From Wikipedia, the free encyclopedia
Content deleted Content added
Line 1:
Line 1:
In [[quantum mechanics]], for systems where the total [[Particle number|number of particles]]
In [[quantum mechanics]], for systems where the total [[Particle number|number of particles]] <math>|\Psi\rangle_\nu</math>composed of single-particle [[Basis (linear algebra)|basis state]]s <math>|\phi_i\rangle</math>:
<math>|\Psi\rangle_\nu</math>composed of single-particle [[Basis (linear algebra)|basis state]]s <math>|\phi_i\rangle</math>:
:<math>|\Psi\rangle_\nu=|\phi_1,\phi_2,\cdots,\phi_n\rangle_\nu</math>
:<math>|\Psi\rangle_\nu=|\phi_1,\phi_2,\cdots,\phi_n\rangle_\nu</math>
Revision as of 17:02, 25 April 2015
In quantum mechanics , for systems where the total number of particles
|
Ψ
⟩
ν
{\displaystyle |\Psi \rangle _{\nu }}
composed of single-particle basis states
|
ϕ
i
⟩
{\displaystyle |\phi _{i}\rangle }
:
|
Ψ
⟩
ν
=
|
ϕ
1
,
ϕ
2
,
⋯
,
ϕ
n
⟩
ν
{\displaystyle |\Psi \rangle _{\nu }=|\phi _{1},\phi _{2},\cdots ,\phi _{n}\rangle _{\nu }}
with 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
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{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}}}
References