Past exam of the theoretical physics II course of the University of Cambridge / 2019 / 3 / a / Solution

For bosonic creation operators and bosonic annihilation operators, the canonical commutation relations areFor fermionic annihilation operators and their adjoints, the canonical anticommutation relations areThe respective number operators are $n_i=a_i^{†}{a}_i$ and $n_i=f_i^{†}{f}_i$. Their allowed eigenvalues are $\boxed{0,1,2,\dots \text{ for bosons};0,1\text{ for fermions}}$. In the fermionic case $n_i^2=n_i$, so its eigenvalues must satisfy $n^2=n$; this is the single-mode Pauli exclusion principle.