ρ2 ) = ω 2 ρ42 ψK
(ρ2 ) ,
−
2
2
2
2
dρ2 ρ2 dρ2
ρ2
#
(6.25)
which in the integral form could be written with the help of the Green function (6.24) with α = 0:
n
ψK
(ρ2 )
2
ω2 Z ∞ 0
4
n
GK2 (ρ2 , ρ′2 )ψK
(ρ′2 )ρ′2 dρ′2 ,
= 2
2
ρ2 0
(6.26)
We note that the term proportional to α in the Green’s function entering Eq. (6.26) has to be
excluded in order for impose proper asymptotic behavior at large ρ2 . Since in view of (2.27),
Y20000 = YK2 =2,L2 =0,M2 =0,ℓ1 =0,ℓ2 =0 (Ω(2) , ϕ2 ) =
2
1
(1,1)
= N2 (2; 00) P1 2 2 (cos(2ϕ2 ))
= √ (2 cos ϕ2 2 − 1) ,
4π
π3
(6.27)
Eq.(6.23) could be written in the following form
1 n
ψ (ρ2 )δK2′ K2
3 K2√
Z ∞
2 ω2 π3
4
n
′
(ρ′2 )ρ′2 dρ′2 ,
GαK2′ (ρ2 , ρ′2 )ψK
+
<
K
|Y
|K
>
2
ℓ1 ℓ2
2 20000
2
2
3 ρ2 4
0
φnK2′ ,ℓ1 ,ℓ2 (ρ2 ) =
100
(6.28)
where
1
4
K′ K
γℓ1 ℓ22 2 − δK2′ K2
< K2′ |Y20000 |K2 >ℓ1 ℓ2 ≡ √
(6.29)
2
π3
Substitution of Eq.(6.28) into Eq. (6.2) finally produces the final expression for the Faddeev wave
·
¸
function corresponding to the nth excited state of the three identical particles interacting via
harmonic oscillator two-body potentials:
1 n
ψ (ρ2 )YK2 L2 M2 ℓ1 ,ℓ2 (Ω(2) , ϕ2 )
3 K2
√
π3
2 ω2 X
′
(Ω
,
ϕ
)
Y
+
< K2′ |Y20000 |K2 >ℓ1 ℓ2
2
(2)
K2 L2 M2 ℓ1 ,ℓ2
3 ρ22 K ′
4
Φ(ρ2 , Ω(2) , ϕ2 ) =
2
×
Z
0
∞
4
n
(ρ′2 )ρ′2 dρ′2 .
GαK2′ (ρ2 , ρ′2 )ψK
2
(6.30)
n
Here ψK
(ρ2 ) and GαK ′ (ρ2 , ρ′2 ) are given by Eqs. (6.16) and (6.24), respectively, and the sum is on
2
2
all K2′ for which the matrix elements < K2′ |Y20000 |K2 >ℓ1 ℓ2 are not vanishing.
For the scalar ground state (L2 = 0, n = 0) with the energy eigenvalue E = 3ω Eqs. (6.13) and
n
(6.14) demand K2 = 0 and ℓ1 = ℓ2 = 0. The radial wave function, ψK
(ρ2 ), in this case reduces to
2
exp (−ωρ22 /2). Since Y00000 =
√1 ,
π3
because of the orthonormality of the hyperspherical functions,
one has
1
(6.31)
< K2′ |Y20000 |0 >00 = √ δK2′ ,2 .
π3
From Eq.(6.30) follows, that up to a constant factor √1π3 the Faddeev component of the ground
state wave function, in view of Eqs.(6.24) and (6.27), has a form
√
ω2
cos (2ϕ2 ) [ βJ4 ( 6ωρ2 )
2
2ρ
Z ∞2 √
√
2
5
+ π { J4 ( 6ωρ2 )
N4 ( 6ωρ′2 ) exp (−ωρ′2 /2)ρ′2 dρ′2
Z ∞0 √
√
2
5
+ N4 ( 6ωρ2 )
J4 ( 6ωρ′2 ) exp (−ωρ′2 /2)ρ′2 dρ′2 }] ,
Φ(ρ2 , Ω(2) , ϕ2 ) = exp (−ωρ22 /2) +
(6.32)
0
where β is an arbitrary constant. This coincides with the scalar ground state Faddeev solution
given in Ref. [Fri80]. One can see, that the general oscillator Faddeev solution (6.30) corresponding
to arbitrary total orbital momenta and excitations has a complicated angular dependence given
by a sum of many hyperspherical harmonics with different K2′ even though the Schroedinger wave
function has a simpler one described by a single hyperspherical harmonic function. It contains
a component of arbitrary strength β, is regular at small distances and oscillates as ρ2 increases
−5/2
with the amplitude proportional to ρ2
, the properties which were already demonstrated by
101
Friar, Gibson and Payne [Fri80] for the ground state case. One has to point out, however, that
the presence of the arbitrary strength component in the Green function (6.24) and its oscillatory
behavior in ρ2 , which leads to a spurious component and oscillations in the Faddeev wave function,
are consequences of the positiveness of the energy eigenvalues in case of the harmonic-oscillator
forces [Fri80]. These features disappear for more realistic potentials whose energy eigenvalues are
√
negative. Indeed, in case of negative En , x = 2En ρ2 has imaginary values. As a result, the
√
√
solution of Eq. (6.20) instead of the Bessel and Neumann functions Jν ( 2En ρ2 ) and Nν ( 2En ρ2 ),
q
q
has to be expressed through the modified Bessel functions Iν ( 2|En |ρ2 ) and Kν ( 2|En |ρ2 ) which
q
√
display no oscillatory behavior. Also, since IK2 +2 ( 2|En |ρ2 ) which replaces JK2 +2 ( 2En ρ2 ) in the
first term of Eq. (6.24) diverges exponentially at large ρ2 , this term does not satisfy anymore the
boundary condition at the infinity and has to be excluded.
Summing up, in this subsection I have presented an analytically solutions of the Faddeev equation for any excitation and for an arbitrary value of the total angular momentum in case of three
identical spinless particles in three dimensional space interacting via harmonic oscillator two-body
potentials. These general solutions have a complicated angular dependence, which, however, could
be conveniently expressed with the help of the hyperspherical harmonic functions. The radial
properties of the solutions are similar to the radial properties of the ground state.
6.1.2
Non-Identical Oscillators
In order to achieve an intuitive feeling for the structure and properties of the Faddeev solutions,
a few analytically solvable examples were found, mostly in one dimension [Bar92, Dod70, Fri80,
Him74, Him77, Maj78], all of them dealing with identical particles for which the three coupled
Faddeev equations are reduced to a single equation. Attempts to find analytically solvable examples for a case of non-identical particles, where one’s intuition about the Faddeev components and
amplitudes is the least developed, have turned out to be unsuccessful. This is not astonishing since
analytic solutions of the three body Schroedinger equation for non-identical particles in most cases
are also not available. For example, the Schroedinger solution is unknown for a case of three nonidentical particles in one dimension interacting via delta potentials of different strengths [Kia76],
although the Schroedinger wave function and the S-matrix for identical particles interacting via
102
delta potentials are well known even for the N-particle case. The purpose of the present subsection is to derive the Faddeev wave function for the case of non-identical particles interacting via
harmonic-oscillator two body forces with non equal coupling constants. In the previous subsection
it was shown that, since in case of three identical oscillators the Hamiltonian commutes with the
generalized angular momentum operator K̂2 , one can express the correspondent Schroedinger solution Ψ in a very elegant and natural way through just one hyperspherical harmonic function. The
Faddeev components Φi = Φ are then easily calculated in the hyperspherical harmonic formalism
using the integral form of the Faddeev equations
Φi = GVi Ψ , i = 1, 2, 3 ;
(6.33)
where Vi are pair potentials in the ”odd man out” notation. For non-identical particles, the Hamiltonian no longer commutes with the generalized angular momentum operator and the Schroedinger
wave function acquires a more complicated hyperspherical structure which could be found by expanding it into the hyperspherical harmonic series. The Faddeev amplitudes are then eventually
derived following the same method as in the previous subsection. Unlike Subsection 6.1.1 where I
gave a detailed derivation of the Faddeev amplitudes for identical particles, in this subsection I will
briefly review the derivation and focus on the results. The full derivation is given in Ref. [Bar94].
In the center of mass (CM) frame the Hamiltonian for three non-identical particles with masses
mi interacting via pair oscillator potentials with different nonnegative coupling constants gij =
gji ≥ 0 is given by
3
X
1 2
1
H=
pi −
M
i=1 2mi
Ã
3
X
pi
i=1
!2
+
3
X
i<j=1
gij (ri − rj )2 ,
(6.34)
where M is the total mass. The mathematical difficulty we face when attempting to solve the
Schroedinger equation, is to diagonalize simultaneously both the potential and the kinetic energy
operators in terms of a suitable set of translation-invariant relative coordinates. Following Hall and
Schwesinger [Hal79] we have shown in [Bar94] that the translation invariant Schroedinger equation
can be written as
1
1
[− ∆ξ − ∆ξ + ω12 ξ12 + ω22 ξ22 ]Ψ = EΨ ,
1
2
2 2
(6.35)
where ξ 1 , ξ 2 are Jacobi coordinates, i.e. translation-invariant relative coordinates. The oscillator
√
frequencies ω1 , ω2 are given in [Bar94] by ωi = 2di , where di are the eigenvalues of the “coupling”
103
matrix A,

A=

−g12
g31 + g12
−g12
g23 + g12


 g2 + g 3

=
−g3
−g3
g1 + g3


 .
(6.36)
In this form, the Schroedinger equation could be separated into two uncoupled harmonic oscillator
equations
1
[− ∆ξ + ωi2 ξi2 ]Ψi = EΨi ,
2 i
(6.37)
where Ψ = Ψ1 Ψ2 . The solutions of equation (6.37) are given by [Flu71]
Ψi = Ni ξiℓi exp (−
3
ωi ξi2
)F (−nri , ℓi + , ωi ξi2 )Yℓi , mi (Ωi ) ,
2
2
(6.38)
where Yℓi , mi (Ωi ) are the spherical harmonics, Ωi are the angels associated with the vector ξ i , nri
are the radial excitation quantum numbers, Ni are normalization factors, and F (a, b, z) is the
confluent hypergeometric function. In order to ensure the proper asymptotic behavior at infinity,
one demands F to be a polynomial which restricts nri to integer numbers. The energy associated
with Ψi is given by Ei = ωi (ni + 32 ), where ni = 2nri + ℓi . The Schroedinger wave function
corresponding to orbital momentum L2 and projection M2 is obtain with the help of the ClebschGordan coefficients < L2 M2 |ℓ1 m1 ℓ2 m2 >,
ΨLn12nM22ℓ1 ℓ2
ω1 ξ12 + ω2 ξ22
)
=
exp (−
2
3
3
×F (−nr1 , ℓ1 + , ω1 ξ12 )F (−nr2 , ℓ2 + , ω2 ξ22 )
2
2
X
×
< L2 M2 |ℓ1 m1 ℓ2 m2 > Yℓ1 , m1 (Ω1 )Yℓ2 , m2 (Ω2 ) .
N1 N2 ξ1ℓ1 ξ2ℓ2
(6.39)
m1 m2
In order to solve the Faddeev equation, we will transform the Jacobi coordinates ξ 1 , ξ 2 into the
hyperspherical coordinates, through the parameterization
ξ 1 = ρ2 n1 cos ϕ2 ,
(6.40)
ξ 2 = ρ2 n2 sin ϕ2 .
(6.41)
In this parameterization the rest frame solution Ψ of the Schroedinger equation
1
1
HΨ = ∆(2) + ρ22 [ω12 cos ϕ2 2 + ω22 sin ϕ2 2 ] Ψ = EΨ ,
2
2
·
¸
104
(6.42)
corresponding to the total angular momentum L2 and its projection M2 , is represented by the
expansion
ΨLn12nM22ℓ1 ℓ2 =
X
ψnK12n2 ℓ1 ℓ2 (ρ2 )YK2 L2 M2 ℓ1 ,ℓ2 (Ω(2) , ϕ2 ) .
(6.43)
K2
Here ∆(2) is the six dimensional Laplace operator defined in Eq. (2.14), ρ2 is the hyperspherical
radius and Ω(2) , ϕ2 are the hyperspherical angels, where Ω(2) stands for angles (Ω1 , Ω2 ) defined
in Section 2.1. YK2 L2 M2 ℓ1 ,ℓ2 (Ω(2) , ϕ2 ) are the hyperspherical harmonic functions presented in 2.3.
In order to obtain ψnK12n2 ℓ1 ℓ2 (ρ2 ) in Eq. (6.43), one has to multiply ΨLn12nM22ℓ1 ℓ2 , Eq. (6.39) by
[YK2 L2 M2 ℓ1 ,ℓ2 (Ω(2) , ϕ2 )]∗ and integrate over the six-dimensional sphere:
ψnK12n2 ℓ1 ℓ2 (ρ2 )
=
Z
1
∗
sin 2ϕ2 2 dϕ2 dΩ1 dΩ2 YK
(Ω(2) , ϕ2 )ΨLn12nM22ℓ1 ℓ2 (ρ2 , Ω(2) , ϕ2 ) .
2 L2 M2 ℓ1 ,ℓ2
4
(6.44)
Rewriting the Schroedinger wave function (6.39) in hyperspherical coordinates (6.40) and performing the integral, one obtains that
ψnK12n2 ℓ1 ℓ2 (ρ2 ) = N1 N2 N2 (K2 ; ℓ2 ℓ1 )ρℓ21 +ℓ2 exp (−
nr1 nr2
×



X X  nr 1   nr 2 



i=0 j=0
i
j
s
X
r=0

1
(ωs + 2ωa ) 2
ρ2 )
2
2Γ(q + 3)


 s + ℓ2 + 1/2   s + ℓ1 + 1/2 
s−r
 (−1)


r
s−r
3
3
×Γ(nr1 − i + j + ℓ1 + r + )Γ(nr2 + i − j + ℓ2 + s − r + )
2
2
3
3
2
2
×F (−nr1 + i, ℓ1 + , ω1 ρ2 )F (−nr2 + j, ℓ2 + , ω2 ρ2 )
2
2
3
2
×F (nr2 + i − j + ℓ2 + s − r + , q + 3, ωa ρ2 ) .
2
Here, s = (K2 − ℓ1 − ℓ2 )/2 and q = nr1 + nr2 + ℓ1 + ℓ2 + s, ωs =
(ω1 +ω2 )
,
2
and ωa =
(6.45)
(ω1 −ω2 )
.
2
So far, I have considered the Schroedinger equation and its solution. The Faddeev equation
corresponding to the Hamiltonian (6.34) is given by

3
X

1 2
1
pi −
M
i=1 2mi
Ã
3
X
pi
i=1
!2

− E  Φj = −gj r2ik Ψ .
(6.46)
2 M2
Here, gj = gik where (ijk) = (123) and Ψ ≡ ΨLn12nM22ℓ1 ℓ2 (ρ2 , Ω(2) , ϕ2 ), Φj ≡ ΦjL
n1 n2 ℓ1 ℓ2 (ρ2 , Ω(2) , ϕ2 ) are
the Schroedinger wave function (6.39) and the corresponding Faddeev components, respectively.
In terms of the hyperspherical coordinates the Faddeev equation may be written as
·
1
∆(2) + E Φj = −gj r2ik Ψ .
2
¸
105
(6.47)
Expanding the Faddeev constituents into the hyperspherical harmonic series
2 M2
ΦjL
n1 n2 ℓ′ ℓ′ =
1 2
X
φnjK1 n22Lℓ21 ℓ2 ;ℓ′ ℓ′ (ρ2 )YK2 L2 M2 ℓ1 ,ℓ2 (Ω(2) , ϕ2 ) ,
(6.48)
1 2
K2 ℓ1 ℓ2
and substituting Eqs. (6.43,6.45) and (6.48) into Eq. (6.47), one obtains the following equation
for the radial parts φnjK1 n22Lℓ21 ℓ2 ;ℓ′ ℓ′ of the Faddeev wave functions:
1 2
"
#
X jL2 K2 K ′ K
1 d 5 d
K2 (K2 + 4)
+ 2E φnjK1 n22Lℓ21 ℓ2 ;ℓ′ ℓ′ (ρ2 ) = 2gj ρ22
ρ2
−
γℓ1 ℓ2 ;ℓ′ ℓ′ 2 ψn12n2 ℓ′ ℓ′ (ρ2 ) .
5
2
1 2
1 2
1 2
ρ2 dρ2 dρ2
ρ2
K′
(6.49)
2
Here we used the notation
jL K K ′
γℓ1 ℓ22 ;ℓ′2ℓ′ 2 ≡< K2 L2 M2 ℓ1 ℓ2 |
1 2
r2ik ′
|K L2 M2 ℓ′1 ℓ′2 > ,
ρ22 2
(6.50)
jL K K ′
for the matrix elements of the two body potential. The γℓ1 ℓ22 ;ℓ′2ℓ′ 2 ’s are explicitly given in Eqs.
1 2
(4.13) and (4.14) of Ref. [Bar94].
Eq. (6.49) is practically identical to Eq. (6.17) in the previous subsection. Solving Eq. (6.49)
exactly in the same way as in Subsection 6.1.1,one easily finds the Faddeev components for the
case of three non identical harmonic oscillators:
φnjK1 n22Lℓ21 ℓ2 ;ℓ′ ℓ′ (ρ2 )
1 2
2gj X jL2 K2 K2′ Z ∞ α
K′
4
= 2
γℓ1 ℓ2 ;ℓ′ ℓ′
GK2 (ρ2 , ρ′2 )ψn12n2 ℓ′ ℓ′ (ρ′2 )ρ′2 dρ′2 ,
1
2
1
2
ρ2 K ′
0
(6.51)
2
where the Green’s function GαK2 (ρ2 , ρ′2 ) is given in Eq. (6.24). Substituting of (6.51) into (6.48) produces the final expression for the Faddeev wave function of three non-identical particles interacting
via different two-body harmonic forces:
2 M2
ΦjL
n1 n2 ℓ1 ℓ2 (ρ2 , Ω(2) , ϕ2 ) =
2gj
ρ22
X
jL K ′ K
YK2′ L2 M2 ℓ′1 ,ℓ′2 (Ω(2) , ϕ2 )γℓ′ ℓ2′ ;ℓ12ℓ2 2
1 2
K2 K2′ ℓ′1 ℓ′2
Z ∞
4
×
GαK2′ (ρ2 , ρ′2 )ψnK12n2 ℓ1 ℓ2 (ρ′2 )ρ′2 dρ′2
0
,
(6.52)
Comparing the Faddeev wave function (6.52) for three non-identical particles to the Faddeev wave
function of three identical particles, given in Eq. (6.30), we can see that the general structure
of the Faddeev components for both cases is very similar. The only difference between two wave
functions is that the Faddeev solution (6.52) has a much more complicated angular dependence
given by an infinite sum of hyperspherical harmonics with different K2 which reflects the more
complicated structure of the Schroedinger wave function in the case of non-identical oscillators.
106
The behavior of the Faddeev constituents (6.52) is therefore similar to that for identical oscillators
[Bar94, Fri80]. They contain a component of arbitrary strength ,are regular at small distances and
−5/2
oscillate as ρ2 increases with an amplitude proportional to ρ2
. One has to point out, as before
[Bar94, Fri80], that the presence of an arbitrary strength component in the Green’s function (6.24)
and its oscillatory behavior in ρ2 , which leads to a spurious component and oscillations in the
Faddeev wave function, are consequences of the positiveness of the energy eigenvalues for harmonic
forces. These features disappear for more realistic potentials whose energy eigenvalues are negative.
Summing up, in this subsection I have presented analytic solutions of the Faddeev equation for any
excitation and for an arbitrary value of the total angular momentum for three non-identical spinless
particles in three dimensional space interacting via harmonic two-body forces. The solutions have
a very complicated angular dependence which, however, could be conveniently expressed with the
help of the hyperspherical harmonic functions. This is the first time that the analytic solutions of
three coupled Faddeev equations were derived.
6.2
The N -body Schroedinger equation
Consider now a system of N particles interacting via two-body spin-isospin independent potential
Vij (rij ). The Schroedinger equation describing the dynamics of this system in the center of mass
frame is given by


N
X
1
− ∆(N −1) +
Vij (rij ) Ψ(η 1 , . . . , η N −1 ) = EΨ(η 1 , . . . , η N −1 ) ,
2
i<j=1
where
∆(N −1) =
N
−1
X
i=1
∆η i =
(6.53)
∂2
1
3N − 4 ∂
− 2 K̂N2 −1 ,
+
2
∂ρN −1
ρN −1 ∂ρN −1 ρN −1
(6.54)
is the N -body Laplace operator defined in Eqs. (2.18), (2.20), and (2.21), and K̂N2 −1 is the N particle hyperspherical angular momentum operator, Eq. (2.22). Expanding the Schroedinger wave
function Ψ into the hyperspherical-harmonic series
Ψ=
KX
max
ΨKN −1 λN −1 αN −1 βN (ρN −1 )YKN −1 LN −1 MN −1 λN −1 αN −1 YN βN (Ω(N −1) , ϕ(N −1) )
KN −1 λN −1 αN −1 βN
,
107
(6.55)
and substituting into Eq. (6.53) we get the following equation for the hyper radial part of the wave
function
"
d2
KN −1 (KN −1 + 3N − 5)
3N − 4 d
−
+
+ 2E ΨKN −1 λN −1 αN −1 βN (ρN −1 )
2
dρN −1
ρN −1 dρN −1
ρ2N −1
#
KX
max
=
L
M
Y
−1 N −1 N
VKNN−1
λN −1 αN −1 βN ;K ′
′
′
′
N −1 λN −1 αN −1 βN
′
′
′
′
KN
−1 λN −1 αN −1 βN
(ρN −1 )ΨKN′ −1 λ′N −1 α′N −1 βN′ (ρN −1 ) ,
(6.56)
where
L
M
Y
−1 N −1 N
VKNN−1
λN −1 αN −1 βN ;K ′
′
′
′
N −1 λN −1 αN −1 βN
(ρN −1 )
′
′
=< KN −1 LN −1 MN −1 λN −1 αN −1 YN βN |V |KN′ −1 LN −1 MN −1 λ′N −1 αN
−1 YN βN > .
Here V is the sum of the two-body interactions V =
PN
i<j=1
(6.57)
Vij (rij ), and Kmax is the maximum
value of KN −1 used in the expansion of the wave function Ψ. The sum in Eq. (6.55) should be
understood as a sum over all the possible values of λN −1 , αN −1 , βN for a given value of KN −1 , LN −1 .
Substituting z = κρN −1 we may rewrite Eq. (6.56) in dimensionless form:
"
=
3N − 4 d
KN −1 (KN −1 + 3N − 5)
d2
+
−
+ 2ε ΨKN −1 λN −1 αN −1 βN (z)
2
dz
z
dz
z2
#
KX
max
L
M
Y
−1 N −1 N
VKNN−1
λN −1 αN −1 βN ;K ′
′
′
′
′
KN
−1 λN −1 αN −1 βN
′
′
′ (
N −1 λN −1 αN −1 βN
z
)ΨKN′ −1 λ′N −1 α′N −1 βN′ (z) ,
κ
(6.58)
where ε =
E
κ2
and κ ≈
q
|2E|. The hyper radial equation (6.58) could be solved by expanding
ΨKN −1 λN −1 αN −1 βN (z) into a complete set of basis functions,
ΨKN −1 λN −1 αN −1 βN (z) =
nX
max
n
CK
Ln (z) ,
N −1 λN −1 αN −1 βN
(6.59)
n=0
where Ln (z) ; n = 0, 1, 2, . . . are orthogonal functions that together with the hyperspherical harmonics form a complete set of basis functions for the N body configuration space. Denoting this
basis function by |n >= Ln (z), expanding the potential V into a power series
V (η 1 , . . . , η N −1 ) =
∞
X
p=0
Vp (Ω(N −1) , ϕ(N −1) )ρpN −1
=
∞
X
p=0
Vp (Ω(N −1) , ϕ(N −1) )
µ ¶p
z
κ
(6.60)
and substituting Eqs. (6.59), and (6.60) into Eq. (6.58) we finally get the eigenvalue equation:
nX
max
n′
"
3N − 4 d ′
KN −1 (KN −1 + 3N − 5) ′
d2
n′
|n > − < n|
|n > CK
< n| 2 +
N −1 λN −1 αN −1 βN
2
dz
z
dz
z
#
108
−
nX
max
max pX
n′
p
1
κp+2
< n|z p |n′ >
KX
max
p;L
M
Y
n′
′
′
′
′
′
′ CK ′
λ
α
β
N −1 λN −1 αN −1 βN
N −1 N −1 N −1 N
N −1 N −1 N
VKN −1
λN −1 αN −1 βN ;K ′
′
′
′
′
KN
−1 λN −1 αN −1 βN
n
,
= 2ε CK
N −1 λN −1 αN −1 βN
(6.61)
which we then solve using the Lanczos diagonalization method.
In the calculations presented in the following subsections we have used the generalized Laguerre
functions, Lνn (z), to expand the hyper radial part of the wave function. These functions obey the
orthogonality relation [Mag66],
Z
∞
e−z z ν Lνn (z)Lνn′ (z)dz =
0
(n + ν)!
δn,n′ .
n!
(6.62)
The radial matrix elements of the potential power expansion is given by
νp
Rn,n
= < n|z p |n′ >
′
=
s
Z ∞
n!n′ !
e−z z ν Lνn (z)Lνn′ (z)z p dz
(n + ν)!(n′ + ν)! 0
v
u
n
u n!(n + ν)! X
m (ν + p + m)!(−p − m)n′
t
(−)
,
=
′
′
(n − m)!(ν + m)!m!
n !(n + ν)! m=0
(6.63)
and the kinetic energy terms are
d2
3N − 4 d ′
|n >
(6.64)
+
2
z
dz
s dz
#
"
Z ∞
n!n′ !
3N − 4 d −z/2 ν
e
Ln′ (z)dz
e−z/2 z ν Lνn (z)
=
′
(n + ν)!(n + ν)! 0
z
dz
·
¸
q
3N − 4 + 2n′ ν−1
1
ν−2
′ ν−2
′
′
δn,n′ −
Rn,n′ + (3n − ν − 5) n Rn,n′ − n (n + ν)Rn,n′ −1 .
=
4
2
ν
Dn,n
= < n|
′
Substituting these results, Eq. (6.61) may be written as
Xh
n′
i
′
ν−2
n
ν
Dn,n
′ − KN −1 (KN −1 + 3N − 5)Rn,n′ CK
N −1 λN −1 αN −1 βN
−
nX
max
max pX
n′
p
KX
max
p;LN −1 MN −1 YN
νp
n′
VKN −1
R
′
′
′
′
′
′ CK ′
n,n
λN −1 αN −1 βN ;KN
λ′N −1 α′N −1 βN
p+2
N −1 λN −1 αN −1 βN
−1
κ
′
′
′
′
KN
−1 λN −1 αN −1 βN
1
n
= 2ε CK
.
N −1 λN −1 αN −1 βN
(6.65)
This is the eigenvalue problem we actually solve. Note that so far we didn’t define ν. The most
natural choice for ν is ν = 3N − 4 as it recovers the volume element dV = (ρN −1 )3N −4 dS3N −4 ,
ν
However, choosing ν = 3N − 5 will reduce Dn,n
′ to a much simpler form.
109
The actual implementation of this method requires three computer codes. The first code,
’GELFAND’, calculates the transformation coefficients from the “tree” hyperspherical functions
defined in Chapter 2 to the Gel’fand-Zetlin hyperspherical harmonic states, the rotational hyperspherical harmonics, using the algorithm described in Chapter 3. The output of this code are
the rotational states expressed in terms of the hyperspherical orthogonal parentage coefficients,
hsopcs. The second code, ’ONSN’, which is independent of the first code, deals with the reduction
problem ON −1 ↓ SN , implementing the recursive algorithm described in Chapter 4. The output of
this program are the symmetric states obtained by the transformation of the GZ basis functions
into states that belong to an irrep of the symmetric group, SN . This states are given in terms of
the ocfps which are the elements of this transformation matrix. The third code, ’NBODY’, uses
the output of these two codes to build the desired N -body states with well defined permutational
symmetry, YN , rotational symmetry, λN −1 , total angular momentum, LN −1 , and hyperspherical
angular momentum KN −1 according to the method described in Section 4.2. Then, using the results of Chapter 5, the program calculates matrix elements of the power expansion of the potential,
p;L
M
Y
N −1 N −1 N
VKN −1
λN −1 αN −1 βN ;K ′
′
′
′
N −1 λN −1 αN −1 βN
, between these basis functions. Once the program as finished
calculating the hyperspherical matrix elements and the matrix elements of the radial functions
are taken care of, the Hamiltonian is diagonalized using the QMR look-ahead Lanczos algorithm
[Fre93] or other diagonalization techniques. In order to check the method and the derivations I
present in the following sections few examples of actual calculations of few body systems.
6.2.1
4-Nucleons interacting via an s-wave potential
Since four-body calculations require an enormous numerical effort, most calculations have been
performed using very simple interaction models, including at best just a spin dependence. The
simplest models are purely central or even s-wave projected. These models are used as simple test
cases for the few-body bound state calculations. One of the most commonly used models is the
model of Malfliet and Tjon [Mal69] which includes two Yukawa-type interactions:
Vij =
4
X
i,j=1
Vr
e−µr rij
e−µa rij
− Va
.
rij
rij
(6.66)
In this subsection I will limit myself to the spin independent potential denoted by MT-V, using
the following parameters Vr = 1458.0470 [MeV fm], µr = 3.110 [ fm−1 ] for the repulsive part of the
110
Table 6.1: Binding energy of four-nucleon system with effective s-wave interaction MT-V. ’CHH’ the binding energy obtained by Rosati, Kievsky and Viviani, using the Correlation Hyperspherical
Harmonic method.
NHH
E [MeV]
Kmax = 0
1
7.178042
Kmax = 2
1
7.178042
Kmax = 4
3
9.805167
Kmax = 6
6
14.10561
Kmax = 8
11
20.67197
Kmax = 10
18
23.68968
Kmax = 12
32
26.68968
Kmax = 14
48
28.12495
Kmax = 16
75
29.07420
Kmax = 20
160
30.36575
CHH [Ros92]
31.355
potential and, Va = 578.0890 [MeV fm], µa = 1.550 [ fm−1 ] for the attractive part of the potential.
The nucleon mass is defined by h̄2 /m = 41.47 [MeV fm2 ]. The calculated binding energy (in MeV)
for the 4 nucleon system with the effective s-wave potential MT-V is presented in table 6.1. As
the spin-isospin degrees of freedom takes care for the anti-symmetric part of the wave function the
spatial part of the function must be symmetric, the number of symmetric hyperspherical functions
taken into account for each value of Kmax is presented in Table 6.1 under NHH . The numerical
calculations were done using power expansion for the potential, with pmax = 15. In the hyper
radial expansion we have used the generalized Laguerre polynomials with ν = 8, and used the
cutoff nmax = 25.
111
6.2.2
3H like Li
Applying the method to atomic systems I first considered the simplified case of three non-interacting
electrons moving in the field of the nucleus. Assuming that the nucleus is the forth particle, the
potential in this problem is:
Vne =
3
X
i=1
−
3
Ze2 X
Ze2
−
=
,
rni
r4i
i=1
(6.67)
where in this subsection I have used Z = 1. Due to the electron spin, the symmetry of the spatial
part of the ground state wave function of the 3-electron system must be of mixed symmetry, i.e.,
.
Γ3 =
(6.68)
In order to accelerate the convergence of the hyperspherical expansion we look for wave function
Ψ in the form [Haf89]:
Ψ = χψ = ef ψ(η 1 , . . . , η N −1 )
(6.69)
where f , the correlation function, is given by:
f = −Z
3
X
rni .
(6.70)
i=1
Substituting (6.69) into the Schroedinger equation we get the following equation for ψ:
·
where
W =
1
− ∆(N −1) + W ψ(η 1 , . . . , η N −1 ) = Eψ(η 1 , . . . , η N −1 ) ,
2
(6.71)
N
X
(6.72)
¸
i<j=1
Vij (rij ) −
−1
−1
−1
1 NX
1 NX
1 NX
∇2i f −
(∇i f )2 −
(∇i f ) · ∇i .
2 i=1
2 i=1
2 i=1
∂
. Although Eq. (6.71) is not an hermitian equation any more, it is solved exactly
Here ∇i = ∂ η
i
in the same way that we presented in the beginning of this section, i.e. by expanding ψ into
the symmetrized hyperspherical harmonics times the Laguerre polynomials. The main differences
between the calculation presented in this subsection and the preceding one are: here we have only
three identical particles in a four body system, therefore we must sum over all the S4 irreps that
contain Γ3 of Eq. (6.68). The second difference is the appearance of velocity dependent terms and
three body terms in the effective potential W , Eq. (6.72). The evaluation of these matrix elements
is somewhat more complicated then the calculation of the matrix elements presented in Chapter
5, as it goes more or less along the same lines I won’t give here their derivations.
112
Table 6.2: The ground state energy, calculated for three non-interacting electrons moving in
Coloumb potential with Z = 1, exact value = −1.125 [a.u.]. Note that due to the correlation
function we loss hermiticity and the eigenvalues are not variational any more.
E [a.u]
NHH
n=5
n = 10
n = 20
K=2
2
-1.068982 -1.070270 -1.070271
K=4
7
-1.109285 -1.110135 -1.110135
K=6
18
-1.122540 -1.122786 -1.122786
K=8
42
-1.124797 -1.125308
K = 10
84
-1.124476 -1.124934 -1.124934
K = 12
153
-1.124760 -1.125211 -1.125211
Using the well known spectrum of the hydrogen atom, one could easily see that for L = 0, and
mixed symmetry configuration the exact value of the ground state should be
1
1
E0 = − (1 + 1 + ) = −1.125 [a.u.]
2
4
and for the first excited state
1
1
E1 = − (1 + 1 + ) = −1.055555 [a.u.] .
2
9
In Tables 6.2, and 6.3 I present the calculated results for the three non-interacting electrons system.
These calculations where done using the generalized Laguerre polynomials taking ν = 7 and κ = 2.
Due to the non-hermiticity of Eq. (6.71), the eigenvalues of the effective Hamiltonian do not
coincide with its expectation values. As a result the eigenvalues do not obey the variational
principle, and we do not get a smooth convergence. Still one can see that we get an accuracy of
about 10−5 in the ground state energy and 10−4 in the first excited state energy.
113
Table 6.3: The first excited state energy, Calculated for three non-interacting electrons moving in
Coloumb potential with Z = 1, exact value = −1.055555 [a.u.]
6.2.3
E [a.u]
NHH
n=5
n = 10
n = 20
K=2
2
-0.914817 -0.854402 -0.853502
K=4
7
-0.965529
K=6
18
-1.039345 -1.034156
K = 10
84
-1.053408 -1.048834 -1.048805
K = 12
153
-1.051864 -1.053311 -1.053286
-0.926797
Ground State of the Lithium Atom
Using the method described in the previous subsection the full atomic problem was considered.
Asuming that the nucleus is the forth particle, the potential in this case is given by,
V = Vne + Vee =
3
X
i=1
−
3
3
3
X
X
X
Ze2
e2
e2
Ze2
−
+
=
+
,
rni
r4i
j>i=1 rij
i=1
j>i=1 rij
(6.73)
where for the Li atom Z = 3. Due to the electron spin, the symmetry of the spatial part of
the ground state wave function of the 3-electron system must be of mixed symmetry, like in the
previous case. In order to accelerate the convergence of the hyperspherical expansion we look for
wave function Ψ in the form [Haf89]:
Ψ = χψ = ef ψ(η 1 , . . . , η N −1 )
(6.74)
where in order to remove the electron-electron cusp f , the correlation function, is taken as
f = −αN e
3
X
rni + αee
i=1
3
X
rij .
(6.75)
j>i=1
The best variational calculation for the Lithium ground state was recently accomplished by
Yan and Drake [Yan95], who calculated this ground state with 10 significant figures to be
Egs = −7.478060323 [a.u.]. The results of our calculations are given in Tables 6.4, 6.2.3, and
6.6 for different values of the correlation factors αN e and αee . The eigenvalues presented in table
6.6, where checked to have hyper radial convergence one order of magnitude better then given
values. Comparing the results with the variational result one can see that with the full correlation,
i.e. nuclei-electron and electron-electron, the accuracy of our calculation is of the order of 10−3 .
114
Table 6.4: The effect of nucleus-electron correlation (no electron-electron correlation), on the calculated ground state energy of the Lithium atom. n = 10, ν = 7, κ = 6.
E [a.u]
αN e = −2.735 αN e = −2.8
αN e = −2.85 αN e = −2.9
αN e = −3.0
K=2
-6.5492
-6.6397
-6.7111
-6.7839
K=4
-6.9026
-7.0057
-7.0876
-7.1712
-7.3437
K=6
-7.2987
-7.3940
-7.4714
-7.5525
-7.7266
K=8
-7.2621
-7.3653
-7.4530
-7.5479
K = 10
-7.3775
-7.4681
-7.5527
K = 12
-7.3159
-7.4130
K = 14
-7.4609
K = 16
-7.3965
Table 6.5:
Lithium ground state energy, calculated without electron-electron correlation,
αN e = −2.8, αee = 0.0, ν = 7, κ = 6
E [a.u]
n = 10
n = 15
K=2
-6.63969
K=4
-7.00573
K=6
-7.39402
K=8
-7.36531
K = 10
-7.46814 -7.46815
K = 12
-7.41304 -7.41471
K = 14
-7.46090 -7.47492 -7.47513
K = 16
-7.39652 -7.43013 -7.43622 -7.43784
115
n = 17
n = 20
Table 6.6: Lithium ground state energy, calculated with electron correlation, αN e = −3.0, αee = 0.5,
ν = 7, κ = 6
E [a.u.]
K=2
-7.21801
K=4
-7.34062
K=6
-7.49726
K=8
-7.42431
K = 10
-7.45762
K = 12
-7.48112
K = 14
-7.46266
116
Chapter 7
Conclusions
117
In this thesis I introduced a general algorithm for the efficient construction of N -body basis
functions that belong to a given irrep of the orthogonal group O(N − 1) and are at the same time
characterized by a well-defined permutational symmetry SN . Since the hyperspherical functions
are among the most useful basis functions for describing N -body systems, I applied the algorithm
to these functions. However, it is straightforward to apply this algorithm to other basis states of
quantum systems like non-spurious harmonic oscillator states [Nov89]. Actually, two independent
algorithms are presented. In Chapter 3 an algorithm for constructing hyperspherical functions with
well defined O(N ) irrep is described, and in Chapter 4 the algorithm for constructing O(N − 1)
irrep states with well defined SN irreps is introduced. The main idea, to diagonalize the second
Casimir operator, is common to these two algorithms. However, they drastically differ in their
mathematical details. In Chapter 4 we also combine these two algorithms in order to obtain
hyperspherical states with permutational and orthogonal symmetry. The computational efficiency
of the procedure developed here has been demonstrated by means of computer codes implementing
it for the evaluation of the symmetrized hyperspherical functions with well defined O(N ) symmetry.
The complementary method for the calculation of the matrix elements of the two-body operators
was also presented. It was shown that for the symmetrized hyperspherical functions the calculation
of the matrix elements is relatively simple. It does not require the the Raynal-Revai and the Tcoefficients. One just have to apply an appropriate rotation on the relative coordinate. This
rotation is realized in terms of the appropriate representation matrices of the orthogonal group.
For systems consisting of identical particles it is sufficient to calculate the matrix elements for
only one two-body operator. For simplicity, we usually choose to calculate the matrix elements
of the potential between the last two particles. However, for system with different particles we
must calculate also the matrix elements of the potential between others pairs of particles. (For
example, to describe an atom composed of a nucleus and electrons we have to calculate the potential
between two electrons and the potential between the nucleus and one electron.) These calculations,
especially in the hyperspherical basis, are difficult to implement. The approach discussed in this
thesis greatly simplifies these tedious calculations.
In Section 6.1 I used the hyperspherical harmonic method to solve analytically the coupled Faddeev equations for three identical or non-identical particles in three-dimensional space interacting
via different harmonic oscillator two-body potentials. The solution is applicable to any excitation
118
and to any arbitrary value of the total angular momentum. These solutions are the only known
analytical solutions of the Faddeev equations, in the three dimensional space.
The solution of the Schroedinger equation for realistic few-body systems was considered in
Section 6.2. In order to check the method and the derivations, I wrote few computer codes implementing it. This codes where applied to study 4-body systems. We found that the convergence
rate of the calculations as a function of KN is quite slow where as the size of the basis set grows
very fast. This results relatively low accuracy of the calculations. In order to improve this results
or to study larger systems one must present either better correlation functions or some controlled
approximation which limit the size of the hyperspherical basis set. However, the development of
such methods is beyond the scope of this thesis and is a subject of future research.
119
Appendix A
The Orthogonal Group and the
Gel’fand-Zetlin States
120
The orthogonal group O(N ) is the set of all linear transformations in the N -dimensional
real Euclidean space which conserves the quadratic form i.e., the sum over all the squares of the N
coordinates. This group is actually the set of all N × N real orthogonal matrices with determinant
±1. For the special case where the determinant of these matrices is 1 we obtain the SO(N ) group.
The group O(N ) has N (N − 1)/2 infinitesimal generators. Let η 1 ,η 2 , . . . η N be a set of in-
dependent coordinates in the d-dimensional space ℜd . (In what follows I will deal only with the
3-dimensional space ℜ3 .) The natural infinitesimal generators are the skew-symmetric, Hermitian
operators X̂m,n
X̂m,n = i(η m ·∇n − η n ·∇m ) .
(A.1)
These operators obey the commutation relations
[X̂m,n , X̂k,l ] = i(δn,k X̂m,l + δm,l X̂n,k − δm,k X̂n,l − δn,l X̂m,k )
(A.2)
where m, n, k, and l run from 1 to N . The operator X̂m,n is the generator of the rotations in the
plane of the coordinates η n and η m . The generators (A.1) are also the generators of the SO(N )
group. Therefore the O(N ) irreps are closely related to the SO(N ) irreps as I will present explicitly
in the following. For convenience, we first describe the SO(N ) irreps and then it is straightforward
to obtain the O(N ) irreps.
The basis of the irreps of the SO(N ) group can be completely specified according to the canonical group-subgroup chain SO(N ) ⊃ SO(N − 1) ⊃ · · · ⊃ SO(3) ⊃ SO(2). In this case the GZ
approach [Gel50] uniquely specifies the basis vectors of the irreps of SO(N ), where N = 2k + 1
(odd number), by










=













λ2k−1 

λ
 2k+1 


 λ2k



ΛN
·
·
λ5
λ4
λ3
λ2

λ
λ2k+1,2 . . . λ2k+1,k−1 λ2k+1,k
 2k+1,1

 λ2k,1
λ2k,2
. . . λ2k,k−1
λ2k,k














=
























λ2k−1,1 λ2k−1,2 . . . λ2k−1,k−1
·
·
λ5,1
λ5,2
λ4,1
λ4,2
·
·
λ3,1
λ2,1
121
...
...


























(A.3)
and where N = 2k (even number) by

λ
 2k

 λ2k−1









=












ΛN
λ2k−2
·
·
λ5
λ4
λ3
λ2


λ
λ2k,2
. . . λ2k,k−1
λ2k,k
 2k,1

 λ2k−1,1 λ2k−1,2 . . . λ2k−1,k−1










 λ2k−2,1





 ·




= ·







 λ5,1





 λ4,1





 λ

 3,1


λ2k−2,2 . . . λ2k−2,k−1
·
·
...
...
λ5,2
λ4,2
λ2,1


























(A.4)
where all the λm,j ’s are simultaneously either integers or half-integers, The following restrictions
are given by the GZ rules [Gel50]
λ2j+1,1 ≥ λ2j,1 ≥ λ2j+1,2 ≥ λ2j,2 ≥ . . . ≥ λ2j,j−1 ≥ λ2j+1,j ≥ |λ2j,j |
(A.5)
λ2j,1 ≥ λ2j−1,1 ≥ λ2j,2 ≥ λ2j−1,2 ≥ . . . ≥ λ2j,j−1 ≥ λ2j−1,j−1 ≥ |λ2j,j |
(A.6)
and
where j = 1, 2, . . .
h
N
2
i
.
The ΛN symbols in (A.3) and (A.4) consist of N − 1 lines where the
h i
j
2
numbers in line j
(= 2, 3, . . . , N ), which we denote by λj , characterize the irrep of the group SO(j). Actually, the
irrep λj−1 is one of the possible irreps of SO(j − 1) which are contained in the irrep λj of SO(j).
Note that the notation of GZ [Gel50] was shifted down by one unit compared to ours, i.e., they
denoted by λj the irreps of the group SO(j + 1). We adopted the more straightforward notation
of Ref. [Pan67].
The GZ basis is related to the highest weights of the appropriate irreps. Starting from the
top line in (A.3) or (A.4), the number λN,1 , (N = 2k + 1 or N = 2k) is the maximum possible
eigenvalue of X̂1,2 in SO(N ). For this eigenvalue of X̂1,2 , the maximum eigenvalue of X̂3,4 is λN,2 ,
and so on, i.e., for the group SO(N ), λN,k is the maximum possible eigenvalue of X̂N −1,N for even
N and X̂N −2,N −1 for odd N , where the eigenvalues of X̂2j−1,2j are equal to λN,j for all 1 ≤ j ≤ k − 1
[Pan67]. These rules are valid for each line in (A.3) and (A.4).
122
Actually, there are some more restrictions on the possible values of the parameters λj,k in (A.3)
and (A.4). For polynomial representations [Ham62] all these values must be integers. In addition,
since we are dealing with three dimensional vectors there are at most three non-zero values in the
irrep λj for j ≥ 6 i.e., λj,1 , λj,2 , and λj,3 [Sur67]. (In general, for d-dimensional space there are at
most d non-zero values in any irrep of the orthogonal group.)
Gel’fand and Zetlin, who introduced the orthogonal group basis states (A.3) and (A.4), also
derived [Gel50] analytic expressions for the matrix elements of the infinitesimal generators (A.1)
between their basis vectors. These expressions were obtained by Louck [Lou60] without using the
group theory approach. Louck started from the generalized orbital angular momentum operators
in N dimensional polar coordinates and calculated analytically, by algebraic techniques, the matrix
elements of the generalized orbital angular momentum component, that are actually the operators
(A.1). Pang and Hecht [Pan67] introduced the normalized lowering and raising operators for the
orthogonal group in the canonical group-subgroup chain O(N ) ⊃ O(N − 1) ⊃ · · · ⊃ O(3) ⊃ O(2)
and used them to calculate analytically the matrix elements of the infinitesimal operators (A.1).
In our method, for construction of N -particle basis functions (in hyperspherical coordinates)
that belong to irreps of the orthogonal group, we need the expression for the matrix element of the
infinitesimal operator X̂N −1,N . Since we have already used the convenient notation of Ref. [Pan67]
for the GZ Basis, we will adopt their notation for the matrix elements of the operator X̂N −1,N as
well. This operator commutes with all the other infinitesimal operators X̂m,n if m, n < N − 1 and
it is a scalar operator with respect to SO(N − 2). Therefore, the matrix element of X̂N −1,N is
independent of the irreps of SO(n), where n < N − 2, and is diagonal in the irreps of SO(N − 2).
This matrix element is also diagonal in the irreps of SO(N ) and can only change one of the
λN −1,j , j = 1, 2, . . .
h
N −1
2
i
by ±1. For an even N it also has a non-vanishing diagonal matrix
element. Consequently, there are three different cases for the non-vanishing matrix elements of the
X̂N −1,N operator. Using the notation in (A.3) and (A.4) and denoting t2k,j = λ2k,j + k − j and
t2k−1,j = λ2k−1,j + k − j we obtain for an odd N (= 2k + 1) the expression ( Eqs. (5.46) in Ref.
[Pan67] )
< λ2k,j + 1|X̂2k,2k+1 |λ2k,j >
#1
k
2
−i Πk−1
α=1 (t2k−1,α − t2k,j − 1)(t2k−1,α + t2k,j )Πβ=1 (t2k+1,β − t2k,j − 1)(t2k+1,β + t2k,j )
=
(A.7)
2
Πkα6=j (t22k,α − t22k,j )(t22k,α − (t2k,j + 1)2 )
"
123
where for brevity only the index that suffers change is indicated. For even N (= 2k) the diagonal
matrix element is ( Eqs. (5.45) in Ref. [Pan67] )
k
Πk−1
α=1 t2k−2,α Πβ=1 t2k,β
< ΛN |X̂2k−1,2k |ΛN >= k−1
Πα=1 t2k−1,α (t2k−1,α − 1)
(A.8)
and the non-diagonal matrix element is ( Eqs. (5.44) in Ref. [Pan67] )
< λ2k−1,j + 1|X̂2k−1,2k |λ2k−1.j >
2
2
k
2
2
Πk−1
α=1 (t2k−2,α − t2k−1,j )Πβ=1 (t2k,β − t2k−1,j )
= −i 2
2
2
2
2
t2k−1,j (4t22k−1,j − 1)Πk−1
α6=j (t2k−1,α − t2k−1,j )[(t2k−1,α − 1) − t2k−1,j ]
"
# 21
(A.9)
Note that for k = 1 there are undefined products in Eqs. (A.7)-(A.9) ( for example Πk−1
α=1 ). These
products are equal to 1 in this case.
It is instructive and useful (see Chapter 3) to derive from Eqs. (A.7)-(A.9) the explicit expressions for the matrix elements of the operators X̂1,2 , X̂2,3 , X̂3,4 , and X̂4,5 . Starting from the
definitions given above we obtain that
t2,1 = λ2,1 : t3,1 = λ3,1 + 1 : t4,1 = λ4,1 + 1 : t4,2 = λ4,2 : t5,1 = λ5,1 + 2 : t5,2 = λ5,2 + 1 .
Substituting these relations in Eqs. (A.7)-(A.9) we obtain that the matrix element of the operator
X̂1,2 is
< Λ2 |X̂1,2 |Λ2 >= λ2,1 ,
(A.10)
and the matrix element of the operator X̂2,3 is
< λ2,1 + 1|X̂2,3 |λ2,1 >=
−i q
(λ3,1 − λ2,1 )(λ3,1 + λ2,1 + 1)
2
(A.11)
Note that for the SO(3) group the parameter λ3,1 is just the angular momentum ℓ and λ2,1 of SO(2)
is the z-projection m. So, Eqs.(A.10)-(A.11) are the well known angular momentum relations. For
the operator X̂3,4 there are the following diagonal and non-diagonal matrix elements,
< Λ4 |X̂3,4 |Λ4 >=
λ2,1 (λ4,1 + 1)λ4,2
(λ3,1 + 1)λ3,1
< λ3,1 + 1|X̂3,4 |λ3,1 >
(λ22,1 − (λ3,1 + 1)2 )((λ4,1 + 1)2 − (λ3,1 + 1)2 )(λ24,2 − (λ3,1 + 1)2 )
= −i
(λ3,1 + 1)2 (4(λ3,1 + 1)2 − 1)
"
124
(A.12)
#1
2
(A.13)
and for the operator X̂4,5 there are two non-diagonal matrix elements because it can change the
value of λ4,1 or λ4,2 by ±1. The matrix element between states with different λ4,1 values is
1
−i
[(λ3,1 − λ4,1 − 1)(λ3,1 + λ4,1 + 2)] 2
2
"
#1
(λ5,1 − λ4,1 )(λ5,1 + λ4,1 + 3)(λ5,2 − λ4,1 − 1)(λ5,2 + λ4,1 + 2) 2
×
(λ24,2 − (λ4,1 + 1)2 )(λ24,2 − (λ4,1 + 2)2 )
< λ4,1 + 1|X̂4,5 |λ4,1 >=
(A.14)
and between states of different λ4,2 is
1
−i
[(λ3,1 − λ4,2 )(λ3,1 + λ4,2 + 1)] 2
2
#1
"
(λ5,1 − λ4,2 + 1)(λ5,1 + λ4,2 + 2)(λ5,2 − λ4,2 )(λ5,2 + λ4,2 + 1) 2
×
.
((λ4,1 + 1)2 − λ24,2 )((λ4,1 + 1)2 − (λ4,2 + 1)2 )
< λ4,2 + 1|X̂4,5 |λ4,2 >=
(A.15)
The sum over all the squares of the infinitesimal generators of SO(N ) (Eq. (A.1)) is known as
the quadratic Casimir operator Ĉ2 (N ). For N > 3 this operator can be constructed recursively
2
as a sum over the X̂i,N
generators ( i = 1, 2, . . . , N − 1 ) and the quadratic Casimir operator of
SO(N − 1) i.e.,
Ĉ2 (N ) =
N
X
i<j=2
2
X̂i,j
= Ĉ2 (N − 1) +
N
−1
X
2
X̂i,N
(A.16)
i=1
Analytic expressions for the eigenvalues of these operators were calculated in Ref. [Pan67] yielding
the expression
C2 (N ) =
k
X
i=1
where k =
h
N
2
(λN,i + N − 2i)λN,i
(A.17)
i
.
In my thesis I present a method to construct basis functions which are eigenvectors of the
quadratic Casimir operator (A.16). The various irreps are identified by comparing the SO(N )
eigenvalues, given by Eq. (A.17) with the calculated ones. In most of the cases this comparison is
unique. For the other cases, ( for example C2 (N = 4) = 24 for the irrep λ4,1 = λ4,2 = 3 and for the
irrep λ4,1 = 4, λ4,2 = 0), the irreps are identified by applying the algorithm described in Chapter
3 in which all the SO(N − 1) irreps which yield these degenerate SO(N ) irreps are considered.
The basis of the O(N ) irreps are specified by using the GZ notations. However, in the case
N = 2k, the O(N ) irrep (λN,1 , λN,2 , . . . λN,k ) splits into the two SO(N ) irreps (λN,1 , λN,2 , . . . λN,k )
and (λN,1 , λN,2 , . . .−λN,k ) if λN,k 6= 0, while for λN,k = 0 or odd N the irreps of O(N ) are irreducible
125
under SO(N ). It is clear that the Casimir operator cannot separate the two SO(N ) irreps that
originate from the same O(N ) irrep. In this case we can split the O(N ) irrep by calculating the
eigenvalues of the operator X̂N −1,N , in the highest weight states.
126
Appendix B
The mass weighted Jacobi coordinates
for the N-body problem
127
As it is well known, in order to separate the motion of the center of mass coordinate, η 0 = Rcm ,
in the N-body problem one has to work with the Jacobi coordinates [Jac42] η 1 , η 2 , . . . , η N −1 . The
general expressions for the mass weighted Jacobi coordinates were given by Hirshfelder and Dahler
[Hir56]
η k−1 =
s
Where Mk =
´
1
Mk−1 mk ³
(m1 r1 + m2 r2 + · · · + mk−1 rk−1 ) ;
rk −
Mk
Mk−1
Pk
i=1
k = 2, 3, · · · , N
(B.1)
mi . This transformation can be written as an orthogonal transformation Sij ,
η k−1 =
N
X
√
Ski mi ri
N
1 X
ri = √
Ski η k−1
mi k=1
,
i=1
Here the matrix Ski has the following form:
Sij
s

mj







s MN



mi mj


 −
MM
=
s i i−1



Mi−1





Mi





0
(B.2)
for i = 1
for i > 1
(B.3)
for i = j
for j > i
One can easily check that the matrix Sij is orthogonal
N
X
Sij Skj = δik
,
detS = 1.
(B.4)
j=1
In view of (B.2), the expressions for the gradients are
N
X
√
∂
∂
Ski mi
=
∂ri k=1
∂η k−1
The corresponding momentum operators
∂
pi = −i
∂ri
are therefore transformed as
N
X
√
Ski mi qk−1
pi =
N
X
1 ∂
∂
Ski √
=
∂η k−1 k=1
mi ∂ri
,
∂
∂η k−1
(B.6)
N
X
(B.7)
qk−1 = −i
,
qk−1 =
,
k=1
(B.5)
1
Ski √ pi
mi
k=1
Because of orthogonality of the matrix S the bilinear combinations are given by:
Iαβ =
Tαβ =
N
X
N
X
mi (ri )α (ri )β =
N
X
(η i−1 )α (η i−1 )β
i=1
N
X
i=1
1
1
(pi )α (pi )β =
(qi−1 )α (qi−1 )β
2m
2
i
i=1
i=1
Lαβ =
N
X
mi (ri )α (pi )β =
N
X
i=1
i=1
128
(η i−1 )α (qi−1 )β
(B.8)
for α, β = 1, 2, 3, which defines the inertia tensor Iαβ , the kinetic energy operator Tαβ and the
angular momentum operator Lαβ , respectively, in the new variables. One can check that in view
of Eqs. (B.2) and (B.8) the moment of inertia around the center of motion could be expressed
through the relative particle distances
I=
N
−1
X
(η i−1 )2 =
i=1
N
1 X
2
mi mj rij
2MN i,j=1
(B.9)
The moment of inertia is thus expressed through sum of squares of mass weighted inter-particle
√
distances mi mj rij and therefore is usually denoted as ρ2 . The variable ρ is called the hyperspherical radius. Finally using Eqs. (B.2), and (B.4) one obtains the expressions for volume elements in
the coordinate and momentum spaces
N
Y
i=1
where µ =
QN √
i=1
dri =
N
1Y
dη
µ i=1 i−1
,
N
Y
i=1
mi .
129
dpi = µ
N
Y
i=1
dqi−1
(B.10)
Bibliography
[Ave89] J. Avery, Hyperspherical Harmonics, Kluwer, Dordrecht (1989).
[Bad66] A.M. Badalyan and Yu.A. Simonov, Sov. J. Nucl. Phys.,3, 755 (1966).
[Bar90] N. Barnea and V. B. Mandelzweig, Phys. Rev. A41, 5209 (1990).
[Bar91] N. Barnea and V. B. Mandelzweig, Phys. Rev. A44, 7053 (1991).
[Bar92] N. Barnea and V. B. Mandelzweig, Phys. Rev. C45, 1458 (1992).
[Bar94] N. Barnea and V. B. Mandelzweig, Phys. Rev. C49, 2910 (1994).
[Bar96] N. Barnea and A. Novoselsky, “Hyperspherical functions with well defined orthogonal
and permutational symmetries”, Proceedings of the 21’th international colloquium on group
theoretical methods in physics, July 15-19, Goslar, Germany (1996).
[Bar97] N. Barnea, “A Recursive method for the construction of O(n) Gel’fand-Zetlin states”, (in
preparation).
[Bar97a] N. Barnea and A. Novoselsky, Ann. Phys. (N.Y.) 256, 192 (1997).
[Bar97b] N. Barnea and A. Novoselsky, “Hyperspherical wave functions with orthogonal and permutational symmetry”, Phys. Rev. A, (submitted for publication).
[Cas84] O. Castanos, E. Chacon and M. Moshinsky, J. Math. Phys. 25, 2815 (1984).
[Cha84] E. Chacon, P. Hess and M. Moshinsky, J. Math. Phys. 25, 1565 (1984).
[Che89] J.-Q. Chen, Group Representation Theory for Physicists ,World Scientific, Singapore,
(1989).
130
[Che89a] J.-Q. Chen, A. Novoselsky, M. Vallieres and R. Gilmore, Phys. Rev. C39, 1088 (1989).
[Cla80] C. W. Clark and C. H. Greene, Phys. Rev. A21, 1786 (1980).
[Cor90] T. Cornelius, W. Gloeckle, J. Heidenbauer, Y. Koike, W. Plessas and H. Witala, Phys.
Rev. C41, 2538 (1990).
[Del59] L.M. Delves, Nucl. Phys. 9, 391 (1959); 20, 275 (1960).
[Dod70] L.R. Dodd, J. Math. Phys. 11, 207 (1970).
[Edm60] A.R. Edmons, Angular Momentum in Quantum Mechanics, Princeton University Press,
Princeton, NJ, (1960).
[Efr72] V. D. Efros, Yad. Fiz. 15, 226 (1972). [Sov. J. Nucl. Phys. 15, 128 (1972)].
[Efr95] V. D. Efros Few-Body Syst. 19, 167 (1995).
[Fab83] M. Fabre de la Ripelle, Ann. Phys. 147, 281 (1983).
[Fad61] L.D. Faddeev, Sov. Phys. JETP 12, 1014 (1961).
[Fad61a] L.D. Faddeev, Sov. Phys. Docl. 6, 384 (1961).
[Fad63] L.D. Faddeev, Sov. Phys. Docl. 7, 600 (1963).
[Fad65] L.D. Faddeev, Mathematical aspects of the three body problem in the quantum scattering
theory, Israel Program for Scientific Translation, Jerusalem, (1965).
[Fil74] G. F. Fillipov, Sov. J. Part. Nucl. 4, 405 (1974).
[Flu71] S. Fluegge, Practical Quantum Mechanics ,Vol. 1, Springer-Verlag, Berlin- Heidelberg-NewYork, (1971).
[Fre93] R.W. Freund, M.H. Gutknecht, and N.M. Nachtigal, SIAM J. Sci. Com.,15, 313 (1994).
[Fri80] J.L. Friar,B.F. Gibson and G.L. Payne, Phys. Rev.,C22, 284 (1980).
[Fri91] J.L. Friar, B.F.Gibson, H.C. Jean and G.L.Payne, Phys. Rev. Lett.,66, 1827 (1991).
131
[Gel50] I. M. Gel’fand and M. L. Zetlin, Dokl. Akad. Nauk.USSR 71, 825 (1950); I. M. Gel’fand,
R. A. Minols, and Z. Ya. Shapiro. in Representations of the rotation and Lorentz groups and
their application p. 353, Pergamon, New York, (1963).
[Gra80] I.S. Gradshtein and I.M. Ryzhik, Tables of Integrals, Series and Products, edited by A.
Jeffrey, Academic Press, New York, (1980).
[Haf87] M. I. Haftel and V. B. Mandelzweig, Phys. Lett. A120, 232 (1987).
[Haf89] M. I. Haftel and V. B. Mandelzweig, Ann. Phys. (N.Y.) 189, 29 (1989); 195, 420 (1989).
[Haf88] M.I. Haftel and V.B. Mandelzweig, Phys. Rev. A38, 5995 (1988).
[Haf89b] M.I. Haftel and V.B. Mandelzweig, Phys. Rev. A39, 2813 (1989).
[Haf90] M.I. Haftel and V.B. Mandelzweig, Phys. Rev. A41, 2339 (1990).
[Haf90b] M.I. Haftel and V.B. Mandelzweig, Phys. Rev. A42, 6324 (1990).
[Hal79] R. L. Hall, B. Schwesinger, J. Math. Phys. 20, 2481 (1979).
[Ham62] M. Hamermesh, Group Theory and its Application to Physical Problems, Addison-Wesley,
Reading, MA, (1962).
[Him74] J.E. van Himbergen and J.A. Tjon, Physica 76, 503 (1974).
[Him77] J.E. van Himbergen, Physica A86, 93 (1977).
[Hir56] J. O. Hirshfelder and J. Dahler, Proc. Natl. Acad. Sci. USA 42, 363 (1956).
[Jac42] C. J. Jacobi, C. R. Acad. Sci. 15, 236 (1842).
[Jah51] H.A. Jahn, Proc. Roy. Soc. (London) A205, 192 (1951); H.A. Jahn and H. van Wieringen,
ibid. A209, 502 (1951).
[Jib77] R.I. Jibuti, N.B. Krupennikova, and N.I. Shubitidze, Theor. Math. Phys. 32, 704 (1977).
[Kia76] D. Kiang and A. Neigawa, Phys. Rev. A14, 911 (1976).
[Kil72] M. S. Kil’dyushov, Yad. Fiz. 15, 197 (1972), [Sov. J. Nucl. Phys. 15, 113 (1972)]
[Kil73] M.S. Kil’dyushov, Yad. Fiz. 16, 217 (1972) [Sov. J. Nucl. Phys. 16, 117 (1973)].
132
[Kor61] G. A. Korn and T. M. Korn , Mathematical handbook for scientists and engineers, McGrawHill, New York, (1961).
[Kri90] R. Krivec and V. B. Mandelzweig, Phys. Rev. A42, 3779 (1990).
[Lan60] L.D. Landau and E. M. Lifshitz, Mechanics, Pergamon Press ,New York , (1960).
[Lev65] J.M. Levy-Leblond and M. Levy-Nahas, J. Math. Phys. 6, 1571 (1965).
[Lou60] J. D. Louck, Jou. Molec. Spec. 4, 298 (1960).
[Lou72] J. D. Louck and H. W. Galbraith, Rev. Mod. Phys. 44, 540 (1972).
[Maj78] C.K. Majumdar and I. Bose, J. Math. Phys. 19, 2187 (1978).
[Mal69] R. A. Malfliet and J. A. Tjon, Nucl. Phys. A127, 161 (1969).
[Man90] V.B. Mandelzweig, Nucl. Phys. A508, 63c (1990).
[Man80] V.B. Mandelzweig, Phys. Lett. A80, 361 (1980); A82, 471 (1981).
[Mag66] W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, Berlin, (1966).
[May75] H. Mayer J. Phys. A8, 1562 (1975).
[Mos71] M. Moshinsky and C. Quesne, J. Math. Phys. 12, 1772 (1971).
[Mos84] M. Moshinsky, J. Math. Phys. 25, 1555 (1984).
[Muk93] S. N. Mukherjee et al., Phys. Rep. 231, 201 (1993).
[Nik91] A. F. Nikiforov, S. K. Suslov and V. B. Uvarov, “Classical Orthogonal Polynomials of a
Discrete Variable”, Springer, Berlin (1991).
[Nov88] A. Novoselsky, J. Katriel and R. Gilmore, J. Math. Phys. 29, 1368 (1988).
[Nov88a] A. Novoselsky, M. Vallieres and R. Gilmore, Phys. Rev. C38, 1440 (1988).
[Nov89] A. Novoselsky and J. Katriel, Ann. Phys. (N.Y.) 196, 135 (1989).
[Nov94] A. Novoselsky and J. Katriel, Phys. Rev. A49, 833 (1994).
133
[Nov95] A. Novoselsky and N. Barnea, Phys. Rev. A51, 2777 (1995).
[Nyi69] J. Nyiri and Ya.A. Smorodinskii, Sov. J. Nucl . Phys. 9, 515 (1969).
[Pan67] S. C. Pang and K. T. Hecht, J. Math. Phys. 8, 1233 (1967).
[Pus66] V.V. Pustovalov and Yu. A. Simonov, Sov. Phys. JETP 24, 230 (1966).
[Ray70] J. Raynal and J. Revai, Nuovo Cimento A68, 612 (1970).
[Ray73] J. Raynal, Nucl. Phys. A202, 631 (1973); A259, 272 (1976).
[Ray73b] J. Raynal, Phys. lett. 45, 89 (1973).
[Ros92] S. Rosati, A. Kievsky and M. Viviani, Proceedings of the XIIIth European conference on
Few-Body problems in physics, Sept. 9-14, Marciana Marina, Isola d’Elba, Italy, 563 (1992).
[Row80] D. J. Rowe and G. Rosensteel, Ann. Phys. (N.Y.) 126, 198 (1980).
[Sim66] Yu. A. Simonov, Sov. J. Nucl. Phys. 3, 461 (1966).
[Smi60] F.T. Smith, Phys. Rev. 120, 1058 (1960); J. Math. Phys. 3, 735 (1962).
[Sur67] E. L. Surkov, Yad. Fiz. 5 (1967), 908. [Sov. J. Nucl. Phys. 5, 644 (1967)].
[Tob81] W. Tobocman, Nucl. Phys. A357, 293 (1981).
[Val94] M. Vallieres and A. Novoselsky, Nucl. Phys. A570, 345c (1994).
[Vil65] N.Ya. Vilenkin, G.I. Kuznetsov and Ya.A. Smorodinskii, Yad. Fiz. 2, 906 (1965) [Sov. J.
Nucl. Phys. 2, 645 (1966)].
[Vil68] N. Ya. Vilenkin, “Special Functions and the Theory of Group Representations”, American
Mathematical Society, Providence, RI, (1968).
[Vil95] N. Ya. Vilenkin and A. U. Klimyk, Representations of Lie Groups and Special Functions,
Kluwer Academic Publishers, London, England, (1995).
[Viv92] M. Viviani, A. Kievsky and S. Rosati, Il Nov. Cim. A105, 1473 (1992).
[Viv95] M. Viviani, A. Kievsky and S. Rosati, Few-Body Sys. 18, 25 (1995).
134
[Wan95] Y. Wang, C. Deng and D. Feng, Phys. Rev. A51, 73 (1995).
[Whi68] R.C. Whitten and F.T. Smith , J. Math. Phys. 9, 1103 (1968).
[Whi68] R.C. Whitten , J. Math. Phys. 10, 1631 (1969).
[Xia90] Xian-hui Liu, J. Math. Phys. 31, 1621 (1990).
[Yan95] Z-C Yan and G. W. F. Drake, Phys. Rev. A52, 3711 (1995).
[Zer35] F. Zernike and H.C. Brinkman, Proc. K. Ned. Akad. Wett. 38, 161 (1935).
[Zic65] W. Zickendraht, Ann. Phys. 35, 18 (1965).
135