For Pauli's exclusion principle to be followed by fermions, we need these anti-commutators $$[a_{\lambda},a_{\lambda}]_+=0 $$ and $$[a_{\lambda}^{\dagger},a_{\lambda}^{\dagger}]_+=0 $$ Then $$n_{\lambda}^{2}=a_{\lambda}^{\dagger}a_{\lambda}a_{\lambda}^{\dagger}a_{\lambda}=a_{\lambda}^{\dagger}\left(1-a_{\lambda}^{\dagger}a_{\lambda}\right)a_{\lambda}=a_{\lambda}^{\dagger}a_{\lambda}=n_{\lambda}.$$
which gives $ n_{\lambda}=0,1 $. Here we used the anti-commutator $$[a_{\lambda},a_{\lambda^{\prime}}^{\dagger}]_+= \delta_{\lambda,\lambda^{\prime}} $$ But we could have used even a commutator instead of the anti-commutator and still got the same result i.e. if we choose $[a_{\lambda},a_{\lambda^{\prime}}^{\dagger}]_{-}=\delta_{\lambda,\lambda^{\prime}} $
then $n_{\lambda}^{2}=a_{\lambda}^{\dagger}a_{\lambda}a_{\lambda}^{\dagger}a_{\lambda}=a_{\lambda}^{\dagger}\left(1+a_{\lambda}^{\dagger}a_{\lambda}\right)a_{\lambda}=a_{\lambda}^{\dagger}a_{\lambda}=n_{\lambda}
$
which also gives $n_{\lambda}=0,1 $
What conditions make us impose the last anti-commutation relation $$[a_{\lambda},a_{\lambda^{\prime}}^{\dagger}]_+= \delta_{\lambda,\lambda^{\prime}} $$ instead of $[a_{\lambda},a_{\lambda^{\prime}}^{\dagger}]_{-}=\delta_{\lambda,\lambda^{\prime}} $ ?
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
Best Answer
Suppose $a$ and $a^{+}$ operators satisfy $$ \left\{ a,a\right\} =0\mbox{ and }\left[a,a^{+}\right]=1 $$ We have basically $a^{2}=0$ and $aa^{+}=a^{+}a+1$.
Now consider $aaa^{+}$.
$$ 0=aaa^{+}=a\left(a^{+}a+1\right)=aa^{+}a+a=a^{+}aa+2a=2a. $$
So we get $a=0$.