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8.1 Permutations of identical particles 215
^
P 31 Ø(1, 2, 3) ë 3 Ø(1, 2, 3)
where ë 1 Ð 1, ë 2 Ð 1, ë 3 Ð 1 are the respective eigenvalues. From equa-
tions (8.21) and (8.25), we obtain
^ ^ ^ ^ ^ ^ ^
P 231 Ø(1, 2, 3) P 23 P 12 Ø(1, 2, 3) P 31 P 23 Ø(1, 2, 3) P 12 P 31 Ø(1, 2, 3)
Ø(2, 3, 1)
or
ë 2 ë 1 Ø(1, 2, 3) ë 3 ë 2 Ø(1, 2, 3) ë 1 ë 3 Ø(1, 2, 3)
from which it follows that
ë 1 ë 2 ë 3
Thus, the simultaneous eigenfunctions Ø(1, 2, 3) are either symmetric
(ë 1 ë 2 ë 3 1) or antisymmetric (ë 1 ë 2 ë 3 ÿ1).
The symmetric Ø S or antisymmetric Ø A eigenfunctions may be constructed
from Ø(1, 2, 3) by the relations
Ø S 6 ÿ1=2 [Ø(1, 2, 3) Ø(1, 3, 2) Ø(2, 3, 1) Ø(2, 1, 3) Ø(3, 1, 2)
Ø(3, 2, 1)] (8:26a)
Ø A 6 ÿ1=2 [Ø(1, 2, 3) ÿ Ø(1, 3, 2) Ø(2, 3, 1) ÿ Ø(2, 1, 3) Ø(3, 1, 2)
ÿ Ø(3, 2, 1)] (8:26b)
where the factor 6 ÿ1=2 normalizes Ø S and Ø A if Ø(1, 2, 3) is normalized. As
in the two-particle case, the functions Ø S and Ø A are orthogonal. Moreover, a
wave function which is initially symmetric (antisymmetric) remains symmetric
(antisymmetric) over time. The probability densities Ø Ø S and Ø Ø A are
S A
independent of how the three particles are labeled. The two functions Ø S and
^
Ø A are, therefore, the eigenfunctions of H(1, 2, 3) that we are seeking.
^
Equations (8.26) may be expressed in another, equivalent way. If we let P be
^
any one of the permutation operators P áâã in equation (8.17), then we may
write
X
^
Ø S,A 6 ÿ1=2 ä P PØ(1, 2, 3) (8:27)
P
^
where the summation is taken over the six different operators P áâã , and ä P is
either 1or ÿ1. For the symmetric wave function Ø S , ä P is always 1, but for
the antisymmetric wave function Ø A , ä P is 1(ÿ1) if the permutation
^
operator P involves the exchange of an even (odd) number of pairs of particles.
^
^
^
Thus, ä P is ÿ1 for P 132 , P 213 and P 321 .
N-particle systems
The treatment of a three-particle system may be generalized to an N-particle
