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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.

F*FΦ = λΦ
and
(F*F)(F*FΦ) = λ(F*F)Φ = λ²Φ
however
(F*F)(F*FΦ) = FF*Φ = λΦ

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.


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