The One-Electron Atom
105
ρ =
1
1
2 ,
Zr a
na
=
2
0
2
4
r
m
e
πε






where a 1 is the radius of the first Bohr orbit for Z = 1, and
2 1
1
l
n
L
+
+
are the
associated Laguerre polynomials given by
L j
i
(ρ) =
( )
i
j
i
d L
d
ρ
ρ
(4.21)
L j (ρ) being the Laguerre polynomials,
L j (r) =
(
)
j
j
p
j
d
e
e
d
ρ
−
ρ
ρ
(4.22)
Here R n, l (r) are normalized to satisfy the condition
∫ [R n,l (r)]
2
r
2
dr = 1
(4.23)
Collecting the radial and angular parts, the solutions are
φ n,l,m (ρ, θ, φ) = R n,l (r) Y l
m
(θ, φ)
(4.24)
with
E n = –
2
2
2
2
0
1
4
2
r
m
Ze
n




πε


(4.25)
The first four solutions are
φ 1,0,0 (r) =
3/ 2
1
1/ 2
1
1
exp (
/ )
Z
Zr a
a
 
−
 
π  
φ 2,0,0 (r) =
(
)
3/ 2
1
1/ 2
1
1
1
2
exp
/2
(32 )
Z
Z r
Zr a
a
a
  

−
−
  

π
  

φ 2,1,0 (r) =
(
)
3/ 2
1
1/ 2
1
1
1
exp
/ 2 cos
(32 )
Z
Zr
Zr a
a
a
 
−
θ
 
π
 
(4.26)
φ 2,1,1 (r) =
3/ 2
1
12
1
1
1
exp (
/ 2 ) sin exp ( )
(64 )
Z
Zr
Zr a
i
a
a
 
−
θ
φ
 
π  
The wave functions of one-electron atom are characterized by three quantum
numbers. These are the principal quantum number n, the angular momentum
quantum number l which determines the angular momentum, and the magnetic