# Particle number operator

(Redirected from Number operator)

In quantum mechanics, for systems where the total number of particles

$|\Psi\rangle_\nu$composed of single-particle basis states $|\phi_i\rangle$:

$|\Psi\rangle_\nu=|\phi_1,\phi_2,\cdots,\phi_n\rangle_\nu$

with creation and annihilation operators $a^{\dagger}(\phi_i)$ and $a(\phi_i)\,$ we define the number operator $\hat{N_i} \ \stackrel{\mathrm{def}}{=}\ a^{\dagger}(\phi_i)a(\phi_i)$ and we have:

$\hat{N_i}|\Psi\rangle_\nu=N_i|\Psi\rangle_\nu$

where $N_i$ is the number of particles in state $|\phi_i\rangle$. The above equality can be proven by noting that

$\begin{matrix} a(\phi_i) |\phi_1,\phi_2,\cdots,\phi_{i-1},\phi_i,\phi_{i+1},\cdots,\phi_n\rangle_\nu &=& \sqrt{N_i} |\phi_1,\phi_2,\cdots,\phi_{i-1},\phi_{i+1},\cdots,\phi_n\rangle_\nu \\ a^{\dagger}(\phi_i) |\phi_1,\phi_2,\cdots,\phi_{i-1},\phi_{i+1},\cdots,\phi_n\rangle_\nu &=& \sqrt{N_i} |\phi_1,\phi_2,\cdots,\phi_{i-1},\phi_{i},\phi_{i+1},\cdots,\phi_n\rangle_\nu \end{matrix}$

then

$\begin{matrix} \hat{N_i}|\Psi\rangle_\nu = a^{\dagger}(\phi_i)a(\phi_i) |\phi_1,\phi_2,\cdots,\phi_{i-1},\phi_i,\phi_{i+1},\cdots,\phi_n\rangle_\nu &=& \sqrt{N_i} a^{\dagger}(\phi_i) |\phi_1,\phi_2,\cdots,\phi_{i-1},\phi_{i+1},\cdots,\phi_n\rangle_\nu \\ &=& \sqrt{N_i} \sqrt{N_i} |\phi_1,\phi_2,\cdots,\phi_{i-1},\phi_{i},\phi_{i+1},\cdots,\phi_n\rangle_\nu \\&=& N_i|\Psi\rangle_\nu\\ \end{matrix}$