Top View Operator in quantum mechanics
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 following is in bra–ket notation : The number operator acts on Fock space . Let
| Ψ ⟩ ν = | ϕ 1 , ϕ 2 , ⋯ , ϕ n ⟩ ν {\displaystyle |\Psi \rangle _{\nu }=|\phi _{1},\phi _{2},\cdots ,\phi _{n}\rangle _{\nu }}
be a Fock state , composed of single-particle states | ϕ i ⟩ {\displaystyle |\phi _{i}\rangle } basis of the underlying Hilbert space of the Fock space. Given the corresponding creation and annihilation operators a † ( ϕ i ) {\displaystyle a^{\dagger }(\phi _{i})} a ( ϕ i ) {\displaystyle a(\phi _{i})\,}
N i ^ = d e f a † ( ϕ i ) a ( ϕ i ) {\displaystyle {\hat {N_{i}}}\ {\stackrel {\mathrm {def} }{=}}\ a^{\dagger }(\phi _{i})a(\phi _{i})}
and we have
N i ^ | Ψ ⟩ ν = N i | Ψ ⟩ ν {\displaystyle {\hat {N_{i}}}|\Psi \rangle _{\nu }=N_{i}|\Psi \rangle _{\nu }}
where N i {\displaystyle N_{i}} | ϕ i ⟩ {\displaystyle |\phi _{i}\rangle } a ( ϕ i ) | ϕ 1 , ϕ 2 , ⋯ , ϕ i − 1 , ϕ i , ϕ i + 1 , ⋯ , ϕ n ⟩ ν = N i | ϕ 1 , ϕ 2 , ⋯ , ϕ i − 1 , ϕ i + 1 , ⋯ , ϕ n ⟩ ν a † ( ϕ i ) | ϕ 1 , ϕ 2 , ⋯ , ϕ i − 1 , ϕ i + 1 , ⋯ , ϕ n ⟩ ν = N i | ϕ 1 , ϕ 2 , ⋯ , ϕ i − 1 , ϕ i , ϕ i + 1 , ⋯ , ϕ n ⟩ ν {\displaystyle {\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}}} N i ^ | Ψ ⟩ ν = a † ( ϕ i ) a ( ϕ i ) | ϕ 1 , ϕ 2 , ⋯ , ϕ i − 1 , ϕ i , ϕ i + 1 , ⋯ , ϕ n ⟩ ν = N i a † ( ϕ i ) | ϕ 1 , ϕ 2 , ⋯ , ϕ i − 1 , ϕ i + 1 , ⋯ , ϕ n ⟩ ν = N i N i | ϕ 1 , ϕ 2 , ⋯ , ϕ i − 1 , ϕ i , ϕ i + 1 , ⋯ , ϕ n ⟩ ν = N i | Ψ ⟩ ν {\displaystyle {\begin{array}{rcl}{\hat {N_{i}}}|\Psi \rangle _{\nu }&=&a^{\dagger }(\phi _{i})a(\phi _{i})\left|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i},\phi _{i+1},\cdots ,\phi _{n}\right\rangle _{\nu }\\[1ex]&=&{\sqrt {N_{i}}}a^{\dagger }(\phi _{i})\left|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i+1},\cdots ,\phi _{n}\right\rangle _{\nu }\\[1ex]&=&{\sqrt {N_{i}}}{\sqrt {N_{i}}}\left|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i},\phi _{i+1},\cdots ,\phi _{n}\right\rangle _{\nu }\\[1ex]&=&N_{i}|\Psi \rangle _{\nu }\\[1ex]\end{array}}}
![{\displaystyle {\begin{array}{rcl}{\hat {N_{i}}}|\Psi \rangle _{\nu }&=&a^{\dagger }(\phi _{i})a(\phi _{i})\left|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i},\phi _{i+1},\cdots ,\phi _{n}\right\rangle _{\nu }\\[1ex]&=&{\sqrt {N_{i}}}a^{\dagger }(\phi _{i})\left|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i+1},\cdots ,\phi _{n}\right\rangle _{\nu }\\[1ex]&=&{\sqrt {N_{i}}}{\sqrt {N_{i}}}\left|\phi _{1},\phi _{2},\cdots ,\phi _{i-1},\phi _{i},\phi _{i+1},\cdots ,\phi _{n}\right\rangle _{\nu }\\[1ex]&=&N_{i}|\Psi \rangle _{\nu }\\[1ex]\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b165f253ce6c2ce7f529d1aaaf7d9cccf84023)