On a classical limit for electronic degrees of freedom that satisfies the Pauli exclusion principle

Fermions need to satisfy the Pauli exclusion principle: no two can be in the same state. This restriction is most compactly expressed in a second quantization formalism by the requirement that the creation and annihilation operators of the electrons satisfy anticommutation relations. The usual classical limit of quantum mechanics corresponds to creation and annihilation operators that satisfy commutation relations, as for a harmonic oscillator. We discuss a simple classical limit for Fermions. This limit is shown to correspond to an anharmonic oscillator, with just one bound excited state. The vibrational quantum number of this anharmonic oscillator, which is therefore limited to the range 0 to 1, is the classical analog of the quantum mechanical occupancy. This interpretation is also true for Bosons, except that they correspond to a harmonic oscillator so that the occupancy is from 0 up. The formalism is intended to be useful for simulating the behavior of highly correlated Fermionic systems, so the extension to many electron states is also discussed.