 Number operator

The number operator counts oscillator quanta in a bosonic mode. For creation and annihilation operators with $[A,A^{†}]=1$, it is $N=A^{†}A$ and satisfies

The number states obey $N|n⟩=n|n⟩$, $n=0,1,2,\dots$. For the quantum harmonic oscillator, $H=\hslash \omega (N+\frac{1}{2})$, so counting quanta also determines the energy. Positivity follows from $⟨\psi |N|\psi ⟩=\Vert A|\psi ⟩{\Vert }^2\ge 0$ on the operator domain. ^[ref]

An expectation value need not be a possible sharp count. In $(|0⟩+|2⟩)/\sqrt{2}$, $⟨N⟩=1$, but a quantum measurement of $N$ gives only $0$ or $2$, each with probability one half. For a general quantum field, a particle number operator also depends on the chosen modes and need not commute with an interacting Hamiltonian.