Page 219 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

210 Systems of identical particles
e
Ø(1, 2) ˆ e ÿar 1 ÿbr 2 (br 2 ÿ 1)
where r 1 ˆjr 1 j and r 2 ˆjr 2 j, then Ø(2, 1) would be
Ø(2, 1) ˆ e ÿar 2 ÿbr 1 (br 1 ÿ 1) 6ˆ Ø(1, 2)
e
Thus, the probability density of the pair of particles depends on how we label
the two particles. Since the two particles are indistinguishable, we conclude
that neither Ø(1, 2) nor Ø(2, 1) are desirable wave functions. We seek a wave
function that does not make a distinction between the two particles and,
therefore, does not designate which particle is at r 1 and which is at r 2 .
^
To that end, we now introduce the linear hermitian exchange operator P,
which has the property
^
Pf (1, 2) ˆ f (2, 1) (8:4)
^
where f (1, 2) is an arbitrary function of q 1 and q 2 .If P operates on
^
H(1, 2)Ø(1, 2), we have
^ ^ ^ ^ ^ ^
P[H(1, 2)Ø(1, 2)] ˆ H(2, 1)Ø(2, 1) ˆ H(1, 2)Ø(2, 1) ˆ H(1, 2)PØ(1, 2)
(8:5)
^
where we have used the fact that H(1, 2) is symmetric. From equation (8.5) we
^
^
see that P and H(1, 2) commute
^ ^
[P, H(1, 2)] ˆ 0, (8:6)
^
^
Consequently, the operators P and H(1, 2) have simultaneous eigenfunctions.
^
If Ö(1, 2) is an eigenfunction of P, the corresponding eigenvalue ë is given
by
^
PÖ(1, 2) ˆ ëÖ(1, 2) (8:7)
We then have
^ 2 ^ ^ ^ ^ 2 (8:8)
P Ö(1, 2) ˆ P[PÖ(1, 2)] ˆ P[ëÖ(1, 2)] ˆ ëPÖ(1, 2) ˆ ë Ö(1, 2)
^
Moreover, operating on Ö(1, 2) twice in succession by P returns the two
particles to their original order, so that
^
2
^
P Ö(1, 2) ˆ PÖ(2, 1) ˆ Ö(1, 2) (8:9)
^
^ 2
2
From equations (8.8) and (8.9), we see that P ˆ 1 and that ë ˆ 1. Since P is
hermitian, the eigenvalue ë is real and we obtain ë ˆÐ 1.
There are only two functions which are simultaneous eigenfunctions of
^
^
H(1, 2) and P with respective eigenvalues E and 1. These functions are the
combinations
Ø S ˆ 2 ÿ1=2 [Ø(1, 2) ‡ Ø(2, 1)] (8:10a)
Ø A ˆ 2 ÿ1=2 [Ø(1, 2) ÿ Ø(2, 1)] (8:10b)
which satisfy the relations