Resonances and quantum scattering for the Morse potential as a barrier

Transcription

1 Resonances and quantum scattering for the Morse potential as a barrier G. Rawitscher, Cory Merow, Matthew Nguyen, and Ionel Simbotin Department of Physics, University of Connecticut, Storrs, Connecticut Received 2 November 2001; accepted 22 April 2002 Quantum scattering in the presence of a potential valley followed by a barrier is examined for a Morse potential, for which exact analytical solutions are known. For our application the sign of the potential is reversed, and the wave function is required to vanish at the origin. This condition requires a special combination of hypergeometric functions, and can lead to resonances for incident energies below the top of the barrier. Numerical values for the analytical phase shifts are presented in and outside the resonant regions, and the corresponding properties of the scattering S matrix are examined in the complex momentum plane. The validity of the Breit Wigner approximation to the resonant part of the phase shifts is tested, and a new method for finding the location of narrow resonances is described. The time decay of a resonant wave packet slowly leaking out of the valley region on a time scale proportional to the inverse of the width of the resonance is compared with theoretical predictions, and complete agreement is not found American Association of Physics Teachers. DOI: / I. INTRODUCTION The Morse potential 1 is one of a class of potentials 2 for which analytical solutions of the Schrödinger equation are known. The Morse potential is widely used as a model for bound states, such as the vibrational states of molecules. For these applications the potential is defined as a function of x which ranges from to, and is given in terms of two parameters V 0 and by U x V 0 e x 2 e x, where V 0 0. At x 0 it has a minimum of depth V 0, and it goes smoothly to zero in the limit of large x. For x ln(2)/, the potential U(x) is positive, and as x decreases further, the potential increases. This potential will be denoted as the regular potential in what follows. In the present application the sign of the potential is reversed, V 0 0, so that the negative valley turns into a positive barrier. In addition, the variable x is replaced by the radial variable r x r e, defined only in the interval 0 r, V M r V 0 e (r r e ) 2 e (r r e ), where r e is the location of the top of the barrier. This threeparameter form of the Morse potential has been given before, 3 and an example is illustrated in Fig. 1. This case, where there is a valley followed by a barrier, will be denoted as the inverted Morse potential. The inverted Morse potential is much closer than a square well to applications that have physical content, such as the interactions between nuclei at low astrophysical energies, and between atoms where barriers and resonances are involved. Much work has been done with the regular Morse potential shown in Eq. 1. Algebraic methods have been used to generate bound eigenstates 4 and scattering states; 5 the analytical connection between the states in the regular and the inverted Morse potentials has been studied; 6 and approximations to barrier penetration, 7,8 and shape resonances 9 have been found. Analytical solutions of the Schrödinger equation for the Morse potential can be found in several textbooks, but not in the same detail as in the references cited. Most of these investigations examine the analytical properties of the bound states, and the ladder operators that generate one state from the previous one, but do not illustrate the results by particular numerical values. This paper differs from the ones cited, in that we use the form of the potential given in Eq. 2 and impose the condition that the wave function vanish at r 0. This valleybarrier combination, together with the reflection of the wave function at the origin, produces conditions that can generate resonances. We numerically illustrate the properties of the resonances by evaluating the analytical results for the phase shifts and the wave functions for real momenta, and the S matrix in complex momentum space. The resonances are described both in terms of the rapid variation of the phase shift as a function of the incident real momentum k, and in terms of the poles of the scattering S matrix in complex momentum space. The real values of k can be interpreted as the wave number of the incident particle. One of the purposes of this paper is to provide pedagogical illustrations of the properties of resonances by means of the analytical solutions of the Schrödinger equation with a reasonably realistic potential. Another aim is to provide precise values for the phase shifts in the vicinity of a resonance which we will use to compare to the results of numerical methods of solving the Schrödinger equation. 11 The resonance region provides a specially stringent numerical test, because at resonance the wave function in the interior of the valley forms a standing wave of large amplitude that decreases through the barrier region, a feature that is difficult to reproduce numerically with great accuracy, because the numerical errors increase exponentially in the barrier region. These errors become larger the more narrow the resonance, because the decrease of the wave function becomes more pronounced; they do not manifest themselves outside of the resonance region, because there the true wave function also increases. The radial Schrödinger equation to be solved is d2 dr 2 k 2 R r V r R r, Am. J. Phys. 70 9, September American Association of Physics Teachers 935

2 vanishes at the origin and obeys the normalization condition 6 is given by an appropriate combination of the functions M in Eq. 11 below, as is shown in Appendix A. As is also shown there, this combination can be written most conveniently in terms of the function M(k,z), defined as M k,z e z/2 M 1/2 ik/ i,1 2ik/ ;z, with the result that 10 R r 1 2i M k,z 0 M k,z /M k,z 0 *e ikr Fig. 1. The Morse potential with A 6fm 2 is illustrated by the thick line. The other parameters are r e 4 fm, and 0.3 fm 1. The bound state and the two resonances are indicated by the horizontal lines. The energy is in units of inverse distance squared (fm 2 ), and the radial distance is in units of fm. where the wave number k of the incident particle is related to the incident energy Ẽ according to k 2 2m/ 2 Ẽ E, 4 and V is the Morse potential V M in Eq. 2 rescaled by the mass factor 2m/ 2, that is, V r Ae (r r e ) 2 e (r r e ). 5 The units of the three parameters A 2m/ 2 V 0, r e, and are inverse length squared, length, and inverse length, respectively. The radial wave function R is required to vanish at the origin and is normalized such that asymptotically it equals R r sin kr, 6 where is the phase shift. In Sec. II we derive the analytical results and in Sec. III we focus on the properties of the resonances. In particular, we study the extent of the validity of the Breit Wigner approximation to both the poles in the complex momentum space, as well as to the k-dependence of the phase shift in real momentum space. Section IV describes the construction of a resonant wave packet in order to illustrate the time delay of the packet in the valley region, and Sec. V contains the summary and conclusions. Suggested homework problems for each section are included at the end of the paper. II. ANALYTICAL RESULTS As is well known, 1,7 solutions of Eq. 3 can be expressed in terms of confluent hypergeometric functions M(a,b;z) of the type 1 F 1, defined in Eq of Ref. 12, where z is a purely imaginary variable defined as z z 0 exp r, 7 with z 0 2i exp r e, 8 and A 1/2 /. 9 The functions M can be defined as a usually convergent series in powers of the variable z, with coefficients that depend on the parameters a and b. The solution R(r) that M k,z /M k,z 0 e ikr. 11 The asterisk denotes complex conjugation. The phase shift is the phase of M(z 0 ), that is, exp i k M k,z 0 / M k,z For real values of k both R and are real, due to the validity of Eq. A11 which is derived in Appendix A. These results illustrate the important role played by the point r 0. This point, represented by z 0, is absent from the expressions for the bound-state eigenvalues and eigenfunctions of the regular noninverted Morse potential in Eq. 1, because these expressions are not subject to the condition that the wave function vanish at x r e. Equations 11 and 12 have not been given before. It is remarkable that the same analytical expression, Eq. 11, can at the same time represent the oscillatory nature of the function R in the valley region, the exponential nature of R in the barrier region, and finally give rise to the correct asymptotic sinusoidal behavior for large values of r. For complex values of k, the phase shift becomes complex, and the absolute value of the S matrix is no longer unity but can have poles in the complex k plane. A convenient expression for the S matrix is then given by S k e 2i M 0 ( ) ( ), 13 M 0 where M 0 ( ) M 1/2 ik/ i,1 2ik/ ;z For complex values of k, Eqs. 11 and 12 are no longer valid, and will be replaced by equations given in Appendix A. It is well known that the S matrix satisfies the two important properties S k* *S k 1 and S k S k It is shown in Appendix A that the S matrix for the Morse potential, Eq. 13, satisfies both properties. Equation 15, which implies that S(k*) S(k) 1, indicates that if S has a pole at the point k 0 k r ik i, then S has a zero at the mirror point about the real k-axis, k r ik i. The combination of Eqs. 15 and 16 implies that S( k*) S(k) *, which shows that if S has a pole at k 0 k r ik i, then it also has a pole at the mirror point about the imaginary axis at ( k*) k r ik i. The latter property was clearly illustrated in a seminal 936 Am. J. Phys., Vol. 70, No. 9, September 2002 Rawitscher et al. 936

3 Fig. 2. The phase shift in units of, as a function of the incident momentum k, for three values of the potential strengths A. The units of A and k are fm 2 and fm 1, respectively. paper by Nussenzweig, 13 who examined the poles of the S matrix for a square well potential. III. RESONANCES Resonances play an important role in many branches of physics. Kukulin, Krasnopol sky, and Horác ek 14 give an extensive description of the theories used to calculate quantum resonances, mainly for coupled systems of equations. Single channel resonances for a potential with an attractive valley and a repulsive barrier of the type described here, occur for certain narrow regions of the real incident energy which is restricted to be positive, but below the top of the barrier. The corresponding wave function at the interface between the valley and the onset of the barrier region that is, the inner turning point has a logarithmic derivative such that the wave function has to decrease in the barrier region. These resonances are thus extensions of the bound states, for which the wave function also has to decrease in the classically forbidden region. The Morse potential with A 6fm 2, r e 4 fm, and 0.3 fm 1, illustrated in Fig. 1, has one bound state at E B fm 2, and two resonance energies at E fm 2, and E fm 2. The latter is already very close to the top of the barrier, 6 fm 2. All the results shown in the figures were obtained by numerically evaluating the analytical expressions given in Sec. II, by using MATHEMATICA 15 and by constructing a FORTRAN program for checking purposes. The key ingredient in the latter is the evaluation of the hypergeometric function M from its power series expansion, using quadruple precision. Care was taken that the cancellations between terms of the series did not introduce errors beyond the desired precision. The results for real values of k were also checked against a numerical method for solving the Schrödinger equation. 11 One of the most striking features of a resonance is the rapid variation of the phase shift with the incident momentum, as it sweeps over the resonance. This behavior is illustrated in Fig. 2, for various values of the potential strength A. As A is changed, the location of the resonances changes accordingly. In the region of the resonances, the phase shifts increase with k. This behavior is intimately related to the time delay between the arrival of an incident wave packet Fig. 3. Wave functions for various values of k in the region of the second resonance for A 6fm 2.Ask increases, the wave functions are pulled toward the origin, as can be seen by observing how a node moves through the barrier to the left. The units of k are fm 1. and the re-emergence of the resonant part of the wave packet from the interior of the potential. That time is related 16 inversely to the width of the resonance, which in turn is related to the distance of the pole of the S matrix from the real axis, as will be illustrated in the following. Another important feature of a resonance is that the magnitude of the wave function in the valley region becomes large for momenta in the vicinity of the resonance. This feature is illustrated in Fig. 3 for the second resonance of the A 6 fm 2 case, which occurs in the vicinity of the momentum k fm 1. The wave function that has the largest amplitude near the inner turning point occurs for k 2.38 fm 1 ; all the other wave functions, either below or above this k value, have smaller amplitudes. The analytical results for the phase shifts for a particular Morse potential the same that is used in our study of wave-packet behavior are displayed in Appendix B for further reference. A closely related feature is the position of the zeros of the wave function. As the value of k increases from below the resonance (2.32 fm 1 ) to above (2.42 fm 1 ), the first minimum of the wave function beyond the barrier region moves to the left across the barrier, until, when k is above the resonance, another half wavelength is added to the wave function in the valley region. This behavior also can be seen by observing the trajectory of the zeros of the wave function displayed in Fig. 3, and is illustrated more clearly in Fig. 4. The shaded region displays the location of the barrier, its upper and lower borders being the locus of the outer and inner turning points, respectively. Curve R 3, for the third zero, is the most interesting one. It shows that the zero of the wave function closest to the right of the outer turning point rapidly sweeps to the left through the barrier region into the valley region, a feature that we will use in our new method for finding the location of a resonance, described below. The simplest method of finding a resonance consists in calculating the wave function and observing the number of nodes inside the valley region. As discussed above, that number changes from one resonance to the next, just like it would for the case of bound states. However, for narrow resonances, this method is quite cumbersome. A speedier numerical method is based on the second observation made above, that is, only near a resonance can a zero of the wave function occur in the barrier region. We make use of this 937 Am. J. Phys., Vol. 70, No. 9, September 2002 Rawitscher et al. 937

4 Fig. 4. Radial positions of the first four nodes of the wave functions in the resonance region, in units of fm, are shown on the vertical axis. The corresponding momentum values k, in units of fm 1, are displayed on the horizontal axis. The shaded area represents the barrier region. It is delimited by the outer and inner turning points of the barrier. Points that fall within this area correspond to zeros located in the barrier region, which is the case for the curve R(3) of the third node. The value of k where d R(3) /dk is largest, approximately at k 2.38 fm 1, corresponds to the center of the resonance. Fig. 5. Contour plot of the pole and adjoining zero of the absolute value of the S matrix in momentum space of the second resonance for the A 6fm 2 case. The horizontal vertical axis contains the real imaginary part of the momentum k, in units of fm 1. The contour lines are spaced in equal increments of the logarithm of S, so as to display the zero more clearly. The latter lies in the upper half of the plane. observation by numerically finding the eigenvalues k 2 of the equation d 2 /dr 2 V(r) R(r) k 2 R(r), constrained by the two-point boundary condition that the wave function vanish both at the origin and at a point r B somewhere in the vicinity of the barrier region. By varying r B and obtaining the corresponding eigenvalue k, we can trace out curves of the type displayed in Fig. 4. For example, if r B 4 fm and A 6 fm 2 we find two resonances, which correspond to k fm 1 and fm The latter is in excellent agreement with the value k fm 1 obtained from Fig. 4, which is the point where the R(3) curve intercepts the r 4 line. The first resonance is very narrow and is discussed below in relation to the poles of the S matrix. This method of finding the resonances has the advantage that only the eigenvalues need to be known, and not the wave function or the asymptotic phase shifts. There are several other methods of locating resonances. A very general one, especially useful for the case of double or triple peaked barriers, makes use of the spectral energy density method, 18 and was recently applied to the resonance levels of atoms in an external electric field. 19 An even more general one makes use of the set of Siegert functions in order to calculate the position of the poles of the S matrix in complex momentum space, by establishing and then solving an eigenvalue equation. 20 A. Poles of the S matrix, and the Breit Wigner approximation In the present study the behavior and location of the S matrix poles in complex momentum space are investigated numerically by evaluating the expression, Eq. 13 repeatedly, until the location of the pole is found to within the desired accuracy. The poles for the first and second resonances in the A 6fm 2 case were thus found at k R (1) fm 1, k I (1) fm 1 17 and k (2) R fm 1, 18 k (2) I fm 1, respectively. Other, more sophisticated, methods could also be considered, such as locating the zeros of the denominator in Eq. 13 by an iterative Newton-type method, or by some other means. 19,20 To demonstrate that the positions of a pole of the S matrix and that of a zero are located at points mirrored across the real axis in k space, the contour lines of the logarithm of the absolute value of the S matrix for the resonance #2 closest to the top of the barrier for the A 6 fm 2 case are shown in Fig. 5, and a three-dimensional view of that pole is shown in Fig. 6. A rich structure of poles is also present at energies above the top of the barrier, as is illustrated in Fig. 7. These poles contribute collectively to the nonresonant value of (k) for k 2.50 fm 1. Similar nonresonant poles are responsible for the shoulders that occur to the right of the resonances for A 2.7 fm 2 and 3.3 fm 2 displayed in Fig. 2. As A increases from 3.3 to 6.0 fm 2, the energy of one of these nonresonant poles becomes submerged below the top of the barrier, giving rise to a new resonance. Next we discuss the validity of the Breit Wigner approximation. 21 In this approximation, the phase shift is decomposed into a nonresonant background part and a resonant part : k k k. 19 The value of changes rapidly with the incident energy E according to the Breit Wigner expression 938 Am. J. Phys., Vol. 70, No. 9, September 2002 Rawitscher et al. 938

5 Fig. 6. The absolute value of the S matrix in the complex momentum plane for the second resonance of the A 6fm 2 case. The horizontal axis, which is nearly parallel orthogonal to the plane of the paper, contains the real imaginary part of k, in units of fm 1. The corresponding contour lines are displayed in Fig. 5. BW arctan /2 E r E. 20 In Eq. 20, E r and are the resonant energy and the width of the resonance, respectively, and E is the incident energy, related to the incident momentum k according to Eq. 4. All three quantities have the same units of inverse length squared. The quality of this approximation for the second resonance for A 6fm 2 is illustrated in Fig. 8, where the solid and dashed lines represent the exact and the Breit Wigner results, respectively. The value of E r and are not adjusted, but are obtained from the values of k (2) R and k (2) I in Eq. 18, making use of Eq. 25 given below. The value of the nonresonant part of is adjusted by bringing the analytical and the Breit Wigner results into agreement at the center of the resonance, at E E r where /2. The agreement near the center of the resonance is very good. The pronounced disagreement for k 2.4 fm 1 is due to the proximity of the poles that lie above the barrier, but that nevertheless are quite close to the top of the barrier, as is shown in Fig. 7. Fig. 7. The absolute value of the S matrix in the part of the momentum plane whose energy is above of the top of the barrier for A 6fm 2 ( ). The horizontal axis, which is nearly parallel orthogonal to the plane of the paper, contains the imaginary real part of k. The units of k are in fm 1. Fig. 8. Accuracy of the Breit Wigner approximation to the second resonance for A 6fm 2. The analytical result solid line was brought into coincidence with the Breit Wigner approximation at the central point k r E r, where /2, by adjusting the nonresonant backgound phase. The width and the position E r in the Breit Wigner expression, Eq. 20, were not adjusted, but were taken from the location of the pole of the S matrix, according to Eqs. 25 and 18. The Breit Wigner approximation 20 can be extended into the complex momentum k plane near the location of the pole, as follows. If we make use of the identities e i sin 1 2i e2i 1 tan 1 i tan, 21 together with Eq. 20, valid for real momenta, we obtain 1 /2 2i e2i 1 E r E i /2 /2 E P E, 22 according to which the Breit Wigner approximation to the resonant part of the S matrix for real momenta becomes e 2i E r E i /2 E r E i /2 E P E i E P E. 23 In the above, E r, and are real constants, and E P is the complex quantity E P E r i /2. 24 When E is real, it follows from Eq. 23 that exp(2i ) 1, and hence is also real. The analytic continuation of Eq. 23 into the complex E-plane shows that E P is the position of a pole of the S matrix, and Eq. 23 becomes the Breit Wigner approximation to the S matrix for complex energies E in the vicinity of the pole. From the location of the pole, we can find the value of the real resonant energy E r by expressing the complex energy in terms of the square of the complex momentum. If in the momentum plane the pole occurs at k (P) R ik (P) I, then using E P k (P) R ik (P) I 2 together with Eq. 24, wefind E r k (P) R 2 k (P) I 2, 25 /2 2k (P) R k (P) I. 26 The Breit Wigner approximation to the S matrix near the pole, Eq. 23, is consistent with the discussion after Eq. 15, according to which in complex momentum space a zero of the S matrix occurs at a mirror point above the pole, as can be seen as follows. According to Eq. 23, a zero of the S matrix occurs at E 0 E r i /2, and, according to Eq. 25, 939 Am. J. Phys., Vol. 70, No. 9, September 2002 Rawitscher et al. 939

6 the corresponding values of the momenta are k (0) R k (P) R and k (0) I k (P) I. However, according to the approximation given by Eqs. 23 and 25, a pole would also occur at k (P) R k (P) R, and k (P) I k (P) I, because then the values of E r and /2 remain unchanged. Such a pole would violate the identity 16, and hence the question arises for how large a region in the complex momentum plane around the pole is the Breit Wigner approximation valid. This question is examined in Appendix C. It is found that the Breit Wigner approximation is reliable for circular regions around the pole out to radii as large as twice the distance k (P) I of the pole to the real k axis. IV. WAVE PACKET We now construct the resonant portion of a wave packet to further demonstrate the properties of a resonance. For this section, the solutions R(r), defined in Eqs. 3 and 11, will be denoted by (k,r) to emphasize the dependence on the momentum k. A general wave packet is obtained by the superposition of the positive energy eigenfunctions for all momenta k from 0 to and has the form r,t C k k,r exp ik 2 t dk In Eq. 27, the coefficients C(k) are to be determined as described below, the energy k 2 is as defined in Eq. 4, and the time t is in units of fm 2, obtained by multiplying the time in units of seconds by /2m. This time unit will be denoted as T, T 1 fm From the given value (r,0) of the wave packet at t 0, the coefficients C(k) are obtained according to C k 2 0 k,r r,0 dr, 29 which follows from the orthogonality of the functions k,r k,r dr k k k k At t 0 the wave packet is chosen to be a Gaussian function centered around the point R r,0 e (r R)2 /b Plots of the corresponding wave packet show that the peak of the packet travels a distance of about 5 fm in the time 10 T. This is much faster than the motion of the resonant part of the wave function, as will now be shown. The resonant part of the wave packet of interest for the present discussion is the contribution to the integral in Eq. 27 from the momenta in the resonant region k R,k R : k R r,t R C k k,r exp ik 2 t dk. 32 kr Our purpose is to demonstrate that the time dependence of this packet is related to the time for which the particle is trapped in the potential well, which in turn is inversely related to the width of the resonance. According to Ref. 14, Fig. 9. The absolute value of the resonance part of the wave packet defined in Eq. 32. The momentum width used to evaluate the integral is k 10 6 fm 1, the time scale is given in Eq. 33. Itis times longer than time scale T for the full wave packet. In this log-plot the curves for the various times turn out to be spaced by approximately equal distances, showing that the magnitude of the wave packet decreases exponentially with time, as expected. the time delay of a wave packet due to a resonance is D 4/, and the lifetime of the resonance as obtained from Heisenberg s uncertainty principle is t 1/(2 ). The latter will be used as a unit of time 1/ 2 fm We show in Problem IV.6 that the time decay constant P in the expression exp( t/ P ) for the probability of finding the particle in the well is 1.88, which is closer to 1/ than to either the Heisenberg time uncertainty or to the time delay D. To translate these time units into more physically meaningful quantities, we will assume that the particle has the mass of a proton, 938 MeV. Then T 1 fm 2 corresponds to the time needed for the particle to traverse a distance of 9.5 fm with the speed of light, and a width of fm 2 corresponds to an energy of MeV. The width b and the position of the center R of the packet of Eq. 31 are chosen to be b 2 fm, and R 20 fm, and the Morse potential parameters are 0.3 fm 1, r e 4 fm, and A 4 fm 2. The values of the first two parameters are the same as used throughout this paper, but the latter is such that for this potential, there is only one resonance and one bound state. The resonance is located between fm 1 k fm 1, and the resonance momentum and width are k r fm 1 and fm The corresponding value of the time delay is D fm 2, and the unit of time is fm 2. The integral in Eq. 32 for the resonance packet R (r,t) was obtained numerically in the interval , by using Bode s rule, given in Eq of Ref. 12, with equispaced momentum steps k 10 6 fm 1, and the result is displayed in Fig. 9. As expected, the resonance wave packet has a much smaller magnitude than the full wave packet, and its motion is determined by the time scale, Eq. 33, which is more than four orders of magnitude longer than the scale T of the full packet. As can be seen from Fig. 9, the radial dependence of the resonance wave packet varies only little with time, in contrast to the full packet. This lack of dependence is due to the fact that in the potential valley region, the contributions to the integral in Eq. 32 are dominated by the wave functions 940 Am. J. Phys., Vol. 70, No. 9, September 2002 Rawitscher et al. 940

7 Fig. 10. Radial wave functions for various values of the momenta k in the resonance region for A 4fm 2. The numbers in the legend are the momentum values that have to be added to k fm 1. For example, the solid curve, labeled k corresponds to k fm 1, and is near the peak of the resonance. The curves to either side of this value of k have smaller amplitudes. (k,r) at the peak of the resonance, which differ little from each other in that region for such a narrow resonance, as is shown in Fig. 10. As a result, the resonance wave packet at t 0, illustrated in Fig. 11, has a radial shape that is very similar to a second bound state wave function if it existed in that it has one node, and has a small magnitude beyond the barrier. In summary, the resonant part of the wave packet behaves differently from the full wave packet. Its main probability distribution is located inside of the resonance valley at t 0, and the magnitude of this distribution decays with time very slowly according to the time scale determined by the width of the resonance. Furthermore, at t 0 the radial shape in the valley region of the resonant portion of the wave packet resembles that of a bound state. V. SUMMARY AND CONCLUSIONS This article is based on the work by two Research Experience for Undergraduate REU students, and was performed during two successive summers. The first stage consisted in showing that the known nature of the analytical solutions of the Schrödinger equation for the Morse potential Fig. 11. The resonant part of the wave packet wave function at t 0. Its absolute value is also illustrated by the top line in Fig. 9. The radial shape of this function is similar to that of a second bound state, which is to be expected, because this Morse potential (A 4fm 2 ) has only one true bound state. could be utilized to construct a solution that vanishes at a particular point, the zero of the radial coordinate. This solution, based on hypergeometric functions with complicated complex parameters, was shown to give rise to the expected behavior for the wave functions and the phase shifts for real values of the wave number k. It was shown independently 11 that this solution agrees with a numerical solution of the Schrödinger equation, and that it serves to provide a severe test for the accuracy of the latter. Our solution further exhibits interesting resonance properties, that could be numerically linked to the properties of the scattering S matrix in complex momentum space. In particular, the good accuracy of the Breit Wigner approximation to the resonance could be confirmed, with the position and width determined entirely from the position of the poles of the S matrix. Furthermore, a simple method was developed for finding the location in momentum space of the resonances, which consists in solving a two-point boundary condition eigenvalue equation in configuration space. That calculation does not require the knowledge of the wave function or of the phase shifts, but only requires the knowledge of the position of the barrier where the wave function is required to vanish. It easily finds resonances, no matter how narrow their width, contrary to the case for other procedures. A resonant wave packet was constructed numerically from momenta that lie entirely within the width of a particular resonance, and its time decay inside the region of the potential well was observed. It was found that the decay constant P of the probability of finding the particle in the well is close to 1/. However, this time is smaller by a factor close to 4) than the time delay of the re-emergence of a particle incident on the potential barrier, a result which serves to emphasize the different meaning of these two time measures. The evaluation of the analytical expressions was done both by utilizing MATHEMATICA, 15 as well as by a FORTRAN program, one serving to check the other. It thus provided an incentive for students to simultaneously learn to appreciate the power of MATHEMATICA and the power of analytical solutions. At the same time it allowed the students to become acquainted with scattering theory by a hands-on procedure. It is thus hoped that this article will provide an interesting topic for courses that teach the art of scientific computing. 22 VI. PROBLEMS The following problems are given to enhance the pedagogical value of this paper, and are ordered according to the section to which they refer, indicated by Roman Numerals. II.1. The funtion M, defined in Eq. 10 in terms of M ( ), is useful because the complex conjugate of M is related to M ( ) according to Eq. A11 in Appendix A. Prove Eq. A11 by making use of Eq. A10. II.2. Prove Eq. A13 in Appendix A, and show that for real values of k, the definition of R(r) according to Eq. A13 agrees with Eq. 11. III.1. a Show that Fig. 4 is compatible with Fig. 3. b Extend the plot in Fig. 4 to k values up to 4 fm 1 so as to include the first resonance. Rather than doing exact numerical evaluations of the analytical formulas, guess based on general principles of the solution of Schrödinger s equation what these functions might look like. III.2. Figure 7 displays an interesting landscape of poles 941 Am. J. Phys., Vol. 70, No. 9, September 2002 Rawitscher et al. 941

8 for momenta above the barrier. Why do these poles not lead to resonancelike rapid variations of the phase shifts for small changes in the momenta? III.3. Use several trigonometric identities to prove Eq. 21, and then show how Eq. 22 follows. IV.1. The coefficients C(k) in Eq. 27 can usually be chosen such that the wave packet (r,0) has any desired form, for example, a Gaussian as given by Eq. 31. Why is it not possible to do the same for the resonant portion, Eq. 32, of the wave packet? For example, the wave packet shown in Fig. 11 is not a Gaussian, but resembles a bound state wave function. IV.2. The numerical value of the coefficients C(k) in Eqs. 27 and 32 was obtained by numerically evaluating Eq. 29. However, the asymptotic value of the functions (k,r) is of the form sin kr (k), for which an exact analytical value for the integral in Eq. 29 exists if the lower limit of integration is set to which for large values of R is a valid approximation. Why would this analytical expression not be used? Hint: convince yourself that the asymptotic expression for (k,r) is not valid for the appropriate range of k and the value of R used in the example in the text. IV.3. The numerical evaluation of the integral in Eqs. 27 and 32 was done using Bode s rule, given in Eq of Ref. 12. In this procedure the continuum values of k are replaced by discrete values, separated by the distance k, and the integral is replaced by a sum. As the time t increases, the mesh size k has to be made smaller to avoid numerical errors. Explain why k has to be made smaller the larger the value of t. This question is difficult. IV.4. According to Fig. 9, the amplitude of the wave packet decreases as the time increases, that is, the probability of finding the particle inside the valley region decreases with time. Is the total probability conserved for a wave packet, that is, is it independent of time? If so, then the loss of probability in the valley regions should be compensated by an increase of the probability outside the valley region. An extended form of Fig. 9, made to include the wave packet outside of the barrier region, should be suitable to illustrate, for example, the decay of an -particle from a heavy nucleus. Note that to produce such an extended picture would be difficult because of numerical errors, described in Problem IV.3. IV.5. Review the arguments that show that the time scale for the resonant part of the wave packet is much longer then the time scale T of the full wave packet. Base your arguments on the discretized numerical evaluation of Eqs. 27 and 32, rather than on arguments given in the literature, such as in Ref. 14, p. 28, which are based on the saddle-point evaluation of the integral in Eq. 27. IV.6. By plotting the maxima of the wave packet amplitude shown in Fig. 9 as a function of time, numerically evaluate the decay constant P which describes the decay of the probability exp( t/ P ) of finding a resonant particle inside the well. Assume that the probability is proportional to the square of the amplitude maxima. Answer: P You will find that the first three points of the amplitude maxima are well represented by the form exp( t/ P ), but the fourth point falls above the expected value due to numerical inaccuracies for the larger time. Use only the first three points. Note added in proof. After our article was written, a method to locate resonances has come to our attention, that is very similar to the one we describe. It is by C. H. Maier, L. S. Cederbaum, and W. Domcke, A spherical-box approach to resonances, in J. Phys. B: Atom. Molec. Phys. 13, L119 L Their method is based on the result that near resonance the energy eigenvalues of the potential placed in a box are nearly independent of the radius of the box, while our method is based on the motion of the zero of a wave function through the radial position of the barrier near resonance. Both methods are very similar in spirit. APPENDIX A: ANALYTICAL SOLUTIONS We show here how an appropriate solution to Eq. 3 can be obtained in terms of the known solutions of a related Whittacker equation, that is, we show that the analytical solution to Eq. 3 is given by Eq. 11 in Sec. II. If we transform the variable r to z according to Eq. 7, and define (z) z 1/2 R(r), as has been done in Refs. 1 and 7, for example, we obtain from Eq. 3 the equation for d 2 dz k/ 2 1/4 z 2 i z 0, A1 which is of the form of a Whittacker equation, as given by Eq of Ref. 12. Two independent solutions of Eq. A1 are given by Ref. 12, which then provide two solutions of Eq. 3 of the form ( ) r e z/2 z ik/ M a ( ),b ( ) ;z, where a k a ( ) 1 2 i k i, b k b ( ) 1 2i k. A2 A3 A4 Note that Eq. A1 can be transformed into one with real terms only, which is very similar to that of a radial Coulomb function for an energy R/4, where R is Rydberg s constant, an effective charge Z /2, and a complex angular momentum l ( ) 1/2 ik/. In view of Eq. 7, we have z e ikr, A5 ik/ z 0 and in the limit r, it follows that M ( ) 1 because z 0. As a result we find that in the limit r z 0 ik/ ( ) r e ikr, A6 which is valid whether k is real or complex, and which leads to the correct sinusoidal behavior at large distances for real values of k. The appropriate boundary conditions for the function R(r) are found from the general solution of Eq. 3 R r C ( ) ( ) r C ( ) ( ) r, A7 by determining the appropriate values of the constants C ( ) as follows. Their ratio is determined by the requirement that R vanishes at r 0, 942 Am. J. Phys., Vol. 70, No. 9, September 2002 Rawitscher et al. 942

9 Table I. Analytical values of A (r max )/ for the Morse potential given in the text, as a function of momentum k in the resonance region. k (fm 1 ) A ( )/ A (100)/ A (50)/ C ( ) C ( ) ( ) 0 ( ) 0 z 0 M ( ) 2ik/ 0 ( ), A8 M 0 where M ( ) 0 M(a ( ),b ( ) ;z 0 ). If we use Eqs. A8 and A7, together with Eq. A5, we obtain R r e z/2 C ( ) z ik/ 0 M ( ) e ikr M ( ) 0 /M ( ) 0 M ( ) e ikr. A9 In the asymptotic limit of r, the S matrix is the negative of the ratio of the coefficients of the outgoing wave exp(ikr) to that of the ingoing wave exp( ikr). Hence, in view of Eq. A6, we obtain the result in Eq. 13. In the above, the phase shift is in general complex, and becomes real when the imaginary part of k is zero. The proof of Eqs. 15 and 16 for the S matrix above is based on Eq of Ref. 12, e z/2 M a,b;z e z/2 M b a,b; z, A10 together with the facts that z is purely imaginary and the quantities a ( ) and b ( ) have the special properties given by Eq. A3. In view of the above, the function M in Eq. 10, has the property according to Eq. A10 that e z/2 M ( ) z M k*,z *, A11 from which it follows that Eq. 13 can also be written in the form S M k,z 0 M k*,z 0 *. A12 For real values of k, Eq. A12 shows that the absolute value of S is unity, and that Eq. 12 is valid. The proof of Eq. 11 also follows from Eq. A11. The normalization requirement, Eq. 6, can be implemented by requiring C ( ) z ik/ 0 e i /2i, which, in view of Eq. 13, leads to the result R r e /2i ( ) z/2 M M 0 ( ) e ikr 1/2 M 0 ( ) M ( ) M 0 ( ) e ikr. 1/2 M 0 ( ) A13 This function vanishes at r 0, is in accordance with Eq. 6 for r, and gives rise to Eq. 11 for real values of k. APPENDIX B: NUMERICAL VALUES OF PHASE SHIFTS The purpose of this appendix is to present accurate numerical values for phase shifts in a narrow resonance region, so as to provide comparison values for accuracy tests of numerical algorithms for solving Eq. 3. The phase shifts obtained from the analytical expression, Eq. 12, assume that the potential extends to. In this appendix we denote these phase shifts as A ( ). However, numerical calculations are carried out to a maximum finite radial distance r max beyond which the potential is set to zero. Therefore, in order to get an analytical value for A (r max ) that can be compared with the numerical value, the contributions to the phase shift from r max to r have to be subtracted from the exact analytical result of Eq. 12, A r max A. B1 This subtraction is accomplished by utilizing a first-order perturbative expression for given by Eq. A11 in Ref. 23. The integrals I s and I c defined there can be evaluated analytically in the present case, because here the potential at large distances can be approximated by a single exponential. The results for A (r max )/, for the parameters 0.3 fm 1, r e 4 fm, and A 4 fm 2, obtained in quadruple precision, are given in Table I for three values of r max, 50, and 100 fm. For r max 50 fm the value of is rad, and therefore the second-order correction in / is Hence the last 10th digit after the decimal point of the values for (50)/ in Table I is uncertain. However for r max 100 fm all significant figures in Table I are reliable. APPENDIX C: REGION OF VALIDITY OF THE BREIT WIGNER APPROXIMATION In how large a region of the complex momentum plane around the pole is the Breit Wigner approximation valid? By looking at Eq. 22, one is tempted to investigate this question by examining the behavior of (S 1) (E P E), and the answer might be expected to be equal to the E-independent quantity i. However, because of the pres- 943 Am. J. Phys., Vol. 70, No. 9, September 2002 Rawitscher et al. 943

10 ence of the unknown background phase, this procedure has to be modified by rewriting Eq. 22 in the form S 1 e 2i e 2i 1 e 2i 1, C1 2 and multiplying both sides of Eq. C1 by (E P E) or k P k 2. In the region of validity of the Breit Wigner approximation, the result should equal S 1 k 2 P k 2 e 2i i e 2i 1 k 2 P k 2, C2 in view of Eq. 22. In the above both k P and k are complex, with k P denoting the known location of the pole. Numerical tests were performed by varying k along a circle centered at k P with a radius, and plotting the left-hand side of Eq. C2 as a function of the angle of the vector along the circle. If we ignore terms of the order of 2 in comparison with k 2 P, then according to the right-hand side of Eq. C2 the result should be a sinusoidal function of, oscillating around a complex constant e 2i i, with an amplitude proportional to the radius. Such behavior was indeed found to be the case out to radii as large as the 2k (P) I. This is twice the distance of the pole to the real k-axis, and encompasses the location where the S matrix has a zero. 1 P. M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels, Phys. Rev. 34, G. Poschl and E. Teller, Bemerkungen zur Quantenmechanik des anharmonischen Oscillators, Z. Phys. 83, S. Flugge, Practical Quantum Mechanics Springer-Verlag, New York, Heidelberg, Berlin, 1974, p Y. Alhassid, F. Gursay, and F. Iachello, Potential scattering, transfer matrix and group theory, Phys. Rev. Lett. 50, ; A. Barut, A. Inomata, and R. Wilson, Algebraic treatment of second-order Poschl Teller, Morse Rosen, and Eckart equations, J. Phys. A 20, Y. Alhassid and J. Wu, An algebraic approach to Morse potential scattering, Chem. Phys. Lett. 109, ; Y. Alhassid, F. Gursay, and F. Iachello, Group theory of the Morse oscillator, ibid. 99, ; Group theory approach to scattering, Ann. Phys. Leipzig 148, A. O. Barut, A. Inomata, and R. Wilson, The generalized Morse oscillator in the SO 4,2 dynamical group scheme, J. Math. Phys. 28, Z. Ahmed, Tunneling through the Morse barrier, Phys. Lett. A 157, ; Tunneling through a one-dimensional potential barrier, ibid. 47, B. Lundborg, Phase-integral treatment of transmission through an inverted Morse potential, Math. Proc. Cambridge Philos. Soc. 81, A. S. Dickinson, An approximate treatment of shape resonances in elastic scattering, Mol. Phys. 18, P. M. Morse and H. Feshbach, Methods of Theoretical Physics McGraw- Hill, New York, 1953, Vol. 2, p. 1672; L. Landau and E. Lifschitz, Quantum Mechanics, 3rd ed. Pergamon, Oxford, 1977, p G. Rawitscher and I. Koltracht unpublished. 12 Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun Dover, New York, H. M. Nussenzweig, The poles of the S-matrix of a rectangular potential well or barrier, Nucl. Phys. 11, V. I. Kukulin, V. M. Krasnopol sky, and J. Horác ek, Theory of Resonances: Principles and Applications Kluwer Academic, Dodrecht, 1989, p S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, 2nd ed. Addison-Wesley, Redwood City, CA, R. Newton, Scattering Theory of Waves and Particles McGraw-Hill, New York, 1966, pp Israel Koltracht, Department of Mathematics, University of Connecticut private communication. 18 The Weyl Titchmarsh Kodaira theorem on spectral energy densities is described in Eigenfunctions Expansion Associated with Second-Order Differential Equations, edited by E. C. Titchmarsh Oxford University Press, Oxford, 1946, Vols. I and II; K. Kodaira, The eigenvalue problem for ordinary differential equations of the second order, and Heisenberg s theory of S-matrices, Am. J. Math. 71, G. N. Gibson, G. Dunne, and K. J. Bergquist, Tunneling ionization rates from arbitrary potential wells, Phys. Rev. Lett. 81, ; K. Barnett and G. N. Gibson, Static field tunneling ionization of H 2, Phys. Rev. A 59, O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, Siegert pseudo-states as a universal tool: resonances, S matrix, Green function, Phys. Rev. Lett. 79, ; Siegert pseudosate formulation of scattering theory: One channel case, Phys. Rev. A 58, G. Breit and E. Wigner, Capture of slow neutrons, Phys. Rev. 49, G. Rawitscher, I. Koltracht, Hong Dai, and C. Ribetti, The vibrating string: a fertile topic for teaching scientific computing, Comp. in Phys. 10, G. H. Rawitscher, B. D. Esry, E. Tiesinga, J. P. Burke, Jr., and I. Koltracht, Comparison of numerical methods for the calculation of cold atom collisions, J. Chem. Phys. 111, Am. J. Phys., Vol. 70, No. 9, September 2002 Rawitscher et al. 944