9.2: The exchange operator and Pauli’s exclusion principle

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We introduce the exchange operator \(\hat{P}_{12}\): an operator which permutes the labels of the particles. This is a rather strange operator, because it only changes the unphysical labels which we have attached to the one-particle wavefunctions in order to make the maths more easy. For a meaningful solution we must have a wavefunction which has a probability amplitude unchanged by \(\hat{P}_{12}\): it must be symmetric or antisymmetric with respect to exchange: \(|\Phi ({\bf r_1}, {\bf r_2}) \rangle = \pm |\Phi ({\bf r_2}, {\bf r_1})\rangle\).

Physical solutions must be eigenfunctions of \(\hat{P}_{12}\) with eigenvalues +1 (bosons) or −1 (fermions). Also, any physically meaningful Hamiltonian must commute with \(\hat{P}_{12}\), otherwise \(\hat{H}\) and \(\hat{P}_{12}\) could not have common eigenfunctions and the system could not remain in an eigenstate of exchange.

A simple product wavefunction \(|a({\bf r_1})b({\bf r_2})\rangle\) does not satisfy this (unless \(a = b\)). A linear combination of all permutations is required, for two particles:

\[|\Phi^− \rangle = |a({\bf r_1})b({\bf r_2}) − a({\bf r_2})b({\bf r_1}) \rangle/ \sqrt{2} \nonumber\]

\[|\Phi^+ \rangle = C_{ab}|a({\bf r_1})b({\bf r_2}) + a({\bf r_2})b({\bf r_1})\rangle + C_{aa}|a({\bf r_2})a({\bf r_1})\rangle + C_{bb}|b({\bf r_2})b({\bf r_1})\rangle \nonumber\]

where the \(C_{ab}\) terms are expansion and normalisation parameters. Note that the antisymmetric combination cannot include terms where both particles are in the same state, but there are three possibilities for the symmetric state. Although any linear combinations of \(C_{ab}\) \(C_{bb}\) and \(C_{aa} = 1\) are possible, \(C_{bb}\) and \(C_{aa}\) correspond to different configurations and are usually set to zero.

Notice that if \(a = b\), then \(|\Phi^− \rangle = 0\). Thus there is no possible antisymmetric combination involving identical states, i.e. two fermions cannot be in the same quantum state: the Pauli exclusion principle.