﻿ Second Quantization for Fermion Particles
San José State University

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Second Quantization for Fermion Particles

## Fermions and Anticommutation

For the case of fermions it is the anticommutator of two operations that must be considered. The anticommutator of two operations, P and Q, is defined as

#### {P, Q} = PQ + QP

Let F be an operator and F* be its adjoint. The canonical quantification conditions to be satisfied by F and F* are

#### {F, F*} = FF* + F*F = I and {F, F} = 0^

where 0^ is the zero operator, the operator that maps any function to the zero function.

Note that {F, F*}=I implies that

#### FF* = I − F*F

The condition that {F, F}=0^ and hence {F*, F*}=0^ is new for the case of the anti-commutator, but [P, P]=0^ is automatically satisfied any operator P for the commutator. The conditions that {F, F}=0^ and {F*, F*}=0^ imply that FF=0^ and F*F*=0^.

Consider now

#### (F*F)(F*F) = F*(FF*)F = F*(I − F*F)F = F*F − (F*F*)(FF) but both F*F* and FF are equal to 0^ Thus (F*F)(F*F) = F*F

Let Φ be an eigenfunction of F*F.

This means that

#### λ² = λ and hence λ must equal 0 or 1

Thus a fermion state can have at most one particle. This is the Pauli Exclusion Principle.