Guide pour comprendre : Physique quantique

•1900
ν hν
•
hν

E=hν h = 6,63.10−34J.s
•~p
E=pc E =hν
~p =~~ k
E=hν
•w=E
~
~
k=~p
~λ=h p
λDB =h
p
•

Ψ(~r, t)
|Ψ(~r, t)|2
Zespace |Ψ(~r, t)|2d3r= 1
i~∂
∂t Ψ(~r, t) = −~2 2m∇2
rΨ(~r, t) + V(~r)Ψ(~r, t)
•Ψ(~r, t)∈L2
•Ψ(~r, t)
•Ψ(~r, t)
•
< φ|ψ >=Zφ∗ψd3~r
i~∂
∂t Ψ(x, t) = −~2 2m
∂2Ψ(x, t)
∂x2
x < 0x>a |Ψ(x, t)|2= 0 Ψ(x, t)=0
0≤x≤a
i~∂
∂t Ψ(x, t) = −~2 2m
∂2Ψ(x,t)
∂x2
Ψ(0, t) = 0
Ψ(a, t)=0
Ψ(x, 0) = Ψ0

Ψ(x, t) = φ(x)g(t)
i~φ(x)∂
∂t g(t) = −~2 2mg(t)∂2φ(x) ∂x2
Ψ(x, t)
Ψ(x, t)φ
i~
∂g(t)
∂t
g(t)=−~2 2m
∂2φ(x)
∂x2
φ(x)
(∂g
∂t (t) + iE ~g(t)=0
∂2φ
∂x2(x) + 2mE ~2φ(x) = 0
g(t) = e−iE
~tφ(x)
k=√2mE
~φ
φ(x) = Aeikx +Be−ikx
Ψ(0, t) = 0 g(t)6= 0 φ(0) = 0 A=−B
Ψ(a, t) = 0 φ(a)=0
A(eika −e−ika)=0
A= 0 eika −e−ika = 0 sin(ka)=0 ka =nπ
Ψ
kn=√2mEn
~≥0kn=√2mEn ~>0
kn=nπ
an∈∗
k=√2mE
~
En=n2~2π2
2ma2n∈∗
Ψn(x, t) = Ansin(nπ
ax)e−iEn
~t
AnRa
x=0 |Ψn(x, t)|2dx = 1
Ψn(x, t) = q2
asin(nπ ax)e−iEn
~t
Ψ(x, t) = ∞
X
n=1
CnΨn(x, t)
Ra
x=0 |Ψ(x, t)|2dx = 1 Ψn
< f, g >=Ra
0fgdx P∞
n=1 |Cn|2= 1

x < 0x > 0
•E > V2
k1=√2mE
~k2=√2m(E−V2) ~
(Ψ1(x, t)=(A1eik1x+B1e−ik1x)e−iE
~t
Ψ2(x, t) = A2ei(k2x−E
~t)
•E < V2
k2=√2m|E−V2|
~
Ψ2(x, t) = A2e−k2x−iE
~t
ek2x+∞
+∞Ψ∈L2
x > 0
x < 0E < V0
φ1(x) = Aeikx +Be−ikx x < 0
φ2(x) = Ceβx +De−βx 0< x < a
φ3(x) = F eik(x−a)+Ge−ik(x−a)x>a