二次量子化

(2)[ai, a†j ]± i = j:
[a†i , a†j ]± = 0
aia†j |n1,
n2
·
·
·
,
ni,
·
·
·
>=
√ ni
η
νi
nj + 1ηνj |n1, n2 · · · , ni − 1, · · · , nj + 1, · · · >
a†jai|n1, n2 · · · , ni, · · · >=
a†i a†i |n1,
n2
·
·
·
,
ni,
·
·
·
>=
√ 1
−
niηνi
√ 1
−
niηνi
|n1,
n2
·
·
·
,
ni
+
1
−
1,
·
·
·
>=
(1
−
ni)|n1,
n2
·
·
·
,
ni
+
1
−
1,
·
·
·
>
(a†i a†i − a†i a†i )|n1, n2 · · · , ni, · · · >= 0
In a word:
n2
·
·
·
,
ni
−
1
+
1,
·
·
·
>=
niη2νi
|n1,
n2
·
·
·
,
ni,
·
·
·
>
η2νi = 1
(aia†i + a†i ai)|n1, n2 · · · , ni, · · · >= 1|n1, n2 · · · , ni, · · · >
In a word:
[ai, a†j]± = δ(i − j)
Many-particles vavefunction:
|λ1,λ2, · · · , λN >=
1
η(1−sgn(j))/2(Seq(j)) (2)
N ! i(ni!) i
Seq(j)represents any (direct product)sequence of (1),j for mark this sequence.sgn(j)is a function about the odevity of the permute number when permute the Seq(j)to
k
V
k
e−ik rc†
k
2
ˆ
ˆ
1 ck = √
V
ddxe−ikxa(x), c† = √1
a standard sequence(|λ1,λ2, · · · , λN >). When permutaion is odd sgn(j) = −1,permutaion is even,sgn(j) = 1.η = −1 for fermions and η = 1for bosons.when η = −1,for fermions,eq.(2) also konwn as Slater determinant.
(ni
+
1)|n1,
n2
·
·
·
,
ni,
·
·
·
>
a†i ai|n1,
n2
·
·
·
,
ni,
·
·
·
>=
√ ni
−
1
+
1η
νi
−1
√ ni
ηνi
|n1
,
n2
·
·
·
,
ni
−
1
+
1,
·
·
·
>=
ni|n1,
n2
·
·
·
,
ni,
·
·
·
>
(aia†i − a†i ai)|n1, n2 · · · , ni, · · · >= 1|n1, n2 · · · , ni, · · · >
Generalizing to many particles(N particles for example): Each particle could choose one of single particle’s eigenstates |λi >:
|λ1 >, |λ2 >, · · · , |λN > (2)
>=
√ 0
+
1η0|1j
>=
|1j
>
a†j a†j |0
>=
√ 1
+
√ 1 1η0η0|2j
>
a†j a†j a†j |0
>=
√ 2
+
√ 11
+
√ 1 1η0η0η0|3j
>
a†j−1a†j a†j a†j |0
>=
√√ 12
+
√ 11
+
√ 1 1η0η0η0η0|1j−1,
3j
>
(2.2) Derivating some algebraic closure relation
Note for Second Quantilization
1. Introduction
(1.1) Quantilization
Uncertainty principle:
Generalizing mechanical quantity to operator. Time representation:
Hˆ |i >= ei|i >
Many particle’s vavefunction could write down directly:
|n1, n2 · · · >
ni repesents the occupation number of i state of single paticle.The seqence of single states could determined arbitrarily. (usually form small energy to large energy) ni = N
ni
+
1,
·
·
·
,
nj
+
1,
·
·
·
>
(a†i a†j − η−1a†j a†i )|n1, n2 · · · , ni, · · · >= (a†i a†j − ηa†j a†i )|n1, n2 · · · , ni, · · · >= 0
i = j (Bosons same as i = j, for fermions merely):
For fermions:
aia†i |n1,
n2
·
·
·
,
ni,
·
·
·
>=
√ 1
−
niηνi
√ 1
−
niηνi
|n1,
n2
·
·
·
,
ni
+
1
−
1,
·
·
·
>=
(1
−
ni)η2νi
|n1,
n2
·
·
·
,
ni,
·
·
·
>
a†i ai|n1,
n2
·
·
·
,
ni,
·
·
·
>=
√ ni
−
1
+
1ηνi
√ ni
η
νi
|n1,
ˆ
ˆ
ai = |r >< r|aidr = |r > ψi(r)dr
ˆ
ˆ
ai = |r > ψi(r)dr = |r > ψ(r)dr
i
i
ˆ
ˆ
a†i = < r| ψi†(r)dr = < r|ψ(r)†dr
i
i
ˆ
a†i ai = |r >< r|ψ(r)†ψ(r)dr = N
i
3.Appling to specific operator
nj
+
1η
(νj
−1)
√ ni
η
νi
|n1
,
n2
·
·
·
,
ni
−
1,
·
·Leabharlann Baidu
·
,
nj
+
1,
·
·
·
>
(aia†j − ηaj a†i ) = 0
i=j: For bosons:
aia†i |n1,
n2
·
·
·
,
ni,
·
·
·
>=
√ ni
+
1ηνi
+1
√ ni
+
1ηνi
|n1,
n2
·
·
·
,
ni
+
1
−
1,
·
·
·
>=
∆r
·
∆pr
≥
,∆ 2
· ∆t ≥ 2
tˆ = t, Hˆ = i ∂t
Coordinate representation:
rˆ = r, pˆr = −i ∂r
(1.2) Applying to many particles
Choose Hamiltonian operator for example.Single particle Hamiltonian’s eigen equation: Hˆ |λi >= λi |λi > (1)
Principles need to obey:
(1)Indistinguishability
(2)Exchange symmetry: symmetrized(bosons) ,anti-symmetrized(fermions)
(3)The sum of all possibility keep invariant.
i
Every states in eq.(2) could be reperented by:
|λ1,λ2, · · · , λN >=
cn1,n2···|n1, n2 · · · >
n1 ,n2 ···
We can also define creation and annihilation oprators.
(3.1) One-body
Hˆ |n1, n2 · · · , ni, · · · >= ( 1n1 + 2n2 + · · · + ini + · · ·)|n1, n2 · · · , ni, · · · >
< n1, n2 · · · , ni, · · · |Hˆ |n1, n2 · · · , ni, · · · >=
a†ja†i |n1, n2 · · · , ni, · · · >=
nj
+
1η(νj
+1)
√ ni
+
1ηνi |n1,
n2
·
·
·
,
ni
+
1,
·
·
·
,
nj
+
1,
·
·
·
>
η−1a†j a†i |n1, n2 · · · , ni, · · · >=
nj
+
1ηνj
√ ni
+
1ηνi |n1,
n2
·
·
·
,
Momentum basis: Fourier transform:
ˆ Bˆ = dra†(r)(Bˆ(r))a(r)
v
ˆ Hˆ =
dra†(r)(
pˆ
ˆ + V (r))a(r) =
dra†(r)( −
2∂r2 + V (r))a(r)
(3)
v
2m
v
2m
a(r) = √1 V
eikrck, a†(r) = √1
(3)Specially:
a†i ai|n1, n2 · · · , ni, · · · >= ni|n1, n2 · · · , ni, · · · >
(2.3) Changing of base
We can recast occupation state basis to an arbitary complete base(Coordinˆate base for example): |r >< r|dr = 1
ni represents the total number of particles in state i . Disadvantage:
N ∼ 1023
2. Second Quantilization
(2.1) Origination and difinition
Also because of indistinguishability,every single particle has the same states.For many-paticle problem we could focus on the occupation number of these single particle states.(If consider about interaction between particles,the single-particle states are different from independent single particle’s but every particle still has the same states) Occupation number representation could be established in any specific representation.we first choose Hamitonian’s eigenstates for example. (We aways study Hamitonian operator). Single particle’s Hamitonian’s eigenstates:|i >(one of particles in many-particle system)
ai|n1,
n2
·
·
·
,
ni,
·
·
·
>=
√ ni
ηνi
|n1,
n2
·
·
·
,
ni
−
1,
·
·
·
>
a†i |n1,
n2
·
·
·
,
ni,
·
·
·
>=
√ ni
+
1ηνi
|n1,
n2
·
·
·
,
ni
+
1,
·
·
·
>
where:
Caculate:
j
√ (a†j )nj
nj !
|0
>
i−1
ν(i) = nj
j=1
a†j |0
(1)[a†i , a†j ]± i = j:
···
j
(a†j )nj nj !
|0
>=
|n1,
n2
·
·
·
nj(a)
>
1
a†i a†j |n1,
n2
·
·
·
,
ni,
·
·
·
>=
√ ni
+
1ηνi
nj + 1ηνj |n1, n2 · · · , ni + 1, · · · , nj + 1, · · · >
ia†i ai < n1, n2 · · · , ni, · · · |n1, n2 · · · , ni, · · · >
i
General basis:
Bˆ = < ν|ˆb|ν > a†ν aν
ν
Bˆ = < µ|ˆb|ν > a†µaν
µν
Coordinate basis:
(r reprsents vector, dr = dx + dy + dz) for example: