Fitts D.D. - Principles of Quantum Mechanics[c] As Applied to Chemistry and Chemical Physics (1999)(en)

162 The hydrogen atom

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 1		(6:19)
Through an explicit integration by parts, we can show that
1		1
…0	REl(r)[H^ l RE9l(r)]r2 dr …0		RE9l(r)[ H^ l REl(r)]r2 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)RE9l(r)r2 dr 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

ë

ìZe92

(6:21)

"(ÿ2ìE)1=2

r

2(ÿ2ìE)1=2 r

2ìZe92 r

2Zr

(6:22)

ë"2

ëaì

"

where aì "2=ìe92. We also make the substitution

REl(r)

2Z

3=2

Sël(r)

(6:23)

ëaì

Equations (6.17) and (6.18) now take the form

r2

d2

2r

d

ër ÿ

r2

!Sël

l(l 1)Sël

(6:24)

dr2

dr

4

where the ®rst term has been expanded and the entire expression has been multiplied by r2.

To be a suitable wave function, Sël(r) must be well-behaved, i.e., it must be continuous, single-valued, and quadratically integrable. Thus, rSël vanishes when r ! 1 because Sël must vanish suf®ciently fast. Since Sël is ®nite everywhere, rSël also vanishes at r 0. Substitution of equations (6.22) and (6.23) into (6.19) shows that Sël(r) is normalized with a weighting function w(r) equal to r2

1
…0	[Sël(r)]2r2 dr 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.

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

^

^

as

angular momentum. We de®ne the operators Aë and Bë

^

d

r

Aë ÿr

dr

ÿ

2

ë ÿ 1

(6:26a)

^

d

ÿ

r

ë

Bë r dr

2

(6:26b)

We now show that the operator

^

^

Aë

is the adjoint of Bë and vice versa. Thus,

^

^

neither Aë nor Bë is hermitian. For any arbitrary well-behaved functions f (r)

and g(r), we consider the integral

f r dr dr …0

f ÿ

2 ë ÿ 1 g dr

…0

f (r)[A^ë g(r)] dr ÿ…0

1

1

dg

1

r

where (6.26a) has been used. Integration by parts of the ®rst term on the righthand side with the realization that the integrated part vanishes yields

1

1

d

1

r

…0

f A^ë g dr …0

g

(rf ) dr …0

f ÿ

ë ÿ 1 g dr

dr

2

1

g r

d

r

ë f dr

…0

ÿ

dr

2

Substitution of (6.26b) gives

1

1

…0

f (r)[A^ë g(r)] dr

…0

g(r)[B^ë f (r)] dr

(6:27)

showing that, according to equation (3.33)

^y

^

^y

^

A ë Bë,

B ë Aë

We readily observe from (6.26a) and (6.26b) that

^ ^

2

d2

d

r2

BëAë

ÿr

dr2

ÿ 2r

dr

ÿ ër

4

ë(ë ÿ 1)

(6:28a)

^ ^

2

d2

d

r2

Aë Bë ÿr

dr2 ÿ

2r dr ÿ (ë ÿ 1)r 4 ë(ë ÿ 1)

(6:28b)

Equation (6.24) can then be written in the form

^ ^

[ë(ë ÿ 1) ÿ l(l 1)]Sël

(6:29)

BëAëSël

showing that

the functions

Sël(r)

are

also

eigenfunctions of

^

^

BëAë. From

equation (6.28b) we obtain

1We 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 aël and bël have been changed.

164	The hydrogen atom
^ ^		ÿ l(l 1)]Sëÿ1,l	(6:30)
Aë BëSëÿ1,l [ë(ë ÿ 1)
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			Aë, we
obtain
^ ^	^	^	(6:31)
Aë BëAëSël [ë(ë ÿ 1)		ÿ l(l 1)]AëSël

Comparison of this result with equation (6.30) leads to the conclusion that

^	and Sëÿ1,l are, except for a multiplicative constant, the same function.
AëSël

We implicitly assume here that Sël is uniquely determined by only two parameters, ë and l. Accordingly, we may write

^	(6:32)
AëSël aël Sëÿ1,l

where aël 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 aël to be real. The function AëSël is an eigenfunction of the operator in equation (6.24) with eigenvalue decreased by

one. Thus, the operator		^	determined by ë, l
		Aë transforms the eigenfunction Sël
into the	eigenfunction	Sëÿ1,l determined by ë ÿ 1, l. For this reason the
operator	^
	Aë is a lowering ladder operator.

Following an analogous procedure, we now operate on both sides of equation

^
(6.30) with the operator Bë to obtain
^ ^ ^	ÿ l(l	^		(6:33)
BëAë BëSëÿ1,l [ë(ë ÿ 1)		1)]BëSëÿ1,l
Comparing equations (6.29) and (6.33) shows that			^	and Sël are
			BëSëÿ1,l
proportional to each other
^				(6:34)
BëSëÿ1,l bël Sël

where bël is the proportionality constant, assumed real, to be determined by the

requirement that Sëÿ1,l and Sël be normalized. The operator					^
					Bë transforms the
eigenfunction S	ëÿ1,l	into the eigenfunction S	ël	with eigenvalue ë increased by
		^
one. Accordingly, the operator Bë is a raising ladder operator.

The next step is to evaluate the numerical constants aël and bël. In order to accomplish these evaluations, we must ®rst investigate some mathematical properties of the eigenfunctions Sël(r).

Orthonormal properties of Sël(r)

Although the functions Rnl(r) according to equation (6.20) form an orthogonal set with w(r) r2, the orthogonal relationships do not apply to the set of functions Sël(r) with w(r) r2. 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 relationships for Sël(r), we express equation (6.24) in the form

6.3 The radial equation

165

^ 9 ÿë

H l Sël Sël

^	is de®ned by
where H9l
	H^ 9		d2		2	d			r		l(l 1)
		r dr2				dr		ÿ	4	ÿ	r
	l

(6:35)

(6:36)

By means of integration by parts, we can readily show that this operator ^ 9 is

H l hermitian for a weighting function w(r) equal to r, thereby implying the

orthogonal relationships

…1

Sël(r)Së9l(r)r dr 0 for ë 6ë9 (6:37)

0

In order to complete the characterization of integrals of Sël(r), we need to consider the case where ë ë9 for w(r) r. Recall that the functions Sël(r) are normalized for w(r) r2 as expressed in equation (6.25). The same result does not apply for w(r) r. We begin by expressing the desired integral in a slightly different form

	1			1
	…0	[Sël(r)]2r dr 21…0			[Sël(r)]2 d(r2)
Integration of the right-hand side by parts gives							…0	r Sël	dr Sël dr
…0	[Sël(r)] r dr 2 r [Sël(r)]			0		ÿ
1	2	1	2	2	1		1	2		d

If Sël(r) is well-behaved, the integrated term vanishes. From equation (6.26a) we may write

		r	d	^		r	ë ÿ 1
			dr	ÿAë ÿ		2
so that
	d			^	1
r	dr	Sël ÿAëSël ÿ			2rSël (ë ÿ 1)Sël

ÿaël Sëÿ1,l ÿ 12rSël (ë ÿ 1)Sël

where equation (6.32) has been introduced. The integral then takes the form

1	1		1
…0	[Sël(r)]2r dr aël…0	Sël Sëÿ1,lr dr 21…0		[Sël]2r2 dr
		1
	ÿ (ë ÿ 1)…0		[Sël]2r dr

Since the ®rst integral on the right-hand side vanishes according to equation (6.37) and the second integral equals unity according to (6.25), the result is

166		The hydrogen atom
	1		1
	…0	[Sël(r)]2r dr			(6:38)
			2ë
Combining equation (6.38) with (6.37), we obtain
1			1
…0	Sël(r)Së9l(r)r dr			äëë9	(6:39)
			2ë

Evaluation of the constants aël and bël

To evaluate the numerical constant aël, which is de®ned in equation (6.32), we square both sides of (6.32), multiply through by r, and integrate with respect to r to obtain

1

1

…0

r(A^ëSël)(A^ëSël) dr aë2l…0

(Sëÿ1,l)2r dr

(6:40)

^

Application of equation (6.27) with f rAëSël and g Sël to the left-hand

side and substitution of equation (6.38) on the right-hand side give

1

…0

Sël B^ë(rA^ëSël) dr aë2l=2(ë ÿ 1)

(6:41)

^

^

The expression Bë(rAëSël) may be simpli®ed as follows

d

r

B^ë(rA^ëSël) r

(rA^ëSël) ÿ

ë rA^ëSël

dr

2

rA^ëSël

r2 dr (A^ëSël) r ÿ

2 ë A^ëSël

d

r

^

^ ^

rAëSël

rBëAëSël

raël Sëÿ1,l [ë(ë ÿ 1) ÿ l(l 1)]rSël

where equations (6.26b), (6.32), and (6.29) have been used. When this result is substituted back into (6.41), we have

1	1
aël…0	Sël Sëÿ1,lr dr [ë(ë ÿ 1) ÿ l(l 1)]…0	Së2lr dr aë2l=2(ë ÿ 1) (6:42)

According to equation (6.39), the ®rst integral vanishes and the second integral equals (2ë)ÿ1, giving the result

a2

ë ÿ 1

[ë(ë

ÿ

1)

ÿ

l(l

1)]

ë

ël

(ë

l

(6:43)

ë ÿ 1

l)(ë

ÿ

ÿ

1)

ë

Substitution into (6.32) gives

6.3

The radial equation

167

A^

S

ë ÿ 1

(ë

l)(ë

ÿ

l

ÿ

1)

1=2 S

ëÿ1,l

(6:44)

ë

ël

ë

where we have arbitrarily taken the positive square root.

The numerical constant bël, de®ned in equation (6.34), may be determined by an analogous procedure, beginning with the square of both sides of equation (6.34) and using equations (6.27), (6.26a), (6.34), (6.30), and (6.39). We obtain

ë							ë
bë2l		[ë(ë ÿ 1) ÿ l(l					1)]		(ë l)(ë ÿ l ÿ 1)	(6:45)
	ë ÿ 1							ë ÿ 1
so that equation (6.34) becomes
	B^ëSëÿ1,l			ë			(ë l)(ë ÿ l ÿ 1) 1=2 Së,l			(6:46)
			ë	ÿ		1

Taking the positive square root here will turn out to be consistent with the choice in equation (6.44).

Quantization of the energy

The parameter ë is positive, since otherwise the radial variable r, which is inversely proportional to ë, would be negative. Furthermore, the parameter ë cannot be zero if the transformations in equations (6.21), (6.22), and (6.23) are to remain valid. To ®nd further restrictions on ë we must consider separately

the cases where l 0 and where l > 1.
For l 0, equation (6.44) takes the form
^	(6:47)
AëSë0 (ë ÿ 1)Sëÿ1,0

Suppose we begin with a suitably large value of ë, say î, and continually apply the lowering operator to both sides of equation (6.47) with ë î

^	^
Aîÿ1AîSî0 (î ÿ 1)(î ÿ 2)Sîÿ2,0
^	^	^	(î ÿ 1)(î ÿ 2)(î ÿ 3)Sîÿ3,0
Aîÿ2Aîÿ1AîSî0

...

Eventually this procedure produces an eigenfunction Sîÿk,0, k being a positive integer, such that 0 ,(î ÿ k) < 1. The next step in the sequence would give a function Sîÿkÿ1,0 or Së0 with ë (î ÿ k ÿ 1) < 0, which is not allowed. Thus, the sequence must terminate with the condition

^	(î ÿ k ÿ 1)Sîÿkÿ1,0	0
Aîÿk Sîÿk,0

which can only occur if (î ÿ k) 1. Thus, î must be an integer and the minimum value of ë for l 0 is ë 1.

For the situations in which l > 1, we note that the quantities a2ël in equation

168 The hydrogen atom

(6.43) and b2ël in equation (6.45), being squares of real numbers, must be positive. Consequently, the factor (ë ÿ l ÿ 1) must be positive, so that ë > (l 1).

We now select some appropriately large value î of the parameter ë in equation (6.44) and continually apply the lowering operator to both sides of the equation in the same manner as in the l 0 case. Eventually we obtain Sîÿk,l such that (l 1) < (î ÿ k) , (l 2). The next step in the sequence would give Sîÿkÿ1,l or Sël with ë (î ÿ k ÿ 1) , (l 1), which is not allowed, so that the sequence must be terminated according to

^

Aîÿk Sîÿk,l aîÿk,l Sîÿkÿ1,l

ÿ

ÿ

ÿ

ÿ

îÿkÿ1,l

î ÿ k

1=2 S

î ÿ k ÿ 1

(î

k

l)(î

k

l

1)

0

for some value of k. Thus, î must be an integer for aîÿk,l to vanish. As k increases during the sequence, the constant aîÿk,l vanishes when

k(î ÿ l ÿ 1) or (î ÿ k) (l 1). The minimum value of ë is then l 1. Combining the conclusions of both cases, we see that the minimum value of

ëis l 1 for l 0, 1, 2, . . . Beginning with the value ë l 1, we can apply equation (6.46) to yield an in®nite progression of eigenfunctions Snl(r) for each

value of l (l 0, 1,	2, . . .),	where ë can take on only integral values,
ë n l 1, l 2,	l 3, . . .	Since î in both cases was chosen arbitrarily

and was shown to be an integer, equation (6.46) generates all of the eigenfunctions Sël(r) for each value of l. There are no eigenfunctions corresponding to non-integral values of ë. Since ë is now shown to be an integer n, in the remainder of this presentation we replace ë by n.

Solving equation (6.21) for the energy E and replacing ë by n, we obtain the

quantized energy levels for the hydrogen-like atom
En ÿ	ìZ2 e94	ÿ	Z2 e92		n 1, 2, 3, . . .	(6:48)
				,
	2"2 n2		2aì n2

These energy levels agree with the values obtained in the earlier Bohr theory. Electronic energies are often expressed in the unit electron volt (eV). An

electron volt is de®ned as the kinetic energy of an electron accelerated through a potential difference of 1 volt. Thus, we have

1 eV (1:602 177 3 10ÿ19 C) 3 (1:000 000 V) 1:602 177 3 10ÿ19 J

The ground-state energy E1 of a hydrogen atom (Z 1) as given by equation (6.48) is

E1 ÿ2:178 68 3 10ÿ18 J ÿ13:598 eV

6.3 The radial equation

169

This is the energy required to remove the electron from the ground state of a hydrogen atom to a state of zero kinetic energy at in®nity and is also known as the ionization potential of the hydrogen atom.

Determination of the eigenfunctions

Equation (6.47) may be used to obtain the ground state (n 1, l 0) eigen-

						^
function S10(r). Introducing the de®nition of An in equation (6.26a), we have
A^1 S10	ÿ r dr 2 S10 0
				d		r
or
		dS10				S10
			ÿ
		dr				2

from which it follows that

S10 ceÿr=2 2ÿ1=2eÿr=2

where the constant c of integration was evaluated by applying equations (6.25), (A.26), and (A.28).

The series of eigenfunctions S20, S30, . . . are readily obtained from equations

(6.46) and (6.26b) with ë n, l 0

B^ n Snÿ1,0 r

d

ÿ

r

n Snÿ1,0 nSn0

dr

2

Thus, S20 is

r

2 2ÿ1=2eÿr=2

S20

1

d

ÿ

r

2

dr

2

1

(2 ÿ r)eÿr=2

p2

and S30 is

2

•••

ÿ 2 3 2p2 (2 ÿ r)eÿr=2

S30

3 r dr

1

d

r

1

1

(6

6

2)eÿr•••=

2

6p2

ÿ r r

and so forth ad in®nitum.

Each eigenfunction is normalized.

•••

The eigenfunctions for l . 0 are determined in a similar manner. A general formula for the eigenfunction Sl 1,l, which is the starting function for evaluating the series Snl with ®xed l, is obtained from equations (6.44) and (6.26a) with l n l 1

170 The hydrogen atom

			d		r
A^l 1 Sl 1,l ÿ r							l Sl 1,l 0
			dr		2
or		l ÿ 2 Sl 1,l
r	dr
	dSl 1,l					r
Integration gives
Sl 1,l [(2l 2)!]ÿ1=2rleÿr=2									(6:49)

where the integration constant was evaluated using equations (6.25), (A.26), and (A.28).

The eigenfunction S21 from equation (6.49) is

1

reÿr=2

S21

p6

•••1 give

and equations (6.46) and (6.26b) for l2

S31 p6 r dr ÿ

2 3 2p6 reÿr=2

1

d

r

1

1•••

r=2

•••

(4 ÿ r)reÿ

12

ÿ

r•••••

dr ÿ 2

12

S41

3

r

d

r

4

1

(4

r)reÿr=2

40

1

(20 ÿ 10r r2)reÿr=2

p30

..

8

•••••

.

The functions S31, S41, . . . are automatically normalized as speci®ed by equation (6.25). The normalized eigenfunctions Snl(r) for l 2, 3, 4, . . . with

n > (l 1) are obtained by the same procedure.	^
A general formula for Snl involves the repeated application of	Bk	for

k l 2, l 3, . . . , n ÿ 1, n to Sl 1,l in equation (6.49). The raising operator must be applied (n ÿ l ÿ 1) times. The result is

nl ( nl)ÿ1( nÿ1,l)ÿ1 . . . ( l 2,l)ÿ1 ^ n ^ nÿ1 . . . ^ l 2 l 1,l

B S

S b b b B B

(l 1)(2l 1)!

1=2

d

r

n

n(n l)!(n ÿ l ÿ 1)!(2l 2)!

r dr

ÿ

2

d

r

d

r

3

r

ÿ

n ÿ 1

r

ÿ

l 2 rleÿr=2 (6:50)

dr

2

dr

2

Just as equation (6.46) can be used to go `up the ladder' to obtain Sn,l from

6.3 The radial equation

171

Snÿ1,l, equation (6.44) allows one to go `down the ladder' and obtain Snÿ1,l from Snl. Taking the positive square root in going from equation (6.43) to (6.44) is consistent with taking the positive square root in going from equation (6.45) to (6.46); the signs of the functions Snl are maintained in the raising and lowering operations. In all cases the ladder operators yield normalized eigenfunctions if the starting eigenfunction is normalized.

The radial factors of the hydrogen-like atom total wave functions ø(r, è, j) are related to the functions Snl(r) by equation (6.23). Thus, we have

		Z					3=2
R10	2								eÿr=2
		aì
	1						Z	3=2
R20	2p2										(2 ÿ r)eÿr=2
					aì
	1 •••						Z	3=2
R30	9p3										(6 ÿ 6r r2)eÿr=2
					aì
..		•••
.
	1						Z	3=2
R21	2p6										reÿr=2
					aì
	1 •••						Z	3=2
R31	9p6										(4 ÿ r)reÿr=2
					aì
R41	1•••								Z		3=2
	32p15						aì				(20 ÿ 10r r2)reÿr=2
..			•••••
.

and so forth.

A more extensive listing appears in Table 6.1.

Radial functions in terms of associated Laguerre polynomials

The radial functions Snl(r) and Rnl(r) may be expressed in terms of the associated Laguerre polynomials Lkj (r), whose de®nition and mathematical properties are discussed in Appendix F. One method for establishing the relationship between Snl(r) and Lkj (r) is to relate Snl(r) in equation (6.50) to the polynomial Lkj (r) in equation (F.15). That process, however, is long and tedious. Instead, we show that both quantities are solutions of the same differential equation.