The number operator of a harmonic oscillator tells us its energy level in terms of an integer $n.$ When applied to a [harmonic oscillator state](Harmonic%20Oscillator%20State%20Vector.md), this gives its associated [eigenvalue](Eigenvalues%20and%20eigenvectors.md), thus making it a quantum mechanical [observable](Observable). The number operator for a [quantum harmonic oscillator](quantum%20harmonic%20oscillator%20Hamiltonian.md) is expressed in terms of the [ladder operators](Harmonic%20Oscillator%20Ladder%20Operators.md) as $\hat{N}=\hat{a}^{\dagger}\hat{a}$ ^4c53b2 This equivalent to the [[number operator]] on a single particle [Fock product space](Fock%20Space.md#Fock%20product%20space). # Derivation This simply follows from how each of the [ladder operators](Harmonic%20Oscillator%20Ladder%20Operators.md) acts on a given [harmonic oscillator state](Harmonic%20Oscillator%20State%20Vector.md), $|n\rangle:$ $\hat{a}^{\dagger}\hat{a}|n\rangle=\sqrt{n}\hat{a}^{\dagger}|n-1\rangle=\sqrt{n}\sqrt{(n-1)+1}|n-1+1\rangle$$=n|n\rangle$ # Properties 1. $\hat{N}^{\dagger}=\hat{N}$ ([hermiticity](Hermitian%20operators.md)) Hermiticity should be expected, as this is an [Observable](Observable). #QuantumMechanics/QuantumMeasurement/QuantumObservables #QuantumMechanics/StationaryStateQuantumSystems/QuantumHarmonicOscillators #QuantumMechanics/QuantumHarmonicOscillators