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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 }
drawn from a
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})}
and
a
(
ϕ
i
)
{\displaystyle a(\phi _{i})\,}
we define the number operator by
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}}
is the number of particles in state
|
ϕ
i
⟩
{\displaystyle |\phi _{i}\rangle }
. The above equality can be proven by noting that
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}}}
then
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}}}
See also
References