14.6 The hydrogen atom 407
Then
and the radial equation becomes
(14.77)
This is identical to equation (13.45) for the associated Laguerre functions when
λ 1 = 1 n, a positive integer. The solutions of the radial equation are therefore given by
equation (13.44)
These functions are finite and continuous for all positive values of ρ, and therefore
of r, and they satisfy the boundary condition for bound states. The functions may
be normalized, using equation (13.46), and the resulting normalized radial wave
functions are*
(14.78)
where, becauseα 1 = 1 Z 2 nwhenλ 1 = 1 n,
(14.79)
The allowed values of the quantum numbers are
n 1 = 1 1, 2, 3, = l 1 = 1 0, 1, 2, =, (n 1 − 1 1) (14.80)
The radial functions form an orthonormal set with respect to the weight functionr
2in the interval 01 ≤ 1 r 1 ≤ 1 ∞:
(14.81)
Some of the radial functions are listed in Table 14.2.
Z
02∞RrR rrdr
nl, n l′,,nn′() () =δ
ρ=
2 Z
n
r
Rr
Z
n
nl
nn l
nl,=−
−−!
+!
()
()
{( )}
123321
2
−++eL
lnlρ lρρ
221()
eL
lnl− l+ρ +ρρ
221()
dR
d
dR
d
ll
R
222211
4
0
ρ
ρρ
ρ
λ
ρ
++−
+−
=
()
dR
dr
dR
d
dR
dr
dR
d
=, 24 =
22222α
ρ
α
ρ
*By convention the radial functions (14.78) are defined with a – sign as part of the normalization constant.