- Source: Particle number operator
- Particle swarm optimization
- Daftar algoritme
- Aljabar
- Pengantar mekanika kuantum
- Supersimetri
- Particle number operator
- Particle number
- Virtual particle
- Fock state
- Creation and annihilation operators
- Ladder operator
- Grand canonical ensemble
- Hamiltonian (quantum mechanics)
- Translation operator (quantum mechanics)
- Phonon
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
Harmonic oscillator
Quantum harmonic oscillator
Second quantization
Quantum field theory
Thermodynamics
(-1)F
References
Bruus, Henrik; Flensberg, Karsten (2004). Many-body Quantum Theory in Condensed Matter Physics: An Introduction. Oxford University Press. ISBN 0-19-856633-6.
Second quantization notes by Fradkin