ex. 2

20 ?
ˆ
H=ˆ H0+λˆ H1
ˆ
H0=ˆp2
2m+1 2Kˆq2
λˆ H1=λˆq4
λ1ˆ H0|ϕni=εn|ϕni
n∈Nεn=~ωn+1
2ω=qK mϕ0(q) =
1
(πσ2)1/4exp −q2
2σ2σ=~ mω 1/2RRe−x2x4dx =3 4√π
? En(λ) = εn+λE(1)
n+λ2E(2) n+. . . n ˆ
H
n= 0
?~= 1, m = 1, ω = 1 E0(λ)
E0(λ)
λ
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
E
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
E
λ
Ε (λ)
0
Exact (numérique)

c
ψc=α|0ci+β|1ci
α, β ∈C|0ci |1ci
c
a
b
φab =1
√2(|0a0bi+|1a1bi)
|0ai |1ai |0bi |1bi
a b b
˜
ψb=α|0bi+β|1bi
c
? φab ⊗ψc
φab ⊗ψc=1
2ϕ1⊗(α|0bi+β|1bi) + 1 2ϕ2⊗(α|0bi − β|1bi)
+1
2ϕ3⊗(β|0bi+α|1bi) + 1 2ϕ4⊗(β|0bi − α|1bi)
a b

ϕ1, ϕ2, ϕ3, ϕ4α, β
{|0a0ci, . . . |1a1ci} a, c
(a, b), c (a, c), b
hϕi|ϕjii, j = 1 . . . 4
?(a, c)
ϕ1, ϕ2, ϕ3, ϕ4ϕi
ϕ1ϕ2ϕ3ϕ4
ϕiψbb
?
ϕiUi
b˜ ψb=α|0bi+β|1bi
U=1 0
0−1(α|0bi − β|1bi)˜
ψb=
(α|0bi+β|1bi)U1, U2, U3, U4
N j = 0,1, . . . (N−1)
x qjj
ˆ
H=
N−1
X
j=0
1
2ˆp2
j+1
2Kˆq2
j+1
2α(ˆqj+1 −ˆqj)2
K≥0α≥0 [ˆqj,ˆpk] =
δj,k ˆˆqj,ˆpj
N+j j j ∈N
ˆ
H
?ˆ H
?(X0,...XN−1)∈CN
N

(Y0,...YN−1)∈CNYk=1
√NPN−1
j=0 e−i2πjk/N Xj
Xj=1
√NPN−1
k=0 ei2πjk/N Yk
PN−1
k=0 Y∗
kYk=PN−1
j=0 X∗
jXj
?ˆ H
k∈ {0, . . . , N −1}
ˆ
φk=1
√N
N−1
X
j=0
e−i2πjk/N ˆqj,ˆπk=1
√N
N−1
X
j=0
e−i2πjk/N ˆpj
ˆ
H=
N−1
X
k=0
1
2ˆπ†
kˆπk+ω2
kˆ
φ†
kˆ
φk
ωkK, α, k, N
ei2πk
Nˆ
φk=1
√NPN−1
j=0 e−i2πjk
Nˆqj+1
ˆ
H
ωkk∈N
K > 0K= 0
? k ∈ {0, . . . , N −1}
ak=1
√2√σkˆ
φk+i
√σk
ˆπk
σk>0σ−k=σk[ak, al]
hak, a†
lik, l
ak=1
√2ˆ
Xk+iˆ Pkˆ Xk,ˆ Pk
hˆ
Xk,ˆ Xlihˆ Pk,ˆ Pli hˆ Xk,ˆ Pli
k, l
σk
ˆ
H=
N−1
X
k=0
ωk
1
2ˆ
P2
k+ˆ
X2
k
φ†
k=φ−kπ† k=π−k
?ˆ H
ˆ
Hϕn=Enϕn
n= (n0,...nN−1)nk∈NEnωk
nk