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Particle number operator

From Wikipedia, the free encyclopedia
(Redirected fromNumber operator)
Operator in quantum mechanics

Inquantum mechanics, for systems where the totalnumber of particles may not be preserved, thenumber operator is theobservable that counts the number of particles.

The following is inbra–ket notation: The number operator acts onFock space. Let

|Ψ⟩ν=|ϕ1,ϕ2,⋯,ϕn⟩ν{\displaystyle |\Psi \rangle _{\nu }=|\phi _{1},\phi _{2},\cdots ,\phi _{n}\rangle _{\nu }}

be aFock state, composed of single-particle states|ϕi⟩{\displaystyle |\phi _{i}\rangle } drawn from abasis of the underlying Hilbert space of the Fock space. Given the correspondingcreation and annihilation operatorsa†(ϕi){\displaystyle a^{\dagger }(\phi _{i})} anda(ϕi){\displaystyle a(\phi _{i})\,} we define the number operator by

Ni^ =def a†(ϕi)a(ϕi){\displaystyle {\hat {N_{i}}}\ {\stackrel {\mathrm {def} }{=}}\ a^{\dagger }(\phi _{i})a(\phi _{i})}

and we have

Ni^|Ψ⟩ν=Ni|Ψ⟩ν{\displaystyle {\hat {N_{i}}}|\Psi \rangle _{\nu }=N_{i}|\Psi \rangle _{\nu }}

whereNi{\displaystyle N_{i}} is the number of particles in state|ϕi⟩{\displaystyle |\phi _{i}\rangle }. The above equality can be proven by noting thata(ϕi)|ϕ1,ϕ2,⋯,ϕi−1,ϕi,ϕi+1,⋯,ϕn⟩ν=Ni|ϕ1,ϕ2,⋯,ϕi−1,ϕi+1,⋯,ϕn⟩νa†(ϕi)|ϕ1,ϕ2,⋯,ϕi−1,ϕi+1,⋯,ϕn⟩ν=Ni|ϕ1,ϕ2,⋯,ϕi−1,ϕi,ϕi+1,⋯,ϕn⟩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}}}thenNi^|Ψ⟩ν=a†(ϕi)a(ϕi)|ϕ1,ϕ2,⋯,ϕi−1,ϕi,ϕi+1,⋯,ϕn⟩ν=Nia†(ϕi)|ϕ1,ϕ2,⋯,ϕi−1,ϕi+1,⋯,ϕn⟩ν=NiNi|ϕ1,ϕ2,⋯,ϕi−1,ϕi,ϕi+1,⋯,ϕn⟩ν=Ni|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}}}

See also

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References

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