Number operator

The number operator $N=A^{†}A$ satisfiesIt is self-adjoint and positive semidefinite.

A number state is a normalized eigenstate of the number operator:For a single quantum harmonic oscillator, it is also an energy eigenstate with energy $\hslash \omega (n+1/2)$.

If $N|\nu ⟩=\nu |\nu ⟩$, then $A$ lowers the eigenvalue by one and $\Vert A|\nu ⟩{\Vert }^2=\nu \Vert \nu ⟩{\Vert }^2$. Positivity forces the lowering ladder to terminate at eigenvalue zero, so every eigenvalue is a nonnegative integer.