Page 226 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

8.2 Bosons and fermions 217

exchange of an even (odd) number of particle pairs. The factor (N!) ÿ1=2
normalizes Ø S and Ø A if Ø(1, 2, ... , N) is normalized.
Using the same arguments as before, we can show that Ø S and Ø A in
equation (8.32) are orthogonal and that, over time, Ø S remains symmetric and

Ø A remains antisymmetric. Since the probability densities Ø Ø S and Ø Ø A
S A
are independent of how the N particles are labeled, the two functions Ø S and
^
Ø A are the only suitable eigenfunctions of H(1, 2, ... , N) to represent a
system of N indistinguishable particles.

8.2 Bosons and fermions
In quantum theory, identical particles must be indistinguishable in order for the
theory to predict results that agree with experimental observations. Conse-
quently, as shown in Section 8.1, the wave functions for a multi-particle system
must be symmetric or antisymmetric with respect to the interchange of any pair
of particles. If the wave functions are not either symmetric or antisymmetric,
then the probability densities for the distribution of the particles over space are
dependent on how the particles are labeled, a property that is inconsistent with
indistinguishability. It turns out that these wave functions must be further
restricted to be either symmetric or antisymmetric, but not both, depending on
the identity of the particles.
In order to accommodate this feature into quantum mechanics, we must add
a seventh postulate to the six postulates stated in Sections 3.7 and 7.2.
7. The wave function for a system of N identical particles is either symmetric or
antisymmetric with respect to the interchange of any pair of the N particles.
Elementary or composite particles with integral spins (s ˆ 0, 1, 2, ...) possess
1 3
symmetric wave functions, while those with half-integral spins (s ˆ , , ...)
2 2
possess antisymmetric wave functions.
The relationship between spin and the symmetry character of the wave function
can be established in relativistic quantum theory. In non-relativistic quantum
mechanics, however, this relationship must be regarded as a postulate.
1
As pointed out in Section 7.2, electrons, protons, and neutrons have spin .
2
Therefore, a system of N electrons, or N protons, or N neutrons possesses an
antisymmetric wave function. A symmetric wave function is not allowed.
2
4
4
Nuclei of He and atoms of He have spin 0, while photons and H nuclei have
spin 1. Accordingly, these particles possess symmetric wave functions, never
antisymmetric wave functions. If a system is composed of several kinds of
particles, then its wave function must be separately symmetric or antisym-
metric with respect to each type of particle. For example, the wave function for