Page 173 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

164 The hydrogen atom
^ ^
A ë B ë S ëÿ1,l ˆ [ë(ë ÿ 1) ÿ l(l ‡ 1)]S ëÿ1,l (6:30)
when ë is replaced by ë ÿ 1 in equation (6.24).
^
If we operate on both sides of equation (6.29) with the operator A ë ,we
obtain
^ ^ ^ ^
A ë B ë A ë S ël ˆ [ë(ë ÿ 1) ÿ l(l ‡ 1)]A ë S ël (6:31)
Comparison of this result with equation (6.30) leads to the conclusion that
^
A ë S ël and S ëÿ1,l are, except for a multiplicative constant, the same function.
We implicitly assume here that S ël is uniquely determined by only two
parameters, ë and l. Accordingly, we may write
^
A ë S ël ˆ a ël S ëÿ1,l (6:32)
where a ël is a numerical constant, dependent in general on the values of ë and
l, to be determined by the requirement that S ël and S ëÿ1,l be normalized.
^
Without loss of generality, we can take a ël to be real. The function A ë S ël is an
eigenfunction of the operator in equation (6.24) with eigenvalue decreased by
^
one. Thus, the operator A ë transforms the eigenfunction S ël determined by ë, l
into the eigenfunction S ëÿ1,l determined by ë ÿ 1, l. For this reason the
^
operator A ë is a lowering ladder operator.
Following an analogous procedure, we now operate on both sides of equation
^
(6.30) with the operator B ë to obtain
^ ^ ^ ^ (6:33)
B ë A ë B ë S ëÿ1,l ˆ [ë(ë ÿ 1) ÿ l(l ‡ 1)]B ë S ëÿ1,l
^
Comparing equations (6.29) and (6.33) shows that B ë S ëÿ1,l and S ël are
proportional to each other
^ (6:34)
B ë S ëÿ1,l ˆ b ël S ël
where b ël is the proportionality constant, assumed real, to be determined by the
^
requirement that S ëÿ1,l and S ël be normalized. The operator B ë transforms the
eigenfunction S ëÿ1,l into the eigenfunction S ël with eigenvalue ë increased by
^
one. Accordingly, the operator B ë is a raising ladder operator.
The next step is to evaluate the numerical constants a ël and b ël . In order to
accomplish these evaluations, we must ®rst investigate some mathematical
properties of the eigenfunctions S ël (r).

Orthonormal properties of S ël (r)
Although the functions R nl (r) according to equation (6.20) form an orthogonal
2
set with w(r) ˆ r , the orthogonal relationships do not apply to the set of
2
functions S ël (r) with w(r) ˆ r . Since the variable r introduced in equation
(6.22) depends not only on r, but also on the eigenvalue E, or equivalently on
ë, the situation is more complex. To determine the proper orthogonal relation-
ships for S ël (r), we express equation (6.24) in the form