Atome d`hydrogène et "hydrogénoïdes"

e+−e−e−−µ+
+ 2+ 91+
1/r
−e=−1,6.10−19 m
+Ze M

~r =~r1−~r2
µ=mM/(m+M)
H=~p2
2µ+V(r)
V(r) = −Ze2
4π²0r
m= 0,9.10−30 M= 1,67.10−27
µ∼m
1/r
E
~
L
E > 0
E < 0
r= 0

1/r
E < 0
2a
T
L= 0
r= 0 r= 2a
a
v
µv2
a=Ze2 4π²0a2
Ecinetique =µv2
2=1 2
Ze2
4π²0a=−1 2|Epotentielle|
Ecinetique Epotentielle
E=Ecinetique +Epotentielle

|E|=µv2
2=1 2
Ze2
4π²0a
2a=e2
4π²0
Z
|E|
v=p2|E|/µ µva
E
Lmax =Ze2
4π²0rµ 2
1
p|E|
T= 2πa/v
T=πe2
4π²0rµ 2
Z
|E|3/2
Lmax ∝Z
|E|1/2
T∝Z
|E|3/2
2a∝Z
|E|
H~ L
|ψnlml>H~ L2Lz
nlml
H|ψnlml>=Enl|ψnlml>
~
L2|ψnlml>=l(l+ 1)¯h2|ψnlml>
Lz|ψnlml>=ml¯h|ψnlml>
Hr
< r|ψnlml>=Rnl(r)Ylml(θ,φ)

n n
l n −1l ml
−l+l l = 0
l= 1 l= 2 l= 3
Enl =−EI(Z)
n2
EI(Z) = µ
2¯h2
e4
(4π²0)2Z2'Z2EH
EHZ= 1
M='
Z2
EH∼13,6
EH= 109737,315709(5)cm−1
EH
EH=1
2µHc2α2
α=e2/((4π²0)¯hc)=1/137,04...
EH/µHc2
Z2
n= 2−n=
1hν ∼Z∼10
Enl ∝Z2
n2