Équation de Schrödinger à une dimension

ψ(x, t)
i~∂ψ
∂t =−
~2
2m
∂2ψ
∂x2+V(x)ψ
V(x)
[0, L]
ZL
0
ψψ∗dx = 1
ψ(x, 0)
[0, L]
V0V(x)
τ=~
V0
λ=√2mV0
~
x
i∂ψ
∂t =−∂2ψ ∂x2+V(x)ψ
H=−∂2
∂x2+V(x)

ψ(x, t) = e−iHtψ(x, 0)
e−iHt
N[0, L]
∆x=L
N−1
∆t
ψn
j=ψ(j∆x, n∆t)
Vj=V(j∆x)
n
∂ψ
∂t →ψn+1
j−ψn
j
∆t
x
∂2ψ
∂x2→ψn
j+1 −2ψn j+ψn+1
j−1
(∆x)2
iψn+1
j−ψn
j
∆t=−ψn
j+1 −2ψn j+ψn j−1
(∆x)2+Vjψn
j
Hψn
j=−ψn
j+1 −2ψn j+ψn j−1
(∆x)2+Vjψn
j
iψn+1
j−ψn
j
∆t=Hψn+1
j
(1 + i∆tH)ψn+1
j=ψn
j

ψ(x, t + ∆t) = e−iH∆tψ(x, t)
e−iH∆t'1−iH∆t
e−iH∆t'1−i1
2H∆t
1 + i1
2H∆t
(1 + i1
2∆tH)ψn+1
j= (1 −i1
2∆tH)ψn
j
ψn+1
j−1+ (iα −(∆x)2Vj−2)ψn+1 j+ψn+1
j+1 =−ψn
j−1+ (iα + (∆x)2Vj+ 2)ψn j−ψn j+1
α=2(∆x)2
∆t
ajψn+1
j−1+bjψn+1 j+cjψn+1 j+1 =djψn
j−1+ejψn j+fjψn j+1 +sn j
ψn
0=ψn N−1= 0
ψ(x, 0) = eik0xe
−(x−x0)2
4σ2
0
x=x0x
ρ(x, 0) = ψ(x, 0)ψ(x, 0)∗=e
−(x−x0)2
2σ2
0

σ0
σ0∆k=1
2
∆k σ
t
σ∆k≥1
2
t= 0
t
σ=σ0s1 + ~2t2
4m2σ4
0
v0=~k0
m
T
L m = 1/2~= 1
T'L
2k0
Tσ0
2σ2
0k0L
L= 1 σ0= 0.05 k0>200
k0∆x1
π
∆k
ω=E
~=~k2
0
2m

k2
0∆t1
∆t= 2∆x2
k0∆x1
k0= 200 L= 1 N= 2000
P=T
∆t=L 4k0∆x2
P= 5000