Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics


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Thus, we may
write \u142(r, \u141, j) as
\u142(r, \u141, j) ˆ R(r)Ylm(\u141, j) (6:16)
Substitution of equations (6.13), (6.14), (6.15b), and (6.16) into (6.15a) gives
H^ l R(r) ˆ ER(r) (6:17)
where
H^ l ˆ ÿ "
2
2ìr2
d
dr
r2
d
dr
\ufffd \ufffd
ÿ l(l ‡ 1)
\ufffd \ufffd
ÿ Ze9
2
r
(6:18)
and where the common factor Ylm(\u141, j) has been divided out.
6.3 The radial equation
Our next task is to solve the radial equation (6.17) to obtain the radial function
R(r) and the energy E. The many solutions of the differential equation (6.17)
depend not only on the value of l, but also on the value of E. Therefore, the
solutions are designated as REl(r). Since the potential energy ÿZe92=r is
always negative, we are interested in solutions with negative total energy, i.e.,
where E < 0. It is customary to require that the functions REl(r) be normal-
6.3 The radial equation 161
ized. Since the radial part of the volume element in spherical coordinates is
r2 dr, the normalization criterion is…1
0
[REl(r)]
2 r2 dr ˆ 1 (6:19)
Through an explicit integration by parts, we can show that…1
0
REl(r)[ H^ l RE9 l(r)]r
2 dr ˆ
…1
0
RE9 l(r)[ H^ l REl(r)]r
2 dr
Thus, the operator H^ l is hermitian and the radial functions REl(r) constitute an
orthonormal set with a weighting function w(r) equal to r2…1
0
REl(r)RE9 l(r)r
2 dr ˆ äEE9 (6:20)
where äEE9 is the Kronecker delta and equation (6.19) has been included.
We next make the following conventional change of variables
º ˆ ìZe9
2
&quot;(ÿ2ìE)1=2 (6:21)
r ˆ 2(ÿ2ìE)
1=2 r
&quot;
ˆ 2ìZe9
2 r
º&quot;2
ˆ 2Zr
ºaì
(6:22)
where aì ˆ &quot;2=ìe92. We also make the substitution
REl(r) ˆ 2Zºaì
\ufffd \ufffd3=2
Sº l(r) (6:23)
Equations (6.17) and (6.18) now take the form
r2
d2
dr2
‡ 2r d
dr
‡ ºrÿ r
2
4
 !
Sº l ˆ l(l ‡ 1)Sº l (6:24)
where the first term has been expanded and the entire expression has been
multiplied by r2.
To be a suitable wave function, Sº l(r) must be well-behaved, i.e., it must be
continuous, single-valued, and quadratically integrable. Thus, rSº l vanishes
when r!1 because Sº l must vanish sufficiently fast. Since Sº l is finite
everywhere, rSº l also vanishes at r ˆ 0. Substitution of equations (6.22) and
(6.23) into (6.19) shows that Sº l(r) is normalized with a weighting function
w(r) equal to r2 …1
0
[Sº l(r)]2r2 dr ˆ 1 (6:25)
Equation (6.24) may be solved by the Frobenius or series solution method as
presented in Appendix G. However, in this chapter we employ the newer
procedure using ladder operators.
162 The hydrogen atom
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
angular momentum.1 We define the operators A^º and B^º as
A^º \ufffd ÿr d
dr
ÿ r
2
‡ ºÿ 1 (6:26a)
B^º \ufffd r d
dr
ÿ r
2
‡ º (6:26b)
We now show that the operator A^º is the adjoint of B^º and vice versa. Thus,
neither A^º nor B^º is hermitian. For any arbitrary well-behaved functions f (r)
and g(r), we consider the integral…1
0
f (r)[A^º g(r)] dr ˆ ÿ
…1
0
f r
dg
dr
dr‡
…1
0
f ÿ r
2
‡ ºÿ 1
\ufffd \ufffd
g dr
where (6.26a) has been used. Integration by parts of the first term on the right-
hand side with the realization that the integrated part vanishes yields…1
0
f A^º g dr ˆ
…1
0
g
d
dr
(r f ) dr‡
…1
0
f ÿ r
2
‡ ºÿ 1
\ufffd \ufffd
g dr
ˆ
…1
0
g r
d
dr
ÿ r
2
‡ º
\ufffd \ufffd
f dr
Substitution of (6.26b) gives…1
0
f (r)[A^º g(r)] dr ˆ
…1
0
g(r)[B^º f (r)] dr (6:27)
showing that, according to equation (3.33)
A^
y
º ˆ B^º, B^yº ˆ A^º
We readily observe from (6.26a) and (6.26b) that
B^ºA^º ˆ ÿr2 d
2
dr2
ÿ 2r d
dr
ÿ ºr‡ r
2
4
‡ º(ºÿ 1) (6:28a)
A^º B^º ˆ ÿr2 d
2
dr2
ÿ 2r d
dr
ÿ (ºÿ 1)r‡ r
2
4
‡ º(ºÿ 1) (6:28b)
Equation (6.24) can then be written in the form
B^ºA^ºSº l ˆ [º(ºÿ 1)ÿ l(l ‡ 1)]Sº l (6:29)
showing that the functions Sº l(r) are also eigenfunctions of B^ºA^º. From
equation (6.28b) we obtain
1 We follow here the treatment by D. D. Fitts (1995) J. Chem. Educ. 72, 1066. However, the definitions of the
lowering operator and the constants aºl and bºl have been changed.
6.3 The radial equation 163
A^º B^ºSºÿ1, l ˆ [º(ºÿ 1)ÿ l(l ‡ 1)]Sºÿ1, l (6:30)
when º is replaced by ºÿ 1 in equation (6.24).
If we operate on both sides of equation (6.29) with the operator A^º, we
obtain
A^º B^ºA^ºSº l ˆ [º(ºÿ 1)ÿ l(l ‡ 1)]A^ºSº l (6:31)
Comparison of this result with equation (6.30) leads to the conclusion that
A^ºSº l and Sºÿ1, l are, except for a multiplicative constant, the same function.
We implicitly assume here that Sº l is uniquely determined by only two
parameters, º and l. Accordingly, we may write
A^ºSº l ˆ aº lSºÿ1, l (6:32)
where aº l is a numerical constant, dependent in general on the values of º and
l, to be determined by the requirement that Sº l and Sºÿ1, l be normalized.
Without loss of generality, we can take aº l to be real. The function A^ºSº l is an
eigenfunction of the operator in equation (6.24) with eigenvalue decreased by
one. Thus, the operator A^º transforms the eigenfunction Sº l determined by º, l
into the eigenfunction Sºÿ1, l determined by ºÿ 1, l. For this reason the
operator A^º is a lowering ladder operator.
Following an analogous procedure, we now operate on both sides of equation
(6.30) with the operator B^º to obtain
B^ºA^º B^ºSºÿ1, l ˆ [º(ºÿ 1)ÿ l(l ‡ 1)]B^ºSºÿ1, l (6:33)
Comparing equations (6.29) and (6.33) shows that B^ºSºÿ1, l and Sº l are
proportional to each other
B^ºSºÿ1, l ˆ bº lSº l (6:34)
where bº l is the proportionality constant, assumed real, to be determined by the
requirement that Sºÿ1, l and Sº l be normalized. The operator B^º transforms the
eigenfunction Sºÿ1, l into the eigenfunction Sº l with eigenvalue º increased by
one. Accordingly, the operator B^º is a raising ladder operator.
The next step is to evaluate the numerical constants aº l and bº l. In order to
accomplish these evaluations, we must first investigate some mathematical
properties of the eigenfunctions Sº l(r).
Orthonormal properties of Sº l(r)
Although the functions Rnl(r) according to equation (6.20) form an orthogonal
set with w(r) ˆ r2, the orthogonal relationships do not apply to the set of
functions Sº l(r) with w(r) ˆ r2. Since the variable r introduced in equation
(6.22) depends not only on r, but also on the eigenvalue E, or equivalently on
º, the situation is more complex. To determine the proper orthogonal relation-
ships for Sº l(r), we express equation (6.24) in the form
164 The hydrogen atom
H^9lSº l ˆ ÿºSº l (6:35)
where H^9l is defined by
H^9l ˆ r d
2
dr2
‡ 2 d
dr
ÿ r
4
ÿ l(l ‡ 1)
r
(6:36)
By means of integration by parts, we can readily show that this operator H^9l is
hermitian for a weighting function w(r) equal to r, thereby implying the
orthogonal relationships…1
0
Sº l(r)Sº9 l(r)r dr ˆ 0 for º 6ˆ º9 (6:37)
In order to complete the characterization of integrals of Sº l(r), we need to
consider the case where º ˆ º9 for w(r) ˆ r. Recall that the functions Sº l(r)
are normalized for w(r) ˆ r2 as expressed in equation (6.25). The same result
does not apply for w(r) ˆ r. We begin by expressing the desired integral in a
slightly different form…1
0
[Sº l(r)]2r dr ˆ 12
…1
0
[Sº l(r)]2 d(r2)
Integration of the right-hand side by parts gives…1
0
[Sº l(r)]2r dr ˆ 12
\ufffd
r2[Sº l(r)]2
\ufffd1
0
ÿ
…1
0
r2Sº l
d
dr
Sº l
\ufffd \ufffd
dr
If Sº l(r) is well-behaved, the integrated term vanishes. From equation (6.26a)
we may write
r
d
dr
ˆ ÿA^º ÿ r
2
‡ ºÿ 1
so that
r
d
dr
Sº l ˆ ÿA^ºSº l ÿ 12rSº l ‡ (ºÿ 1)Sº l
ˆ ÿaº lSºÿ1, l ÿ 12rSº l ‡ (ºÿ 1)Sº l
where equation (6.32) has been introduced.