Page 232 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

8.4 Non-interacting particles 223
2 1 2 2 1 2 2
jØ S j ˆ jø a (q 0 )j jø b (q 0 )j ‡ jø a (q 0 )j jø b (q 0 )j
0
2
2

‡ Re[ø (q 0 )ø (q 0 )ø a (q 0 0ø b (q 0 )]
a b
2 2
ˆ 2jø a (q 0 )j jø b (q 0 )j
2 1 2 2 1 2 2
jØ A j ˆ jø a (q 0 )j jø b (q 0 )j ‡ jø a (q 0 )j jø b (q 0 )j
0 2 2

ÿ Re[ø (q 0 )ø (q 0 )ø a (q 0 )ø b (q 0 )]
a b
ˆ 0
Thus, the two bosons have an increased probability density of being at the same
point in space, while the two fermions have a vanishing probability density of
being at the same point. This conclusion also applies to systems with N
identical particles. Identical bosons (fermions) behave as though they are under
the in¯uence of mutually attractive (repulsive) forces. These apparent forces
are called exchange forces, although they are not forces in the mechanical
sense, but rather statistical results.
The exchange density in equations (8.52) and (8.53) is important only when
the single-particle wave functions ø a (q) and ø b (q) overlap substantially.
2
Suppose that the probability density jø a (q)j is negligibly small except in a
2
region A and that jø b (q)j is negligibly small except in a region B, which does

not overlap with region A. The quantities ø (1)ø b (1) and ø (2)ø a (2) are then
a
b
negligibly small and the exchange density essentially vanishes. For q 1 in region
2
2
A and q 2 in region B, only the ®rst term jø a (1)j jø b (2)j on the right-hand
sides of equations (8.52) and (8.53) is important. This expression is just the
probability density for particle 1 con®ned to region A and particle 2 con®ned
to region B. The two particles become distinguishable by means of their
locations and their joint wave function does not need to be made symmetric or
antisymmetric. Thus, only particles whose probability densities overlap to a
non-negligible extent need to be included in the symmetrization process. For
example, electrons in a non-bonded atom and electrons within a molecule
possess antisymmetric wave functions; electrons in neighboring atoms and
molecules are too remote to be included.

Electron spin and the helium atom
We may express the single-particle wave function ø n (q i ) as the product of a
1
spatial wave function ö n (r i ) and a spin function ÷(i). For a fermion with spin ,
2
such as an electron, there are just two spin states, which we designate by á(i)
1 1
for m s ˆ and â(i) for m s ˆÿ . Therefore, for two particles there are three
2 2
symmetric spin wave functions