Chapitre 2 Une particule quantique sans spin à 1 dimension (II)

ˆx, ˆp, ˆ
H
ˆ
H
d
dt |φ(t)i=−i ~ˆ
H|φ(t)i |φ(0)i
t= 0 |φ(t)i
t

d
dt|φ(t)i=−i ~ˆ
H|φ(t)i
ˆ
H|φ(0)it= 0
|φ(t)it
|φ(t)i=ˆ U(t)|φ(0)i
ˆ
U(t)
dˆ U(t)
dt =−i ~ˆ
H(t)ˆ U(t),ˆ U(0) = ˆ I
ˆ
H(t)
ˆ
U(t) = exp −iˆ
Ht
~!=∞
X
n=0
1
n! −iˆ
Ht
~!n
|φ(t)i
d|φ(t)i/dt =dˆ
U
dt |φ(0)i=−iˆ
H/~ˆ U(t)|φ(0)i=−iˆ H/~|φ(t)i
ˆ
Hˆ H=ˆp2
2m+V(ˆx)
ˆ
H

ˆ
X, ˆ P , ˆ H
ˆ
Hˆ H+=ˆ Hˆ U(t)
∀ϕ, 
ˆ
U(t)ϕ
=kϕk
∀ϕ, ψ, hˆ U(t)ψ|ˆ U(t)ϕi=hψ|ϕi
ˆ
U(t)+=ˆ U(t)−1=ˆ U(−t)
ˆ
H+=ˆ H
ˆ
U+(t) = exp −iˆ Ht/~+= exp iˆ H+t/~= exp iˆ Ht/~
= exp −iˆ H(−t)/~=ˆ U(−t)
ˆ
U+(t)ˆ U(t) = ˆ U(−t)ˆ U(t) = exp iˆ Ht/~exp −iˆ Ht/~=ˆ I
ˆ
U(t)+=ˆ U(−t) = ˆ U(t)−1
(2) : ∀ψ, ϕ, hˆ U(t)ψ|ˆ U(t)ϕi=hψ|ˆ U(t)+ˆ U(t)ϕi=hψ|ϕi
⇔ˆ U+(t)ˆ U(t) = ˆ I
⇔
ˆ
U(t)ϕ
2=Dˆ
U(t)ϕ|ˆ U(t)ϕE=
hϕ|ϕi=kϕk2
⇒ hψ+φ|ψ+φi=hψ|ψi+hφ|φi+2<(hψ|φi)
hψ+iφ|ψ+iφi=hψ|ψi+hφ|φi − 2=(hψ|φi)
hψ|φi=<(hψ|φi) + i=(hψ|φi) =
=1
2(hψ+φ|ψ+φi+ (i−1)hψ|ψi+ (i−1)hφ|φi − ihψ+iφ|ψ+iφi)
hψ|φi
hϕ|ϕi=kϕk2
ˆ
Uhψ|φi
◦
◦

ˆ
U(t)t∈R
ˆ
U(t1)ˆ U(t2) = ˆ U(t1+t2) :
ˆ
U(0) = I:
ˆ
U(t)−1=ˆ U(−t) :
t
ˆ
U(t)
ˆ
Hˆ
H
ˆ
U(t)
R(R2)
R3
R2
SO(2)
R3
SO(3)
ˆ
H

ˆ
X, ˆ P , ˆ H
t1
ˆ
U(t)
ˆ
U(t) = exp −iˆ
H
~t!=ˆ
I−iˆ
H
~t+o(t)
ˆ
Hˆ U(t)
|ψi ∈ H |ψ(t)i=ˆ U(t)|ψi
|ψiˆ H|hψ(t)|ψ(0)i| =
1,∀t∆E= 0
|ψi ∈ H |ψ(t)i=
ˆ
U(t)|ψi
|hψ(0) |ψ(t)i|2= 1 −t
∆t2
+o(t2)
∆t=~
∆E
(∆E)2:= hˆ H−Dˆ HE2i=hˆ H2i − Dˆ HE2
|ψ(t)i |ψ(0)i
∆t