In [[quantum mechanics]], the'''particlenumberoperator'''isan operator whose eigenvalues can be interpreted as the number of particles thatareinagiven state.
In [[quantum mechanics]], for systems where the total [[Particle number|number of particles]] <math>|\Psi\rangle_\nu</math>composed of single-particle [[Basis (linear algebra)|basis state]]s <math>|\phi_i\rangle</math>:
Assume the total [[Particle number|number of particles]] <math>|\Psi\rangle_\nu</math> is composed of single-particle [[basis]] states <math>|\phi_i\rangle</math>:
with [[creation and annihilation operators]] <math>a^{\dagger}(\phi_i)</math> and <math>a(\phi_i)\,</math> we define the number operator <math>\hat{N_i} \ \stackrel{\mathrm{def}}{=}\ a^{\dagger}(\phi_i)a(\phi_i)</math> and we have:
Given [[creation and annihilation operators]] <math>a^{\dagger}(\phi_i)</math> and <math>a(\phi_i)\,</math> we define the number operator <math>\hat{N_i} \ \stackrel{\mathrm{def}}{=}\ a^{\dagger}(\phi_i)a(\phi_i)</math> and we have: