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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
composed of single-particle basis states
:
![{\displaystyle |\Psi \rangle _{\nu }=|\phi _{1},\phi _{2},\cdots ,\phi _{n}\rangle _{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0ff9891622072b0ce8bfd3e50d8f4b64247d22)
with creation and annihilation operators
and
we define the number operator
and we have:
![{\displaystyle {\hat {N_{i}}}|\Psi \rangle _{\nu }=N_{i}|\Psi \rangle _{\nu }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5dacc7e23cd9d056e35b4dfebaa4b7b17af0204)
where
is the number of particles in state
. The above equality can be proven by noting that
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d384eb190f81a040dea5c10578ab02c35e38de)
then
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c98310890f4659f69de8c7367a30cdba87a8ff5d)
See also
References