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1 Bosons and Fermions

An N particle quantum system is described by a wavefunction of N arguments \Psi(\mathbf{r}_1,\ldots, \mathbf{r}_N). The starting point of many body quantum mechanics is that:

Systems of indistinguishable particles are described by totally symmetric or totally antisymmetric wavefunctions.

Just to be clear, totally symmetric means the wavefunction is unchanged by exchanging any two coordinates, whereas totally antisymmetric means that it changes sign.

A good fraction of this course is devoted to exploring the ramifications of this fact. Perhaps we should therefore give a very quick summary of why it appears to be true.

The first question is: what are indistinguishable particles? I’ll give a theorist’s answer. Indistinguishable particles are those described by Hamiltonians that are invariant under permuting the particle’s labels. Thus the sum of single particle Hamiltonians

H = \sum_{i=1}^{N} \left[-\frac{\nabla_i^{2}}{2m}+V(\mathbf{r_i})\right]

[did you remember that \hbar=1?] describes indistinguishable particles while