Page 223 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

214 Systems of identical particles
^ ^ ^ ^ ^ ^ ^
P 132 ˆ P 31 P 23 P 12 ˆ P 12 P 23 P 31
^ ^ ^ ^ ^ ^ ^ ^ ^
P 231 ˆ P 12 P 23 P 31 P 12 ˆ P 31 P 12 P 31 P 12
However, the number of pairwise exchanges for a given permutation is always
^
^
^
^
either odd or even, so that P 123 , P 231 , P 312 are even permutations and P 132 ,
^ ^
P 213 , P 321 are odd permutations.
^
Applying the same arguments regarding the exchange operator P for the
two-particle system, we ®nd that
^
^
^ 2 12 ˆ P 2 23 ˆ P 2 31 ˆ 1
P
giving real eigenvalues 1 for each operator. We also ®nd that each exchange
^
operator commutes with the Hamiltonian operator H
^
^
^
^
^
^
[P 12 , H] ˆ [P 23 , H] ˆ [P 31 , H] ˆ 0 (8:23)
^
^
^
^
so that P 12 and H possess simultaneous eigenfunctions, P 23 and H possess
^
^
simultaneous eigenfunctions, and P 31 and H possess simultaneous eigenfunc-
^
^
^
tions. However, the operators P 12 , P 23 , P 31 do not commute with each other.
For example, if we operate on the wave function Ø(1, 2, 3) ®rst with the
^ ^
^ ^
product P 31 P 12 and then with the product P 12 P 31 , we obtain
^ ^ ^
P 31 P 12 Ø(1, 2, 3) ˆ P 31 Ø(2, 1, 3) ˆ Ø(3, 1, 2)
^ ^ ^
P 12 P 31 Ø(1, 2, 3) ˆ P 12 Ø(3, 2, 1) ˆ Ø(2, 3, 1)
The wave function Ø(3, 1, 2) is not the same as Ø(2,3,1), leading to the
conclusion that
^ ^ ^ ^
P 31 P 12 6ˆ P 12 P 31
^
^
Thus, a set of simultaneous eigenfunctions of H(1, 2, 3) and P 12 and a set of
^
^
simultaneous eigenfunctions of H(1, 2, 3) and P 31 are not, in general, the same
^
set. Likewise, neither set are simultaneous eigenfunctions of H(1, 2, 3) and
^
P 23 .
^
There are, however, two eigenfunctions of H(1, 2, 3) which are also simul-
^
^
^
taneous eigenfunctions of all three pair exchange operators P 12 , P 23 , and P 31 .
These eigenfunctions are Ø S and Ø A , which have the property
^ á 6ˆ ⠈ 1, 2 (8:24a)
P áâ Ø S ˆ Ø S ,
^ á 6ˆ ⠈ 1, 2 (8:24b)
P áâ Ø A ˆÿØ A ,
To demonstrate this feature, we assume that Ø(1, 2, 3) is a simultaneous
^
^
^
^
eigenfunction not only of H(1, 2, 3), but also of P 12 , P 23 , and P 31 . Therefore,
we have
^
P 12 Ø(1, 2, 3) ˆ ë 1 Ø(1, 2, 3)
^ (8:25)
P 23 Ø(1, 2, 3) ˆ ë 2 Ø(1, 2, 3)