Potentiel quantique constant Notes de cours

x∆x
pxOx ∆px
∆x∆px≥~
2
~=h
2π= 1,05 ×10−34 J·s
m= 100 g v= 2,0 m ·s−1
Ox
B∆x x ∆pxpx
m≈10−31 kg ∆x≈
10−10 mω≈1017 rad ·s−1
Bpx∆px
◦

B∆x≈1 mm ∆px
px≈0,1% ∆px≈2×10−4kg ·m·s−1
∆px≈2×10−7J·s~
2
Bpx=m r0ω≈10−31 ×10−10 ×1017 ≈10−24 kg ·m·s−1
∆px≈10−34
10−10 ≈10−24 kg ·m·s−1∆px≈px
J·s
~
Ox ˜
ψ(x, t)
x t
Z+∞
−∞ ˜
ψ(x, t)
2dx = 1 ∀t
g
gm=Z+∞
−∞
g˜ ψ(x, t)
2dx
xm=R+∞
−∞ x˜
ψ(x, t)
2dx
˜
ψ(x, t)˜ ψ(x, t)ei α
˜
ψ(x, t)m
i~∂˜
ψ
∂t =−~2 2m
∂2˜ ψ
∂x2+V(x, t)˜
ψ
V(x, t)
◦

V(x)
Bb H
B˜ ψ(x, t) =
˜ϕ(x) ˜χ(t) ˜ϕ(x)E
B˜χ(t) = e−i ω t
BE ω
Bb H=−~2
2m
∂2
∂x2+V(x)
i~∂˜
ψ
∂t =b
H˜ ψ
˜
ψm−1
2
B˜ ψ(x, t) =
˜ϕ(x) ˜χ(t)
i~˜ϕ(x)∂˜χ(t)
∂t = ˜χ(t)b
H˜ϕ(x)⇒i~˜χ0
˜χ=b
H˜ϕ
˜ϕ=cste =E⇒b
H˜ϕ=E˜ϕ
B˜χ0+iE
~˜χ= 0 ˜χ(t) = e−i ω t
BE=~ω
˜
ψ(x, t) = ˜ϕ(x)e−i ω t ˜ϕ(x)
−~2
2m
∂2
∂x2+V(x)˜ϕ(x) = b
H˜ϕ(x) = E˜ϕ(x)
E=~ω
˜
ψ(x, t) = X
n
cn˜ϕn(x)e−i ωnt
◦

E p
•ν ω E =h ν =~ω
•k λ p =~k=h c
λ
◦

dθ
~
ki~
kd
z
x
Oz
B∆px
d
Bθ
B∆px= 0 px=~kx= 0 ∆x→ ∞
B∆x≈d∆px=~∆kx≥~
2d
θ=kx
kz≈1
2d2π
λ≈λ
d
λ
S1
S2
M
z
x
D
S1S2˜ ψ1
˜
ψ2S2S1
BM S1S2
BM S1S2
|˜ ψ1(M, t)+ ˜ ψ2(M, t)|2=|˜ ψ1(M, t)|2+|˜ ψ2(M, t)|2+˜ ψ1(M, t)˜ ψ∗
2(M, t) + ˜
ψ∗
1(M, t)˜
ψ2(M, t)6=|˜ ψ1(M, t)|2+|˜ ψ2(M, t)|2
◦