Page 231 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

222 Systems of identical particles

ging two columns and hence changes the sign of the determinant. Moreover, if
any pair of particles are in the same single-particle state, then two rows of the
Slater determinant are identical and the determinant vanishes, in agreement
with the Pauli exclusion principle.
Although the concept of non-interacting particles is an idealization, the
model may be applied to real systems as an approximation when the inter-
actions between particles are small. Such an approximation is often useful as a
starting point for more extensive calculations, such as those discussed in
Chapter 9.

Probability densities
The difference in behavior between bosons and fermions is clearly demon-
2
2
strated by their probability densities jØ S j and jØ A j . For a pair of non-
interacting bosons, we have from equation (8.49a)
2
2
2
2
2
1
1

jØ S j ˆ jø a (1)j jø b (2)j ‡ jø a (2)j jø b (1)j ‡ Re[ø (1)ø (2)ø a (2)ø b (1)]
2 2 a b
(8:52)
For a pair of non-interacting fermions, equation (8.49b) gives
2
2
1
2

2
2
1
jØ A j ˆ jø a (1)j jø b (2)j ‡ jø a (2)j jø b (1)j ÿ Re[ø (1)ø (2)ø a (2)ø b (1)]
2 2 a b
(8:53)
The probability density for a pair of distinguishable particles with particle 1 in
2
2
state a and particle 2 in state b is jø a (1)j jø b (2)j . If the distinguishable
2
2
particles are interchanged, the probability density is jø a (2)j jø b (1)j . The
probability density for one distinguishable particle (either one) being in state a
and the other in state b is, then
2
1 jø a (1)j jø b (2)j ‡ jø a (2)j jø b (1)j 2
2
2
1
2 2
2
2
which appears in both jØ S j and jØ A j . The last term on the right-hand sides
of equations (8.52) and (8.53) arises because the particles are indistinguishable
and this term is known as the exchange density or overlap density. Since the
2
2
exchange density is added in jØ S j and subtracted in jØ A j , it is responsible
for the different behavior of bosons and fermions.
2
2
The values of jØ S j and jØ A j when the two particles have the same
coordinate value, say q 0 , so that q 1 ˆ q 2 ˆ q 0 , are