For Pauli's exclusion principle to be followed by fermions, we need these anti-commutators [aλ,aλ]+=0
and
[a†λ,a†λ]+=0
Then
n2λ=a†λaλa†λaλ=a†λ(1−a†λaλ)aλ=a†λaλ=nλ.
which gives
nλ=0,1. Here we used the anti-commutator
[aλ,a†λ′]+=δλ,λ′
But we could have used even a commutator instead of the anti-commutator and still got the same result i.e. if we choose
[aλ,a†λ′]−=δλ,λ′
then
n2λ=a†λaλa†λaλ=a†λ(1+a†λaλ)aλ=a†λaλ=nλ
which also gives
nλ=0,1
What conditions make us impose the last anti-commutation relation
[aλ,a†λ′]+=δλ,λ′
instead of
[aλ,a†λ′]−=δλ,λ′ ?
I mean, we do not need all relations to be anti-commuting. I can take 2 of them to be anti-commuting but the third one i.e. relation between creation and annihilation operator to be commuting and still maintain the Pauli's exclusion
This post imported from StackExchange Physics at 2014-05-04 11:38 (UCT), posted by SE-user cleanplay