Ch 4.4 The operator $\hat{N} = \hat{a}^\dagger\hat{a}$, whose eigenvalues are the non-negative integers: $\hat{N}|n\rangle = n|n\rangle$. The Hamiltonian of the harmonic oscillator is $\hat{H} = \hbar\omega(\hat{N} + 1/2)$. *See also:* Fock state, Ladder operators.