Elements of Modern Physics
104
2. Define
u (r) = R (r) exp [(–2m r E/
2
)
1/2
r]
(4.15)
and consider a series solution for u (r),
u (r) =
k
k
k s
b r
∞
=
∑ , k = s, s + 1, ..., b s ≠ 0
(4.16)
3. Impose the condition that the solution for u(r) does not alter the asymptotic
behaviour of R(r). This constraint on the asymptotic behaviour leads to
the result that the series in Eq. (4.16) must terminate for a finite value of
k, say k = s + p, p is an integer, i.e. b k = 0 for k > (s + p).
Substituting the above expressions in Eq. (4.13), and equating the coefficients
of the same powers of r, in particular of r
k–2
, gives
b k
2
[ ( 1)
( 1)]
2 r
l l
k k
m
+ −
+
=
1/ 2
2
2
1
2
0
2
4
r
k
r
m E
Ze
k
b
m
−
−
−
πε
(4.17)
Since b s–1 = 0, we have s = l or s = – l – 1. For l ≠ 0, the s = – l – 1 solutions
are not normalizable and hence are discarded. For l = 0, the first term in
Eq. (4.16) for the s = –l –1 solution, is b –1 r
–1
. However, Eq. (4.17) for k = 0
gives b –1 = 0 which is inconsistent. Hence, only the s = l solution need be
considered. The requirement that the series terminates, i.e.
b k = 0 for k = l + p + 1, p ≥ 0, then leads to
2
2
0
4
r
m Ze
πε
= (l + p + 1)
1/ 2
2
2 r
m E
−
(4.18)
Thus the negative-energy solutions exist only for energies
E n = –
2
2
2
2
0
1 ,
4
2
r
m
Ze
n
n
πε
= 1, 2, ...
(4.19)
n = l + p + 1, p = 0, 1, ...
with l + p being the highest power of r in the series solution for u(r). The wave
functions corresponding to the solutions are related to Laguerre polynomials,
and are given by
R n,l (r) =
(
)
(
)
1/ 2
3
/ 2
2 1
3
1
1 !
2
( )
2
!
−
+
+
− −
ρ
ρ
+
l
p p
l
n l
n l
Z
e
L
na
n n l
(4.20)
104
2. Define
u (r) = R (r) exp [(–2m r E/
2
)
1/2
r]
(4.15)
and consider a series solution for u (r),
u (r) =
k
k
k s
b r
∞
=
∑ , k = s, s + 1, ..., b s ≠ 0
(4.16)
3. Impose the condition that the solution for u(r) does not alter the asymptotic
behaviour of R(r). This constraint on the asymptotic behaviour leads to
the result that the series in Eq. (4.16) must terminate for a finite value of
k, say k = s + p, p is an integer, i.e. b k = 0 for k > (s + p).
Substituting the above expressions in Eq. (4.13), and equating the coefficients
of the same powers of r, in particular of r
k–2
, gives
b k
2
[ ( 1)
( 1)]
2 r
l l
k k
m
+ −
+
=
1/ 2
2
2
1
2
0
2
4
r
k
r
m E
Ze
k
b
m
−
−
−
πε
(4.17)
Since b s–1 = 0, we have s = l or s = – l – 1. For l ≠ 0, the s = – l – 1 solutions
are not normalizable and hence are discarded. For l = 0, the first term in
Eq. (4.16) for the s = –l –1 solution, is b –1 r
–1
. However, Eq. (4.17) for k = 0
gives b –1 = 0 which is inconsistent. Hence, only the s = l solution need be
considered. The requirement that the series terminates, i.e.
b k = 0 for k = l + p + 1, p ≥ 0, then leads to
2
2
0
4
r
m Ze
πε
= (l + p + 1)
1/ 2
2
2 r
m E
−
(4.18)
Thus the negative-energy solutions exist only for energies
E n = –
2
2
2
2
0
1 ,
4
2
r
m
Ze
n
n
πε
= 1, 2, ...
(4.19)
n = l + p + 1, p = 0, 1, ...
with l + p being the highest power of r in the series solution for u(r). The wave
functions corresponding to the solutions are related to Laguerre polynomials,
and are given by
R n,l (r) =
(
)
(
)
1/ 2
3
/ 2
2 1
3
1
1 !
2
( )
2
!
−
+
+
− −
ρ
ρ
+
l
p p
l
n l
n l
Z
e
L
na
n n l
(4.20)