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) 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) ER(r) (6:17) where H^ l 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 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 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 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 ìZe9 2 "(ÿ2ìE)1=2 (6:21) r 2(ÿ2ìE) 1=2 r " 2ìZe9 2 r º"2 2Zr ºaì (6:22) where aì "2=ìe92. We also make the substitution REl(r) 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 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 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 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 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 1 0 g d dr (r f ) dr 1 0 f ÿ r 2 ºÿ 1 \ufffd \ufffd g dr 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 1 0 g(r)[B^º f (r)] dr (6:27) showing that, according to equation (3.33) A^ y º B^º, B^yº A^º We readily observe from (6.26a) and (6.26b) that B^ºA^º ÿr2 d 2 dr2 ÿ 2r d dr ÿ ºr r 2 4 º(ºÿ 1) (6:28a) A^º B^º 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 [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 [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 [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 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 [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 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) r2, the orthogonal relationships do not apply to the set of functions Sº l(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 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 ÿºSº l (6:35) where H^9l is defined by H^9l 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 0 for º 6 º9 (6:37) In order to complete the characterization of integrals of Sº l(r), we need to consider the case where º º9 for w(r) r. Recall that the functions Sº l(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 0 [Sº l(r)]2r dr 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 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 ÿA^º ÿ r 2 ºÿ 1 so that r d dr Sº l ÿA^ºSº l ÿ 12rSº l (ºÿ 1)Sº l ÿaº lSºÿ1, l ÿ 12rSº l (ºÿ 1)Sº l where equation (6.32) has been introduced.