Dimensionalities of weak solutions
in hydrogenic systems
Alejandro López-Castillo
Institute for Theoretical Chemistry University of Vienna
Departamento de Química - Centro
Universitário FIEO (UNIFIEO)
Almost One-Dimensional Hydrogen Atom
The Hydrogen Atom
The Hamiltonian of the 3-D hydrogen atom in spherical coordinates is
separable and its wave function is given by Ψ(r,θ,φ)=R(r)Θ(θ)Φ(φ).
The equation for Θ(θ) is
(1/sinθ) (∂/∂θ)(sinθ ∂Θ/∂θ) - m2Θ/sin2θ + l(l+1)Θ = 0.
(1)
The solution of Eq.(1) converges for all values of |m| and l and for
-1<cos(θ)<1. However, it diverges for cos(θ)=± 1.
The Eq.(1) with l=0 (and |m|=0) is reduced to
(1/sinθ) (∂/∂θ)(sinθ ∂Θ/∂θ) = 0.
(2)
The normalized solutions for Eq.(2) are
Θl=0,m=0(θ) = P0,0 = √2/2
and
Θ‡l=0,m=0(θ) = Q0,0 = (√6/π) ln [tan(θ/2)].
(3)
(4)
Results
Fig.1: Q0,0 as function of θ
Fig.2: Sin(θ)Q0,02 as function of θ
The divergences in the θ coordinate do not imply in divergences on
the Cartesian coordinates.
If we replace Θl,m(θ) = 1/√J(θ) Tl,m(θ) = 1/√sinθ Tl,m(θ) on Eqs.(1) and
(2) the results are, respectively:
∂2T(θ)/∂θ2 + [(l+1/2)2 + ((1/2)2-m2)/sin2θ] T(θ) = 0
(5)
and
∂2T(θ)/∂θ2 + (1/2)2[1+ 1/sin2θ] T(θ) = 0.
(null orbital angular momentum)
(6)
We can interpret classically these equations.
For example, the function T‡0,0 = √(sinθ) Q0,0 is not divergent, the
|T‡0,0|2 = sinθ |Q0,0|2 is shown in Fig.2.
|Ψ0,0,0 (x,y,z)|2
|Ψ‡0,0,0 (x,y,z)|2
1
0
-1
1
0
1
-5
-1
0
-2.5
-1
0
-1
1
0
2.5
1
0
5
1
0
-5
-1
-2.5
1
0
0
2.5
-1
5
-1
Density plot of |Ψ‡1,0,0(x, y=0, z)|2
Non-orthogonality of Ψ‡n,0,0
The Ψ‡n,0,0 is not orthogonal to Ψn,l(odd),0, e.g., Ψ‡1,0,0
∞
<Ψ‡1,0,0| = ∑
n-1
n=1 ∑
l=0 Cn,l <Ψn,l,0|
+ R <ΨR|,
(8)
where <Ψ‡1,0,0| = <R1,0 Θ ‡0,0|, <Ψn,l,0| = <Rn,l Θl,0|, Cn,l = <Ψ‡1,0,0|Ψn,l,0>
= <R1,0|Rn,l><Θ‡0,0|Θl,0>, ΨR is LI wave function remainder of the
Ψ‡1,0,0 and R = <Ψ‡1,0,0|ΨR>.
Integrating Eq.(8)
∞
n-1
2
∑
n=1
l=0 |Cn,l|
<Ψ‡1,0,0| Ψ‡1,0,0> = ∑
∞
with ∑
n-1
n=1 ∑
l=0 |Cn,l|
2
+ R2 = 1
(9)
= 1 - R2 = P2.
We estimated that P < 0.6 (R > 0.8) and the normalized <ΨR| is
N
<ΨR| = (1/R) (<Ψ‡1,0,0| - ∑
n-1
n=1∑
l=0 Cn,l<Ψn,l,0|),
(10)
where N is maximum value of n. The projection P as a function of N
is shown in Fig.3.
If the projection P is not unitary then Ψ‡1,0,0 has also continuum
component.
Fig3: P as function of N
Final Considerations
The 3D hydrogen atom Hamiltonian revealed formal eigenvectors
often discarded in the literature. Although not in its domain, such
eigenvectors belong to the Hilbert space. They are then related to low
dimensionality and it is found that they have continuous components,
meaning that ionization can take place.
The odd symmetry of the wave functions (Ψ‡n,0,0) is the same of that
one-dimensional solution.
There is a strong analogy between those quantum solutions with the
old quantum theory previsions. The n can be defined as n=nr+k, where
nr is the quantum number for radial excitation and k is for angular,
formally k=l+1. (nr=0, k=1) Æ 1s and (nr=1, k=0) Æ “1-D”. The solution
k=0 was excluded, the corresponding orbit is a degenerate line ellipse
which would cause the electron to strike the nucleus [Pauling].
The infinite binding energy of the ground state (n=0) for 1-D H would
have a interpretation. The electron can be in a very short distance of
the proton in a relativistic scenery. If we consider the equilibrium
between Coulomb and relativistic centripetal forces we obtain
d~1.3x10-15 m~radius of neutron.
Acknowledgments
FAPESP and CNPq
References
Pauling L and Wilson Jr E B 1963 Introduction to Quantum
Mechanics with applications to Chemistry (New York:
Dover) pp. 25-150
Loudon R 1959 Amer. J. Phys. 27 649
Lopez-Castillo A and de Oliveira C R 2006 Journal of Physics A
39 3447
Arfken G B and Weber H J 2001 Mathematical Methods for
Physicists 5th ed. (New York: Harcourt-Academic Press)
Reed M and Simon B 1972 Functional Analysis (New York:
Academic Press); 1975 Fourier Analysis, Self-Adjointness
(New York: Academic Press)
Eigenvalues
The eigenvalues of Ψ‡n,0,0 are
En = -ħ2/(2 μ a02 n2),
which are also similar to those of the 1-D hydrogen atom.
Self-Adjoint Domain
The solutions belonging to the self-adjoint domain of the L2 operator
are given by associated Legendre polynomials of the first kind.
However, the solutions of the second kind do not belong to this
domain [Teschl].
A second-order linear Hermitian operator (L) is an operator that
satisfies
b
b
∫a Ψ* L Φ dx = ∫a Φ L Ψ* dx,
where the superscript (*) denotes a complex conjugate. If L is selfadjoint and satisfies the boundary conditions
Ψ* L Φ’ |x=a = Ψ* L Φ’ |x=b ,
then it is automatically Hermitian [Arfken]. Hermitan operators are
complex self-adjoint ones [Butkov].
An outside solution of the self-adjoint domain can be important to
make connections among solutions of a three-dimensional system
with a low-dimensional subsystem.
Sturm-Liouville
d/dx[p(x)dy(x)/dx] – s(x)y(x) + λr(x)y(x) = 0
L≡ d/dx[p(x)d/dx] – s(x)
L{y(x)} = - λr(x)y(x)
ym*(d/dx[p(x)dyn/dx] – s(x)yn + λr(x)yn = 0)
yn*(d/dx[p(x)dym/dx] – s(x)ym + λr(x)ym = 0)
p(x)[yndym/dx - ymdyn/dx]|ab =
(λn - λm) r(x) ym(x) yn(x) dx = 0 (orthog)
Legendre: d/dx[(1-x2)dy(x)/dx] – λy(x) = 0
(1-x2)[P0,0 dQ0,0/dx - Q0,0dP0,0/dx ]|0π =cte≠0
P0,0=cte, Q0,0~ln((1+x)/(1-x)) and Q´~1/(1-x2)
The H operator of Schroedinger equation is essentially self-adjoint
with unique self-adjoint extension, i.e., the H has a unique complete
set of eigenfunctions [Kato].
Different self-adjoint extensions of H corresponding to different
“physical” situations. The problem of a “correct” choice of the selfadjoint extension is not just a question of mathematical “technique”
but it is closely related to the physics of the system under
consideration.
The boundary condition give us the “physical” situation. For
example, the logarithm divergence cannot be a good physical
feature for usual 3-D system. However, some solutions outside of
the self-adjoint domain can have some meaning if different physical
system are considered.
The second solution of Legendre equation can represent the
connection between one and three dimensional solutions with null
angular momentum (l=0). Since this solution is obtained from the
general 3-D problem it could be better justification to the use of the
1-D models. (Many-electron atoms and molecules).