Elements of Modern Physics
60
were p ⊥ and p r are components of momentum perpendicular and parallel to the
radius, respectively, and the integrals are over the complete orbit. The first
condition reduces to the Bohr relation,
L = n
(2.88)
The total energy is
2
2
2
2
0
1
2
4
2
=
+
− πε
r
r
r
L
Ze
E
p
m
r
m r
(2.89)
Using Eqs. (2.89) and (2.88), the quantization condition in Eq. (2.87) becomes
max
min
2 2
2
2
1/2
0
2
(2
/
/2
)
−
+
π ε
=
∫
r
r
r
r
dr m E n
r
m Ze
r
kh
(2.90)
The integral can be evaluated using the theory of complex variables and
leads to
1/ 2
2
0
2
2
r
Ze
m
nh kh
E


−
− =


ε 

(2.91)
so that
2
2
2
2
0
1 , (
) 1
4
2
(
)


= −
+ ≥


πε
+


r
m
Ze
E
kn
k n
(2.92)
Thus, for a given value of the principal quantum number (k + n), states with
the same energy exist, for k = 0, ..., k + n – 1. Thus we have what is called as
a degeneracy of order k + n (see. Sec. 3.4).
Example 8
Suppose in addition to the Coulomb attraction, there is a potential energy terms
g/r
2
. Then the expression for the total energy E is
2
2
2
2
2
0
1
2
4
2
=
+
+
− πε
r
r
r
L
g
Ze
E
p
m
r
m r
r
(2.93)
Since L
2
= n
2 2 , this is equivalent to replacing n by (n
2
2
+ 2m r g)
1/2
.
Therefore, the expression for the quantized energy is
2
2
2
2
2 1 / 2 2
0
1
4
2
( (
2
/ ) )
r
r
m
Ze
E
k n
m g


= −


πε
+
+


(2.94)