Page 227 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

218 Systems of identical particles

the hydrogen molecule must be antisymmetric with respect to the interchange
of the two nuclei (protons) and also antisymmetric with respect to the
interchange of the two electrons. As another example, the wave function for the
oxygen molecule with 16 O nuclei (each with spin 0) must be symmetric with
respect to the interchange of the two nuclei and antisymmetric with respect to
the interchange of any pair of the eight electrons.
The behavior of a multi-particle system with a symmetric wave function
differs markedly from the behavior of a system with an antisymmetric wave
function. Particles with integral spin and therefore symmetric wave functions
satisfy Bose±Einstein statistics and are called bosons, while particles with
antisymmetric wave functions satisfy Fermi±Dirac statistics and are called
4
3
fermions. Systems of He atoms (helium-4) and of He atoms (helium-3)
4
provide an excellent illustration. The He atom is a boson with spin 0 because
the spins of the two protons and the two neutrons in the nucleus and of the two
1
3
electrons are paired. The He atom is a fermion with spin because the single
2
neutron in the nucleus is unpaired. Because these two atoms obey different
statistics, the thermodynamic and other macroscopic properties of liquid
helium-4 and liquid helium-3 are dramatically different.

8.3 Completeness relation

The completeness relation for a multi-dimensional wave function is given by
equation (3.32). However, this expression does not apply to the wave functions
Ø íS,A for a system of identical particles because Ø íS,A are either symmetric or
antisymmetric, whereas the right-hand side of equation (3.32) is neither.
1
Accordingly, we derive here the appropriate expression for the completeness
relation or, as it is often called, the closure property for Ø íS,A.
For compactness of notation, we introduce the 4N-dimensional vector Q
^
with components q i for i ˆ 1, 2, ... , N. The permutation operators P are
allowed to operate on Q directly rather than on the wave functions. Thus, the
^
^
expression PØ(1, 2, ... , N) is identical to Ø(PQ). In this notation, equation
(8.32) takes the form
X
^
Ø íS,A ˆ (N!) ÿ1=2 ä P Ø í (PQ) (8:33)
P
We begin by considering an arbitrary function f (Q) of the 4N-dimensional
vector Q. Following equation (8.33), we can construct from f (Q) a function
F(Q) which is either symmetric or antisymmetric by the relation

1 We follow the derivation of D. D. Fitts (1968) Nuovo Cimento 55B, 557.