Occupation number ($n_k$)

The eigenvalue of a mode number operator counts excitations in that mode. Bosonic occupation numbers range over all nonnegative integers; fermionic occupation numbers are zero or one.

For $\{f,f^{†}{\}}_{+}=1$ and $f^2=0$, the occupation number operator $f^{†}f$ has eigenvalues zero and one. This gives the single-mode exclusion principle.

For $[a,a^{†}]=1$, the occupation number operator $a^{†}a$ has eigenvalues $0,1,2,\dots$ on the bosonic oscillator Fock space.