VOL. 7, NO. 2
CHINESE JOURNAL OF PHYSICS
OCTOBER 1969
Mathematical Structure of the Periodic Hilbert Space and the
One-Dimensional Structure Constants in the
Green Function Method *
DER-RUENN SU (
&f& 24 )
Department of Physics, National Taiwan University,
Taipei
.
(Received September 20, 1969)
The modern theory of Fourier series is used to derive the periodic potential
Schrbdinger equation in (]K+k> j -representation from { \r>) -representation.
The resulting equation is formulated more generally for the case that the
Bloch functions and the periodic potentials are “generalized functions”. After
putting the {jr> j -representation and the {/K-i k>} -representation of the
SchrBdinger equation into the form of a single operator equation, the mathematical structure of the Hilbert space of finite norm periodic “generalized
functions”, including the scalar product and the operators etc., is discussed.
Further, as an application of the operator equation, the basic integral equation
of the Green function method is derived easily. One-dimensional Green function is evaluated in terms of the lattice spacing, and the structure constants
for this one-dimensional case are found explicitly.
I. INTRODUCTION
OR the solids with periodic structure, the modern theory of Fourier series”’
is used in this note to investigate the rigorous mathematical theory in changing the representation of the Schrodinger equation from { jr>}-representation to
{ I K + k>}-representation. In this modern theory, only Fourier transforms (over
infinite domain) are considered rigorously. For the ordinary Fourier series, it
is needed to use the “generalized function” approach”‘. The resultant equation
F
after a change of the representation is the same one as usual difference equation
in the “momentum representation”‘“’ for usual absolutely integrable Bloch
functions and potential, but it is formulated for the more general case that
these periodic functions are “generalized functions”(‘*‘) (cf. (2.91, (2.3) and (2.5) 1.
Because the ordinary Fourier series theory is built for the functions which are
*
Supported, in part, by the National Science Council.
(1) M. J. Lighthill, Fourier Analysis and Generalized finctions (Cambridge Univ. Press, London, 1960)
(2) Ref. 1 p. 4 and Ch. 5
(3)
J. C. Slater: “E lectronic Structure of Solids: The Energy Band Method”, Tech. Rept. 4, Solid
State and Molecular Theory Group M. I. T. July 15, 1953. A quotation from this report can be
“The Energy Band Theory of Solids” in Handbook of Pkysics edited by
found from H.B. Callen:
E. U. Condon and H. Odishow (McGraw-Hill, N. Y. 1958) pp. 8-24-8-42
76
D. R. SU
77
absolutely integrable, this formulism for “generalized functions” is considered
more general. This resultant equation together with the original SchrGdinger
equation are put into the form of an operator equation (Z.l@) with the representations (2.11) and (2.12).
In Sec. III, the operators in (2.10) are discussed, especially in the { I K+ k>}representation. It is concluded that the Hilbert space for this kind of finite norm
periodic “generalized functions” should be defined with a scalar product as given
in (3.3). The mathematical structure of this kind of the Hilbert space is discussed there.
In Sec. IV, one of the applications of (2.10) is given i. e. the basic equation
for the Green function method is derived. Further, the structure constants in the
Green function method for one-dimensional case are obtained. These structure
constants are valid generally for all one-dimensional cases.
II. THE OPERATOR EQUATION
Let the crystal under consideration be N=NlN2Na unit cells, each of them
with a volume ZI and let us denote the Bloch functionC4) as b(k, r). Using the
bases in both the lattice vectcr space ai and the reciprocal lattice vector space bi
r= ZXiai
i
dr=al-az x a3 fI dxi=V IfI dxi
i-1
jsl
k=2rTmibi/‘Ni
k*r=2;rzmixi/Ni
K=2nZiibi
i
K*r=2nZggiXi
i
with gi, mi=integers
(2.1)
and using the “unitary function” U(x) (lv2) 7 b (k, r) can be expanded, due to the
periodicity, in k-space as@)
b(k, r) = ( NV ) -‘~*~e~(K+k)*r,(K+k)
--g<riK+k>n(K+k)
(2.2)
where@)
r(K+k) = (NV)-“‘$“b(k,
-CO
with the normalization
(4)
(51
F. Bloch, Z. Physik 52 555 (1928)
Ref. 1, p. 62. Theorem 22
r) fi U(xi/Ni)e-i(K-~k).rdr
i=l
(2.3)
78
MATHEMATICAL STRUCTURE OF THE PERIODIC HILBERT SPACE
~d(K+k)x,,(K+k) =6,,~,
and the potential is expanded as
V(r) = Z VK,eiK’- r
(2.4)
. .
where f srmrlar to”;2 .3) 7
(2.5)
Because of introducing U(z), (2.3) and (2.5) can be used for generalized functions
b(k, r) and V(r), such as 6-function@) etc.. But if b(k, r) and V(r) are absolutely
integrable as required in usual Fourier series theory, (2.3) and (2.5) are reduced
to usual expressions as expected.“)
Consequently, the Schrodinger equation
L
becomes
O=f(k, r)=(-&~‘- --E~(k) +V(r) )b(k, r)
(K+k)‘- E(k)]lr(K+k) +g,V,_,,x(K’+ k)}
x ei(K+k)-r.
(2.6)
For simplicity, denoting
K + k=2nCnibi/Ni with nizinteger,
i
we see that the periodicity of ei(K+k).r in (2.6) for Xi is Ni.
vector in the reciprocal vector space be
2~y=2~E~ibi
i
and define f(r) = C f(k, r) summing over the reduced zone.
k
Let the general
Then the Fourier
transform g(y) of f(r) can be obtained by using a theorem in the theory of
Fourier series@). The result is
O’S(y) z$IU(NUi--ni)
= {[-&(K+k)‘- E(k)]n(K+k)
+g,&_KrdK’+k))
X filG(yi-ni/‘N i)
which means for any “good function”(‘)G (y)
( 6
(7
( 8
) Ref. 1, p. 48
) Ref. 1, p. 66 and eq. (3.3”) below
) Ref. 1, p. 63, Theorem 23
(2.7)
D. R. SU
79
The change of the argument in G(y) is due to the a-functions in (2.7), i.e.
y=Tyibi=CnibiliZT,= (K+k)/2x.
Since G(Y) in (2.8) is arbitrary, therefore
ig(K+k)‘-E(k)lx(K+k)
+sv,,,x(K’+k) =O.
(2.9)
It is noted that, in (2.9), the coefficients (not Fourier transforms in the “generlized
function” approach! ) n(K+k) and V, can be those of “generalized functions”.
Mathematically, we write (2.9) and the original Schrddinger equation in (2.6)
as one operator equation
(2.10)
(T--E)++V*=O
in which, in{ i r>}-representation,
<ri *jr>
=-2C~z
2m
<riVir>=V(r)
(2.11)
<r I +>=b(k, r)
and in{ I K + k>}-representation,
<K+kiTiK’+k’>=~(K+k)ZdK,Kblr,li
<K+k’/+>=ic(K+k’)6k,k
<K + k I r/-I K’ + k’>= VK_&+
(2.12)
III. DISCUSSION
On closer investigating (2.10) and (2.11) and (2.12), (2.11) is fundamental
and certainly correct. As to (2.12)) if b(k, r) and V(r) are absolutely integrable
and if we define the scalar product in the Hilbert space as
<rsl~>=SNUdr<~lr><rl~>
(3.1)
or equivalently, the closure property
/ Nvdrlr><rl =l,
(3.2)
then it will be shown that the expressions in (2.12) are rigorous. The difference
between (3.1) or (3.2) and the ordinary quantum mechanical case is the domain
of integration, that of (3.1) and (3.2) being NU but in the ordinary case, from
--co to a. Verifying (2.12), illustrations are taken for two cases, the wave
function and the operator representations, i. e.
MATHEMATICAL STRUCTURE OF THE PERIODIC HILBERT SPACE
80
<K + k’ I +> = _/- Nvdr<K f k’ I r><r I@>
= v&&&.
It is noted that it is simpler for 2”.
More generally, if b(k, r) and V(r) are “generalized functions”, it is found
that the integration should be taken as
(3.3)
or equivalently, the closure property
s
(3.3 ’)
mdrlr>i@l U(Xi/N;f<ri =I,
-CC
where (2.1) has been taken into account.
To check this, let us illustrate again
the representations,
x z(K’+ k).
Let us denote K’+k-K-k’=Zxzkibi/Ni
i
=N-’
h 5 JNi~‘i
i.=l
n.=-cc
=N-1 fi
(_J(
0
-c
with ki=integer, then, from (2.1),
+,)eiznkiXi Ni
‘
Nidxiei2,kiIi’Ni
ii-1 _I”
0
3
=N-’ II Ni6kio
i-l
=
‘iK’&k’k
(3.3”)
D. R. SU
81
where the property of any “unitary function”
5
(3.4)
U(x+m) =1
m--CC
for all x, and the interchangeable of the integration and the summation have
been used.. Finally, we obtain the result
<K+k’/ +>=ic(K+k)6&
with x(K+ k) given in (2.2) verifies (2.12).
Similar techniques are applied for
the operator case,
<K+ki V;K’+ k’> =lzdr$I
U(zi/N)<K+kir>V(r)<rIK’+ k’>
= (NV) -lSmdr~ U(2i/N;)e’( K’+ k’- -R-k).’
-CO
x c VK,,&“-r
K”
=
v,_ K&k
with V K given in (2.4).
Mathematical Structure of the Hilbert Space of P e r i o d i c
“G eneralized Functions”
From the discussions above, it is concluded that, if a set of square integrable
periodic “generalized functions” is considered as a function space, then together
wi-th (3.3)) this set forms a Hilbert space. Reviewing the definition of the
Hilbert space”‘), we have for an abstract Hilbert space R
A) H is a linear vector space,
B ) a scalar product is defined in H ,
C ) there is a basis with the number of the basis vectors the same as the dimensionality of H,
D ) H is complete and separable.
Specifying H to be our periodic Hilbert space Hfi which is the set of square
integrable (or more rigorously speaking, finite norm) periodic “generalized functions” (no loss of generality, one dimensional case with period L is considered),
we have
A’ 1 the linearity of H* is obvious,
B' ) the scalar product is defined by (3.3) for any “unitary function” with the
__(9)
(10)
properties (3.4) and U(Z) =O for lxl>l etc.,
Ref. 1, p. 61, Theorem 21 and its proof.
J. Von Neumann, Mathematical Founddions of Quantum Mechanics [Princeton Univ. Press, 1949)
pp. 36-46
MATHEMATICAL STRUCTURE OF THE PERIODIC HILBERT SPACE
82
C’ ) we may choose the linearly independent complete basis as the set of the
Fourier components i. e. {eiznkr} with k=K;/L and Ki =integer,
D’ ) since, by using (3.3) or (3.3’) and expanding
& (cc) =L-1!2$$ (k)&i2zbX
+ (2) =L-l’z F+ (k)eiZrkx
with
<kl~>=~(k)=L-“‘ S_~~~U(x/L)6;(s)e-’? .””
<kii>=~(k)=L-‘! 2S_~~~u(Lc/L)i(~)~-i2.”’ ,
we can prove that
<4I?O=~=~X4*(X)1/1.(2)
(3.5)
=?4*(k)i(k)
for the periodic ordinary functions with which the manipulation (3.3”) is possible.
Then from the similar theory as given in von Neumann’s book(“), we can
show that H* is complete and separable.
Next the operators in Hfi is considered.
By the definition of an operator A
in HP,
with ti and 3 belonging to HD.
Therefore A should be such an operator that
A+ is still a finite norm periodic “generalized function.” As illustrations, the
operators T, V in (2.11) are operators in H$. But the coordinate vector r is not
an operator in Hp. The non-periodic potentials are also not operators. Conclusively, for instance, the fundamental quantum condition
[P, 41= --i+fi
has different sense and the uncertainty relation of P, q need further verifications
although the uncertainty relation for two canonically conjugate operators is still
correct since the uncertainty relation is derived from the Schwarz inequality and
the Schwarz inequality is held also in Hp, etc..
to be developed by the same author.
(11) Ref.
10,
PP.
59-n
Further theory of HP is going
L
D. R. SU
83
IV. APPLICATION
Since T-E in (2.10) is invertable( we write
+=- --J-d+.
(4.1)
T-E
In the { I r>}-representation W) , by (3,5) or (3,3’) and the periodicities,
b(k. r)=-K2+K+k><K+k i&i K+k><K+kjV+>
f
(4.2)
which is the basic integral equation for the Green function method(‘4-‘g).
There-
fore the Green function
G (T _ r’) = _
$_ z --.$%!!?(r-
(4.3)
Fc(K+k)2--eZ
where e2-E. It is noted that since the Green function, must satisfy the equation
( $Vz+E)Gk(r-r’) =O”(r-r’),
(4.4)
where it is required that r and r’ should be inside the same unit cell.
Now let us consider the one-dimensional case in which the coordinates are x
and x’ and the lattice spacing is a. The Green function (4.3) becomes
1 c
G;(R) =lim--
pi(KAk)R
where we have put ii2/2m=1 and Rex-x’.
need the formulaCzO’
(12)
(13)
(14)
(15)
(16)
(17)
(13)
(19)
(20)
-_-.-
(4.5)
’
In order to evaluate G, (R) , we
P.R. Halmos, Finite Dimensional Vector Spaces 2nd ed. (D. Van Nostrand Co. 1958)
In (1 Ki- k> 1 -representation, (4.1) becomes
which can be obtained from (2.2) and (4.2) as expected.
J. Korringa, Phvsica 13 392 (1947)
W. Kohn and N. Rostoker, Phys. Rev. 94 1111 (1954)
B. Segall, Phys. Rev. 105 108 (1957)
F. S. Ham, and B. Segall, Phys. Rev. 124 1786 (1961)
B. Segall, Phys. Rev. 124 1797 (1961)
J. S. Faulkner, H. L. Davis and H. W. Joy, Phys. Rev. 161 656 (1967)
Ref. 1 pp. 67-68
84
MATHEMATICAL STRUCTURE OF THE PERIODIC HILBERT SPACE
and integrals
~_
1
2a
O3
J
eiYcRenn)
eTieiwna
dv=ri
-m g2-e2fiq
where the contour integrals- I’ were evaluated by the contours as shown in Fig. 3.
Therefore the Green function becomes
=lim- &
=lir
1
x
272
= +,i Ceikna
n
$
eikna
JJei(K+k-y)r eiyR dxdr
v2-e2fi7j
eiy(R-nal
dy
y2-e2*t7j
n
eTiel R -nd
2e
‘~
”u
=(2e)- eik (sin(eIR-na/)+icos(e IR-722
n
I)).
y-plane
+
-e
W
+
/
\
+e
) ’
.
*
/
Fig. 1. Contours of integration
Since R is valid only in unit cell and the inhomogeneity or singularity of (4.4)
comes from the term sin(e i x-x’ I ) as we discussed above, the Green function
becomes
GZ(R)=(2e)-‘i2,~
eih”“sin(e(na-RR))+ 2 eik”“sin(e(R--a))
n=--03
*iCeiknucos(e(R-na)) +sin(e/x--‘I)]
=&cYos(e(x-x’))+A$sin(e(x-x’))+sin(eix-x’l)
=A.?(cos(ex)cos(ex’) +sin(ex)sin(ex’))
+Af(sin(ex)cos(ex’) -cos(ex)sin(ex’)) +sin(eIx-x’l).
D. R. SU
85
The structure constants AZ, A& A$, AZ defined similar to the three-dimensional
case are
A$=&=&$
/+-&=A:
since the solutions for homogeneous d.ifferential equation for G(x) are cos(ez) and
sin (ez) . Evaluating A’s , we get
+ -~
sin( *ea)
Z[sin$-(kie)a] [sin+(kFe)a]
1
sin (02)
ZZ_
2 e cos(ka) -cos(ea)
i
sin (ka)
=_
2 e cos(FEa)-cos(ea)
which have the same values for both f cases. It is noted that these structure
constants are generally valid for all one-dimensional cases.
_
._