Page 228 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

8.3 Completeness relation 219
X
^
F(Q) ˆ (N!) ÿ1=2 ä P f (PQ) (8:34)
P
Since F(Q) is symmetric (antisymmetric), it may be expanded in terms of a
complete set of symmetric (antisymmetric) wave functions Ø í (Q) (we omit the
subscript S, A)
X
F(Q) ˆ c í Ø í (Q) (8:35)
í
The coef®cients c í are given by

c í ˆ Ø (Q9)F(Q9)dQ9 (8:36)
í
because the wave functions Ø í (Q) are orthonormal. We use the integral
notation to include summation over the spin coordinates as well as integration
over the spatial coordinates. Substitution of equation (8.36) into (8.35) yields
" #

X

F(Q) ˆ F(Q9) Ø (Q9)Ø í (Q) dQ9 (8:37)
í
í
where the order of summation and the integration over Q9 have been inter-
changed. We next substitute equation (8.34) for F(Q9) into (8.37) to obtain
" #
X X
^

F(Q) ˆ (N!) ÿ1=2 ä P f (PQ9) Ø (Q9)Ø í (Q) dQ9 (8:38)
í
P í
^ ÿ1
We now introduce the reciprocal or inverse operator P to the permutation
^
operator P (see Section 3.1) such that
^
^ ^
^ ÿ1 P ˆ PP ÿ1 ˆ 1
P
We observe that
^
Ø í (P Q) ˆ ä P ÿ1Ø í (Q) ˆ ä P Ø í (Q) (8:39)
ÿ1
^
^ ÿ1
The quantity ä P ÿ1 equals ä P because both P and P involve the interchange
of the same number of particle pairs. We also note that
X 2
ä ˆ N! (8:40)
P
P
because there are N! terms in the summation and each term equals unity.
We next operate on each term on the right-hand side of equation (8.38) by
^ ÿ1 ^
P . Since P in equation (8.38) operates only on the variable Q9 and since the
order of integration over Q9 is immaterial, we obtain
" #

X X
^
ÿ1

F(Q) ˆ (N!) ÿ1=2 ä P f (Q9) Ø (P Q9)Ø í (Q) dQ9 (8:41)
í
P í