Page 338 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Appendix H

Recurrence relation for hydrogen-atom expectation

values

k
The expectation values hr i nl of various powers of the radial variable r for a hydrogen-
like atom with quantum numbers n and l are given by equation (6.69)

1
k k 2 2
hr i nl ˆ r [R nl (r)] r dr (H:1)
0
where R nl (r) are the solutions of the radial differential equation (6.17). In this
appendix, we show that these expectation values are related by the recurrence relation
2 2
k ‡ 1 k a 0 kÿ1 1 ÿ k a 0 kÿ2
hr i nl ÿ (2k ‡ 1) hr i nl ‡ kl(l ‡ 1) ‡ hr i nl ˆ 0 (H:2)
n 2 Z 4 Z 2
To simplify the notation, we de®ne the real function u(r)by u rR nl (r) and denote
the ®rst and second derivatives of u(r)by u9 and u0. Equation (H.1) then takes the
form

1
k k 2
hr i nl ˆ r u dr (H:3)
0
Since we have

dR(r) u9 u d 2 dR(r)
ˆ ÿ , r ˆ ru0
dr r r 2 dr dr
equation (6.17) becomes
" #
l(l ‡ 1) 2Z Z 2
u0 ˆ ÿ ‡ u (H:4)
2 2
r 2 a 0 r n a
0
where equation (6.57) for the energy E n has also been introduced.
Before beginning the direct derivation of equation (H.2), we ®rst derive a useful
relationship. Consider the integral

1
í
r uu9 dr
0
and integrate by parts

1 1 1 d
í
í
r uu9 dr ˆ r u ÿ u (r u)dr
í 2

0 0 0 dr
The integrated part vanishes because R(r) ! 0 exponentially as r !1 and u(r) ! 0
as r ! 0. Expanding the derivative within the integral on the right-hand side, we have
329