CHAPTER 12 THE HYDROGEN ATOM 243
This will be true if l=0 and ml=0. Another possible solution is:
Θ(θ) =Bcosθ (db12.28)
Substitution gives:
sinθ(− 2 Bsinθcosθ) +[l(l+1)sin^2 θ−ml^2 ]Θ(θ) = 0 (db12.29)
This is true if l=1 and ml=0. In general the solutions are polynomials of trigonometric
functions.
Radial solution
The radial equation was found previously (eqn db 12.11) to be:
(db12.30)
Multiplying this by R(r)/ryields:
(db12.31)
These are called Laguerre functions and are eigenfunctions with a series of solutions. To
determine the general functional form of the solutions, let Π(r) =rR(r). Then:
(db12.32)
Consider the case where l=0 then, after multiplying by − 2 m/Z, the equation reduces to:
(db12.33)
To solve this equation, try a solution with exponentials with a constant, α:
Π(r) =re−αr (db12.34)
(db12.35)
(db12.36)
d
d
2
22
2
0
2
4
0
Π
Π
()
()
r
r
me
r
++Er
⎛
⎝
⎜
⎜
⎞
⎠
⎟
Z πε ⎟ =
−+
+
−
⎡
⎣
ZZ^22
2
2
2
2
(^20)
1
mr 24
r
ll
mr
e
r
d
d
Π()
()
πε
⎢⎢
⎢
⎤
⎦
⎥
⎥ΠΠ()rEr= ()
−
+
−
+
Z^22
2
2
0
2
242 m
rR r
r
e
r
rR r
m
l
d
d
[()]
[()]
πε
(( ) [()] [()]l
r
+= 1 rR r rR r E
1
2
Z
−
+
−
−+
Z^22 Z
2
2
0
2
24 m
r
Rr
rR r
r
er
rE
()
d[()]
d πε
2
2
10
m
ll()+=
d
d
2
2
2
r
Π()rreee=− −αα( −−ααrr+ )−α−αr=αrre−−ααrr− 2 αe
d
dr
Π()rr e e=−( )α −−ααrr+