Particle number operator

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

|\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(\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}


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