number operator

The number operator, usually written N, is built by applying the annihilation operator and then the creation operator in succession. Its job is simply to count: acting on a state with n quanta, it returns that state unchanged, multiplied by the whole number n. In that sense it reads off how many rungs up the energy ladder a state sits.

Because it is the creation operator times the annihilation operator, the number operator is Hermitian and so is a genuine observable, with eigenvalues that are the non-negative whole numbers. This is a deep result: it is the algebra of the ladder operators that forces the count to come in integers, which is why energy in an oscillator, and particles in a field, arrive in discrete whole units rather than continuous amounts.

The Hamiltonian of the oscillator is just the number operator scaled by ℏω plus the constant zero-point term, so measuring energy and counting quanta are essentially the same measurement. In quantum field theory the number operator counts how many photons, phonons, or other particles occupy a given mode, making it the bridge between the wave picture of a field and the particle picture of its quanta.