Chapitre 7 Introduction à la théorie de la di usion

◦
◦
~x =~x2−~x1
V(~x2−~x1)
~x1, ~x2∈R3V
|~x2−~x1|
V(~x) = o1
|~x|

~x := ~x2−~x1
m:= m1m2
m1+m2
ˆ
H=~p2
2m+V(~x)
m1m2
~
X=1
m1+m2(m1~x1+m2~x2)'~x1~x1
~x1= 0 ~x (t) = ~x2−~x1=~x2
m'm1V(~x) = −e2
4πε0|~x|
V(~x)

F(K, θ, ϕ)
f(k, θ, ϕ)
E
~x ≡(r, θ, ϕ)
ψ(~x)E=~2k2
2m
ˆ
Hψ =Eψ r =|~x|  1ψ
ψ(~x) = a−(θ, ϕ)e−ikr
r
| {z }
Onde sph´erique entrante
+a+(θ, ϕ)e+ikr
r
| {z }
Onde sph. sortante
+o1
r, r =|~x|,
a±(θ, ϕ)
o1
r1/r
r→ ∞

a±(θ, ϕ)
r=|~x|  1
V(~x)→0
ˆ
H'ˆ H0+o1
|~x|,ˆ
H0=~p2
2m=−~2 2m∆
ˆ
Hψ =Eψ ⇔ − ~2
2m∆ψ=~2k2 2mψ+o1 rψ⇔∆ψ=−k2ψ+o1 rψ
~x ≡(r, θ, ϕ)
∆ψ=1
r
∂2(rψ)
∂r2+1 r21 sin θ
∂
∂θ sin θ∂ψ ∂θ +1
sin2θ
∂2ψ
∂ϕ2
ψ(r, θ, ϕ) = R(r)a(θ, ϕ)
∆ψ=1
r
d2(rR)
dr2a+o1 r=−k2R a
⇔d2(rR)
dr2=−k2(rR) + o(1)

F(K, θ, ϕ)
rR (r) = e±ikr +o(1)
R(r) = e±ikr
r+o1 r
◦∆ = ∂2
∂x2+∂2 ∂y2+∂2 ∂z2
ψ~p (~x) = ei~p.~x
~=ei~
k~x, ~p =~~
k∈R3,
ˆ
H0
ˆ
H0ψ~p =Eψ~p, E =~p2
2m=~2k2 2m:
L2(R3)
E
~p ∈R3|~p|=√2mE
EEE
EE
ψ
E z
V
θ=π
θ= 0