Solutions of the Schrödinger equation for an attractive 1/r 6 potential

Transcription

1 PHYSICAL REVIEW A VOLUME 58, NUMBER 3 SEPTEMBER 998 Solutions of the Schrödinger equation for an attractive /r 6 potential Bo Gao Department of Physics and Astronomy, University of Toledo, Toledo, Ohio Received 3 April 998 Mathematical methods are presented that give pairs of linearly independent solutions for an attractive /r 6 potential. These solutions represent a different class of special functions that are important to the understanding of molecular vibration spectra near the dissociation threshold and slow atomic collisions. S PACS numbers: w, 34.0.x, t, f I. INTRODUCTION The knowledge of Coulomb functions has played a key role in our understanding of atomic spectra and electron-ion collisions. It is the cornerstone of quantum defect theory QDT, which provides a systematic understanding of atomic spectra near thresholds and relate properties of bound or quasibound states to properties of electron-ion scattering,2. The availability of Coulomb functions also greatly reduces the configuration space that has to be treated numerically and leads to powerful computational methods such as the eigenchannel R-matrix method 3. The solutions of the Schrödinger equation for an attractive /r 6 potential, to be presented in this paper, play a similarly important role for molecular vibration spectra and atom-atom collisions 4. Consider the radial Schrödinger equation for a C n /r n potential: d2 dr 2 ll r 2 n2 n u r n l r0, where n (2C n / 2 ) /(n2) and 2/ 2. The solutions of this equation are well known for n,2. They are also easily obtained for arbitrary n at 0. In all these cases, Eq. has only a single irregular singularity. What makes the cases of n2 and 0 fundamentally different is the existence of two irregular singularities, one at zero and the other at infinity. For n4, the solutions can be expressed in terms of Mathieu functions 5,6. For n6, however, Eq. cannot be transformed into one that is satisfied by any known special function and different mathematical methods are required. II. SUMMARY OF THE SOLUTION Motivated by an observation of Cavagnero 7, we have found for an attractive /r 6 potential that a pair of linearly independent solutions with energy-independent normalization near the origin can be written as where f and ḡ are another pair of linearly independent solutions given by and f l r m b m r /2 J m 2 2 r/ 6, ḡ l r m b m r /2 Y m 2 2 r/ 6, l cos 0 /2X l sin 0 /2Y l, l sin 0 /2X l cos 0 /2Y l, X l m m b 2m, Y l m m b 2m, b j j 0 0 j 0 j 0 j c j, 0 b j j 0 j j 0 j c 0 0 j. In Eqs. 0 and, j is a positive integer, is a scaled energy defined by 6 2 /6 6 2 /2/ 6 2, is related to angular momentum l by 0 (2l)/4, and f 0 l r 2 l 2 l l f l r l ḡ l r, 2 c j b 0 QQ Qj. 3 g 0 l r 2 l 2 l l f l r l ḡ l r, 3 The coefficient b 0 is a normalization constant that can be set to and Q() is given by a continued fraction /98/583/7287/$5.00 PRA The American Physical Society

2 PRA 58 SOLUTIONS OF THE SCHRÖDINGER EQUATION FOR AN Q Q Finally, is a root, which can be complex, of a characteristic function. 4 l ; /Q Q, 5 in which Q Q. 6 The pair of solutions f 0 and g 0 have been defined in such a way that they have energy-independent normalization near the origin with asymptotic behaviors given by f 0 l r r 0 4 r/ 6r cos /2 2 r/ , 7 g 0 l r r 0 4 r/ 6r sin /2 2 r/ for both positive and negative energies. Their asymptotic behaviors at large r are given for 0 by f 0 l r r kz 2 ff sin kr l 2 Z fg cos kr l 2, g 0 l r r k 2 Z gf sin kr l 2 Z gg cos kr l 2, 9 20 where k(2/ 2 ) /2 and Z ff X 2 l Y 2 l sin l l sin l cosg l sinl/2/4 l G l cosl/2/4, 2 Z fg X 2 l Y 2 l sin l l sin l cosg l cosl/2/4 l G l sinl/2/4, 22 Z gf X 2 l Y 2 l sin l l sin l cosg l sinl/2/4 l G l cosl/2/4, 23 Z gg X 2 l Y 2 l sin l l sin l cosg l cosl/2/4 l G l sinl/2/4, 24 in which G l 0 0 C 25 and C()lim j c j (). For 0, f 0 and g 0 have asymptotic behaviors given by f 0 l r /2 lim W r r fi I 2 rw fk K 2r, 26

3 730 BO GAO PRA 58 g 0 l r /2 lim W r r gi I 2 rw gk K 2r, 27 where (2/ 2 ) /2 and W fi X 2 l Y 2 l sin l sin l cosg l l G l, 28 W fk X 2 l Y 2 l 2 l sin2 l cos2 l G l, 29 W gi X 2 l Y 2 l sin l sin l cosg l l G l, 30 W gk X 2 l Y 2 l 2 l sin2 l cos2 l G l. 3 It is important to note that both the Z and the W matrices for a specific l are universal functions of that are independent of the specific value of C 6. Different C 6 coefficients only scale the energy differently according to Eq. 2. III. DERIVATION OF THE SOLUTION The key steps in arriving at this solution can be summarized as follows. First, express the solution as a generalized Neumann expansion with proper argument so that the coefficients satisfy a three-term recurrence relation. Second, solve the recurrence relation using continued fractions. Third, sum a corresponding Laurent-type expansion to obtain the asymptotic behavior at infinity. A. Change of variable and Neumann expansion Through a change of variable defined by xr/l, u l rr /2 f x, with 2 and L(/2) /2 6, Eq. can be written as d 2 2 x dx x d 2 dx x2 0 2fx2 x fx, 34 in which is the scaled energy defined by Eq. 2. Expanding f (x) in a Neumann expansion 7,8 f x m b m J m x, 35 where J can be either J or Y, and making use of the property of Bessel functions 8 x J m x2m J m xj m x, 36 one can easily show that the coefficients b m satisfy a threeterm recurrence relation m b m m b m m b m 0. B. Solution of the recurrence relation 37 The three-term recurrence relations for m are solved by defining b j j 0 0 j 0 j 0 j c j for j0 and 38 Q j c j /c j. 39 It is then easily shown that Q j is given by a continued fraction Q j 2 jj j2j Q j 40 and c j Q j Q j2 Q 0 b 0. 4 The recurrence relations for m are solved in terms of b 0 by defining

4 PRA 58 SOLUTIONS OF THE SCHRÖDINGER EQUATION FOR AN b j j 0 j j 0 j c 0 0 j 42 for j0 and Q j c j /c j. 43 Q j is then given by a continued fraction Q j 2 jj j2j Q j 44 and Q j Q j, 52 c j Q j Q j2 Q 0 b From Eqs. 40 and 44, Q j and Q j have the properties Q j, 46 j Q j Q j, 53 where function Q is given by the continued fraction 4. By defining c j as in Eq. 3, c j and c j can be written as c j c j, 54 Q j, 47 j c j c j, 55 Q j Q 0 j, 48 which correspond to the notation used in Eqs. 0 and. Finally, the recurrence relation for m0, Q j Q 0 j, 49 b b 0 b 0, 56 By defining Q j Q j. 50 requires the to be a root of the characteristic function defined by Eq. 5. The same set of coefficients b m gives two linearly independent solutions corresponding to using either J or Y in Eq. 35. QQ 0, 5 one can write Q j and Q j in terms of a single function Q() as C. Asymptotic behaviors The asymptotic behaviors of f and ḡ for small r are straightforward. From the asymptotic expansions of Bessel functions for large arguments 8 we have f l r r 0 /2 4 r/ 6r l cos 2 r/ l sin 2 r/ , 57 ḡ l r r 0 /2 4 r/ 6r l cos 2 r/ l sin 2 r/ , 58

5 732 BO GAO PRA 58 l. TABLE I. Critical scaled energy for different angular momenta l c s p d f g where and are defined in Eqs. 6 and 7. The pair of functions f 0 and g 0 has been defined in Eqs. 2 and 3 such that each has energy-independent normalization near the origin with asymptotic behaviors characterized by Eqs. 7 and 8. For the derivation of asymptotic behaviors at large r, it is more convenient to use the pair of functions l r m b m r /2 J m 2 2 r/ 6, l r m m b m r /2 J m 2 2 r/ 6, which are related to f and ḡ by f l r l r, ḡ l r sin lcos l For 0, k /6 and one can rewrite function as a Laurent-type expansion where p m s0 l rr /2 kr/2 2 m p m kr/2 2m, 63 s s!ms 2s m2s b m2s. 64 FIG.. Movement of the root of the characteristic function for l2. r and i represent the real and imaginary parts of, respectively. The root becomes complex beyond c. l r G l r /2 lim J 2 kr 2 r r k l G l sin l 2 4sin kr l 2 cos l 2 4cos kr l Similarly for the function, we have l r G l r /2 lim J 2 kr r r 2 k lcos G l 2 4sin kr l 2 sin l 2 4cos kr l For 0, /6 and a similar procedure leads to l G l r /2 lim I 2 r r r The key to deriving the asymptotic behavior at large r is to recognize that the asymptotic behavior of a Laurent-type expansion, such as the one in Eq. 63, depends only on the m dependence of p m for large m 9. Making use of the properties of the function for large arguments 8 and the properties of b j for large j as embedded in Eq. 47, one can show that p m G l m m m!2m, 65 where G l () is defined by Eq. 25. Comparing the m dependence of p m with the coefficients of the Bessel function, we have FIG. 2. Z matrix for l2.

6 PRA 58 SOLUTIONS OF THE SCHRÖDINGER EQUATION FOR AN * is also a solution; and iii if is a noninteger solution, n, where n, an integer, is also a solution. To ensure continuity across the threshold, the proper root to take is the one that originates at 0 at 0. Using the property of the roots mentioned above, it can be shown that this root becomes complex beyond a critical scaled energy c determined by l 0; Q The critical scaled energy corresponds to the at which the root that started out at 0 coalesces with another root from n) at FIG. 3. l () function for l2. n l l/2, l/2, l even l odd. 72 G l r /2 lim I r 2 r 2sin2 K 2 r, 68 l G l r /2 lim I 2 r. r r 69 The asymptotic behaviors of other pairs of functions are easily obtained from the asymptotic behaviors of and. IV. DISCUSSION A. Computation The computation is most conveniently carried by starting with Q as defined in Eq. 6. It is given by a continued fraction Q Q, 70 which can be calculated following standard procedures 0. B. Roots of the characteristic function The characteristic function l (; 2 ) defined by Eq. 5 is a function of with 2 as a parameter thus independent of the sign of ). Consistent with the Neumann expansion, the roots of this function have the following properties: i If is a solution, is also a solution; ii if is a solution, This pair of roots becomes a complex pair n l i i and * for c. Both and * give the same physical results and we will take the one with positive imaginary part. Table I lists c for the first few partial waves. Figure illustrates the movement of this root for l2. C. Key functions In a quantum defect theory based on this set of solutions 4, the key functions involved above the threshold is the Z matrix, which is the transformation matrix relating the energy-normalized solution pair defined by f l r r 2 k sin kr l 2, g l r 2 r k cos kr l to the set of solutions with energy-independent normalization near the origin. This relation in matrix form is cf. Eqs. 9 and 20 0 f 0 Z ff Z fg g Z gf Z ggg. f 75 Figure 2 is a plot of the Z matrix for l2. Below threshold, the key function involved in a QDT formulation of bound states is l W fi /W gi lsin l cosg l l G l l sin l cosg l l G l. 76 Figure 3 is a plot of the l () function for l2. W f 0,g 0 4/. 77 D. Wronskians From Eqs. 7 and 8 it is easy to show that the f 0 and g 0 pair has a Wronskian given by Since the Wronskian is a constant that is independent of r, it is a useful check of numerical calculations. In particular, the

7 734 BO GAO PRA 58 asymptotic forms of f 0 and g 0 at large r should give the same Wronskian, which requires detzz ff Z gg Z gf Z fg 2, detww fi W gk W gi W fk 4. These requirements have been verified in our calculations. V. CONCLUSION The exact solutions of the Schrödinger equation for an attractive /r 6 potential have been presented. With this set of solutions, a quantum defect theory for molecular vibration spectra and slow atomic collisions can be set up in a fashion similar 4,2 to the quantum defect theories for other asymptotic potentials with known analytic solutions,2. For example, in the case of a single channel, the bound states of any potential that behaves asymptotically as an attractive /r 6 can be formulated as the crossing points between l () defined earlier and a short-range K matrix K l 0 () 4. Above the threshold, the scattering phase shift can be written in terms of the Z matrix defined earlier and the short-range K matrix K l 0 as 4 K l tan l Z ff K l 0 Z gf K l 0 Z gg Z fg, 80 which provides an analytic description of energy dependences, including shape resonances, of cold-atom collisions. This and other applications are discussed elsewhere 4,3. See, e.g., M. J. Seaton, Rep. Prog. Phys. 46, ; U. Fano and A.R.P. Rau, Atomic Collisions and Spectra Academic, Orlando, 986, and references therein. 2 C. H. Greene, A. R. P. Rau, and U. Fano, Phys. Rev. A 26, C. H. Greene and L. Kim, Phys. Rev. A 38, , and references therein. 4 B. Gao unpublished. 5 E. Vogt and G. H. Wannier, Phys. Rev. 95, ; T.F. O Malley, L. Spruch, and L. Rosenberg, J. Math. Phys. 2, S. Watanabe and C. H. Greene, Phys. Rev. A 22, M. J. Cavagnero, Phys. Rev. A 50, Handbook of Mathematical Functions Natl. Bur. Stand. Appl. Math. Ser. No. 55, edited by M. Abramowitz and I. A. Stegun U.S. GPO, Washington, DC, C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers McGraw-Hill, New York, W. H. Press et al., Numerical Recipes in C Cambridge University Press, Cambridge, 992. It is not difficult to see that the real part of should stay at n l for c. Otherwise, more roots are created since n also have to be roots. 2 B. Gao, Phys. Rev. A 54, B. Gao unpublished.