1 Particule libre

o1
F(x) = 0 F=−dV/dx
V(x) = . V (x)=0 V(0) = V(L)→+∞
ˆ
Hψ =Eψ ˆ H=ˆp2
2m+V(x)
ˆp=−i~d/dx
ˆ
Hψ =−~2
2m
d2ψ
dx2=Eψ : 0 < x < L
ψ(0) = ψ(L) = 0
ψn(x) = Cnsin πn x
L, n = 1,2,3. . .
Cn
1 = kψk ⇔ 1 = C2
nZL
0
sin2πn x
Ldx
=C2
nZL
0
1
21−cos 2πn x Ldx =C2
n
L
2
Cn=q2
L
ˆ
Hψn=−~2
2m
d2ψn
dx2=~2 2mπn L2ψn=Enψn
En=~2
2mπn L2.

ˆ
Id =R|xihx|dx
hψ|ψi=Zhψ|xihx|ψidx =Z|ψ(x)|2dx
=C2Zexp −(x−x0)2
σ2!dx =C2Zexp −x2 σ2dx =C2σ√π
kψk2=hψ|ψi= 1 C= 1/πσ21/4
P(x) = |ψ(x)|2=C2exp −(x−x0)2
σ2!
x0σ P (x)dx
[x, x +dx]
P(x)
˜
ψ(p) = hp|ψi=Zhp|xihx|ψidx
=1
√2π~Zexp −ipx
~Cexp ip0x ~exp −(x−x0)2 2σ2!dx
˜
ψ(p) = Cσ
√~exp ip0x0
~exp −ipx0 ~exp −(p−p0)2
2 (~/σ)2!
˜
P(p) = 
˜
ψ(p)
2=C2σ2
~exp −(p−p0)2
(~/σ)2!
p0~/σ ˜ P(p)dp
[p, p +dp]
σ→ ∞ 1
C√2π~ψ(x)hx|p0i=1
√2π~exp (ip0x/~)
|p0iσ→01
Cσ√2πe−ip0x0/~ψ(x)
hx|x0i=δ(x−x0)|x0i
hxi=ZxP (x)dx =Zxhψ|xihx|ψidx =hψ|ˆxψi
h.i
(∆x)2=D(x− hxi)2E=Dx2+hxi2−2xhxiE=x2+hxi2−2hxihxi=x2− hxi2

hxi=x0,hpi=p0
x2=σ2
2+x2
0∆x=σ
√2∆p=1 √2(~/σ) ∆x∆p=~/2
∆x∆p≥~
2
∆x∆v≥~
2m
∆x∆v t = 0
∆x∆v
∆v∆x
(~/m) = 10 cm2/s t = 0 ∆x≤1cm ∆v≥~
2m∆x≥
0.5cm/s ∆x1cm t = 2s.
(~/m)≥10−18 m2/s ∆x'10−9m
∆v'10−9m/s
m'10−3
∆x(0) ∆v(0) ≥~
2m∼10−32m2/s
∆x(0) '10−16m, ∆v(0) '10−16m/s t ≥16s. ∆x(t)≥
1m.
∆x(TE) = eTE/τ ∆x(0) = 1m, ∆v(TE) = eTE/τ ∆v(0) = 1m/s,
a0= 1m2/s
a0= ∆x(TE) ∆v(TE) = e2TE/τ ~
2m
TE=τ
2log a02m ~= 0.5 log 1032'37
∆x(TE)

i~dψ(t)
dt =ˆ
Hψ (t)
hp|ˆ H|pi=H(p)|pi
i~dhp|ψ(t)i
dt =hp|ˆ
Hψ (t)i=hˆ Hp|ψ(t)i=H(p)hp|ψ(t)i
⇔i~d˜
ψ(p, t)
dt =H(p)˜
ψ(p, t)
⇔˜ ψ(p, t) = −iH(p)t
~˜
ψ(p, 0)
V(x)
˜
ψ(p, t)p'p0p'p0
H(p)'H(p0)+(p−p0)∂H
∂p p=p0
=p2
0
2m+ (p−p0)p0 m=E0+ (p−p0)v0
E0=H(p0) = p2
0
2mv0=p0 m
ψ(x, t) = hx|ψ(t)i=Zdp hx|pihp|ψ(t)i=Zdp eipx
~˜
ψ(p, t)
=Zdp eipx
~−iH(p)t ~˜
ψ(p, 0)
=−iE0t
~ip0v0t ~Zdp eip(x−v0t) ~˜
ψ(p, 0)
=i(p0v0−E0)t
~ψ(x−v0t, 0)
|ψ(x, t)|=|ψ(x−v0t, t)|
v0
ω0=L0
~,L0=p0v0−E0
L0=p0v0−H(p0)
H(p)
H(p) = H(p0)+(p−p0)∂H
∂p p=p0
+1
2(p−p0)2∂2H ∂p2p=p0
=E0+(p−p0)v0+(p−p0)2
2m
ψ(x, t) = Zdp eipx
~−iH(p)t ~˜
ψ(p, 0)
=−iE0t
~eip0x ~Cσ
√~Zdp exp i(p−p0) (x−v0t)
~−i(p−p0)2 2m~t−i(p−p0)x0 ~−(p−p0)2
2 (~/σ)2!
=−iE0t
~eip0x ~Cσ
√~ZdP exp iP
~(x−x0−v0t)−P2 2it m~+σ2 ~2

|ψ(x, t)|2=C2σ2
σ4+~2t2
m21/2exp 
−σ2(x−x0−vt)2
σ4+~2t2
m2
x
∆x(t) = σ2
2+~2t2 2m2σ21/2
="(∆x0)2+∆p0
mt2#1/2
∆p0
mt
∆x0x0
∆v= ∆p0/m
v0=p0/m
(∆v)t t
∆x∆p
(x0, p0)p0∆p0