Page 217 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

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Systems of identical particles

The postulates 1 to 6 of quantum mechanics as stated in Sections 3.7 and 7.2
apply to multi-particle systems provided that each of the particles is distin-
guishable from the others. For example, the nucleus and the electron in a
hydrogen-like atom are readily distinguishable by their differing masses and
charges. When a system contains two or more identical particles, however,
postulates 1 to 6 are not suf®cient to predict the properties of the system. These
postulates must be augmented by an additional postulate. This chapter intro-
duces this new postulate and discusses its consequences.

8.1 Permutations of identical particles

Particles are identical if they cannot be distinguished one from another by any
intrinsic property, such as mass, charge, or spin. There does not exist, in fact
and in principle, any experimental procedure which can identify any one of the
particles. In classical mechanics, even though all particles in the system may
have the same intrinsic properties, each may be identi®ed, at least in principle,
by its precise trajectory as governed by Newton's laws of motion. This
identi®cation is not possible in quantum theory because each particle does not
possess a trajectory; instead, the wave function gives the probability density for
®nding the particle at each point in space. When a particle is found to be in
some small region, there is no way of determining either theoretically or
experimentally which particle it is. Thus, all electrons are identical and there-
fore indistinguishable, as are all protons, all neutrons, all hydrogen atoms with
2
1 H nuclei, all hydrogen atoms with H nuclei, all helium atoms with He
4
3
nuclei, all helium atoms with He nuclei, etc.
Two-particle systems
For simplicity, we ®rst consider a system composed of two identical particles

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