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8.4 Non-interacting particles 221
structed by applying equation (8.32). For example, for a system of two identical
particles, one particle in state ø a , the other in state ø b , the symmetrized two-
particle wave functions are
Ø ab,S (1, 2) 2 ÿ1=2 [ø a (1)ø b (2) ø a (2)ø b (1)] (8:49a)
Ø ab,A (1, 2) 2 ÿ1=2 [ø a (1)ø b (2) ÿ ø a (2)ø b (1)] (8:49b)
The expression (8.49a) for two bosons is not quite right, however, if states ø a
and ø b are the same state (a b), for then the normalization constant is 1 2
rather than 2 ÿ1=2 , so that
Ø aa,S (1, 2) ø a (1)ø a (2)
From equation (8.49b), we see that the wavefunction vanishes for two identical
fermions in the same single-particle state
Ø aa,A (1, 2) 0
In other words, two identical fermions cannot simultaneously be in the same
quantum state. This statement is known as the Pauli exclusion principle
because it was ®rst postulated by W. Pauli (1925) in order to explain the
periodic table of the elements.
For N identical non-interacting bosons, equation (8.32) needs to be modi®ed
in order for Ø S to be normalized when some particles are in identical single-
particle states. The modi®ed expression is
1=2
N a !N b ! X
Pø a (1)ø b (2) ... ø p (N)
Ø S ^ (8:50)
N!
p
where N n indicates the number of times the state n occurs in the product of the
single-particle wave functions. Permutations which give the same product are
included only once in the summation on the right-hand side of equation (8.50).
For example, for three particles, with two in state a and one in state b, the
products ø a (1)ø a (2)ø b (3) and ø a (2)ø a (1)ø b (3) are identical and only one is
included in the summation.
For N identical non-interacting fermions, equation (8.32) may also be
expressed as a Slater determinant
ø a (1) ø a (2) ø a (N)
ø b (1) ø b (2) ø b (N)
Ø A (N!) ÿ1=2 (8:51)
ø p (1) ø p (2) ø p (N)
The expansion of this determinant is identical to equation (8.32) with
Ø(1, 2, ... , N) given by (8.47). The properties of determinants are discussed
in Appendix I. The wave function Ø A in equation (8.51) is clearly antisym-
metric because interchanging any pair of particles is equivalent to interchan-
