Particle number operator

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In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles.

The number operator acts on Fock space. Given a Fock state |\Psi\rangle_\nucomposed 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(φi) 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 Ni 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},\phi_{i+1},\cdots,\phi_n\rangle_\nu \\ a^{\dagger}(\phi_i) |\phi_1,\phi_2,\cdots,\phi_{i-1},\phi_{i+1},\phi_{i+1},\cdots,\phi_n\rangle_\nu &=& \sqrt{N_i+1} |\phi_1,\phi_2,\cdots,\phi_{i-1},\phi_{i+1},\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},\phi_{i+1},\cdots,\phi_n\rangle_\nu \\ &=& 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}

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