Number operator

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

For a bosonic mode satisfying the canonical commutation relation $[a,a^{†}]=1$, the number operator $n=a^{†}a$ obeys $n^2=a^{†}a{a}^{†}a=(a^{†}{)}^2{a}^2+n$. Thus a two-body on-site interaction is proportional to $n(n-1)$, with no self-interaction. In a fixed-number two-mode condensate with probability $q$ of occupying this mode, $⟨n(n-1)⟩=N(N-1)q^2$ by the binomial distribution.

For a hopping operator $X$ with $[\Delta ,X]=2X$, spectral functions obey $Xf(\Delta )=f(\Delta -2)X$. The coefficient on the left of the hop is evaluated on the final state. This ordering determines the shifted energy denominator in virtual-hopping calculations.

For a single electromagnetic field mode, the eigenvalues of the number operator count its photons.

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.