Page 172 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.3 The radial equation 163

Ladder operators
We now solve equation (6.24) by means of ladder operators, analogous to the
method used in Chapter 4 for the harmonic oscillator and in Chapter 5 for the
^
^
1
angular momentum. We de®ne the operators A ë and B ë as
d r
^
A ë ÿr ÿ ‡ ë ÿ 1 (6:26a)
dr 2
d r
^ ÿ ‡ ë (6:26b)
B ë r
dr 2
^
^
We now show that the operator A ë is the adjoint of B ë and vice versa. Thus,
^
^
neither A ë nor B ë is hermitian. For any arbitrary well-behaved functions f (r)
and g(r), we consider the integral

1 1 dg 1 r
^
f (r)[A ë g(r)] dr ˆÿ f r dr ‡ f ÿ ‡ ë ÿ 1 g dr
0 0 dr 0 2
where (6.26a) has been used. Integration by parts of the ®rst term on the right-
hand side with the realization that the integrated part vanishes yields

1 1 d 1 r
^
f A ë g dr ˆ g (rf )dr ‡ f ÿ ‡ ë ÿ 1 g dr
0 0 dr 0 2

1 d r
ˆ g r ÿ ‡ ë f dr
0 dr 2
Substitution of (6.26b) gives

1 1
^ ^
f (r)[A ë g(r)] dr ˆ g(r)[B ë f (r)] dr (6:27)
0 0
showing that, according to equation (3.33)
^
^
^
^
y
y
A ë ˆ B ë , B ë ˆ A ë
We readily observe from (6.26a) and (6.26b) that
d 2 d r 2
^ ^ 2 ÿ 2r ÿ ër ‡ ‡ ë(ë ÿ 1) (6:28a)
B ë A ë ˆÿr
dr 2 dr 4
^ ^
A ë B ë ˆÿr 2 d 2 ÿ 2r d ÿ (ë ÿ 1)r ‡ r 2 ‡ ë(ë ÿ 1) (6:28b)
dr 2 dr 4
Equation (6.24) can then be written in the form
^ ^
B ë A ë S ël ˆ [ë(ë ÿ 1) ÿ l(l ‡ 1)]S ël (6:29)
^ ^
showing that the functions S ël (r) are also eigenfunctions of B ë A ë . From
equation (6.28b) we obtain
1 We follow here the treatment by D. D. Fitts (1995) J. Chem. Educ. 72, 1066. However, the de®nitions of the
lowering operator and the constants a ël and b ël have been changed.