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Semiclassical Approximations in Wave Mechanics Semiclassical Limit K E Mount Reports on Progress in Physics Related content - Potential scattering cross sections in the Semiclassical approximations in wave mechanics semiclassical limit K E Mount To cite this article: M V Berry and K E Mount 1972 Rep. Prog. Phys. 35 315 - Uniform approximation for potential scattering involving a rainbow M V Berry - Semiclassical techniques for treating the one-dimensional Schrodinger equation: View the article online for updates and enhancements. uniform approximations and oscillatory integrals B J B Crowley Recent citations - ERS approximation for solving Schrödinger’s equation and applications Hichem Eleuch and Michael Hilke - A quantum mechanical insight into SN2 reactions: Semiclassical initial value representation calculations of vibrational features of the ClCH3Cl pre-reaction complex with the VENUS suite of codes Xinyou Ma et al - Semiclassical Treatment of High-Lying Electronic States of H2+ T. J. Price and Chris H. Greene This content was downloaded from IP address 169.228.105.170 on 09/01/2019 at 20:44 Semiclassical approximations in wave mechanics M V BERRY AND K E MOUNT H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL Contents Page 1. Introduction , . 316 2. Problems with no classical turning points . 317 2.1. A simple illustrative example . 317 2.2. The basic WKB solutions . 319 2.3. The semiclassical reflected wave. 322 3. The complex method for treating classical turning points , . 325 3.1. The origin of the connection problem . 325 3.2. The connection formulae for the case of one turning point: their reversibility . 328 3.3. More turning points . , 335 4. Uniform approximations for one-dimensional problems . 342 4.1. The method of comparison equations . 342 4.2. Simple turning-point problems . 344 4.3. Scattering lengths . 349 5. The WKB method and the radial equation , . 350 5.1. Origin of the Langer modification . 350 5.2. Semiclassical approximations for singular potentials . 353 5.3. Uniformly approximate solutions of the radial equation . 354 6. Short-wavelength potential scattering , . 355 6.1. Transformation of the eigenfunction expansion . , 355 6.2. Introduction of the classical paths . 360 6.3. The phenomena of semiclassical scattering . 364 7. General semiclassical theory , . 371 7.1. Feynman’s formulation of quantum mechanics . 371 7.2. Semiclassical evaluation of the path integral . 375 7.3. ‘ Sewing the wave flesh on the classical bones’ . 382 7.4. Quantization and the density of states . , 386 8. Conclusions, . 393 Acknowledgments . 393 References . 394 Abstract. We review various methods of deriving expressions for quantum- mechanical quantities in the limit when tL is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turning- point regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semi- classical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman’s path-integral technique and Rep. Prog. Phys. 1972 35 315-397 316 iW V Berry and K E Mount emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972. 1. Introduction Three classes of approximation method are commonly employed in quantum mechanics. Perturbation techniques produce series expansions for quantities of interest in powers of a variable which specifies the departure of the given problem from an exactly soluble case (as in the Born approximation where scattering ampli- tudes, etc are expanded in powers of the strength of the potential). Variational methods produce the best estimate out of a given class of trial solutions. This article will deal with semiclassical approximations where expressions for wave functions, energy levels, phase shifts, scattering cross sections, etc are derived whose analytic forms are correct in the limiting case where Planck’s (reduced) constant F, is small in comparison with the action functions occurring in the corresponding classical problem. (We shall frequently abbreviate this description of the limit by using such phrases as ‘2, is small’ or ‘correction terms are O(k)’ or ‘as &-to’, bearing in mind that F, is a dimensional quantity which can really have only one value, namely that found in nature.) We emphasize that it is not generally possible to express quantum-mechanical quantities as power series in F, whose first terms are the values of the quantities according to classical mechanics. This is because wave-mechanical functions are almost always highly nonanalytic in 7i as k+O, so that ordinary perturbation theory cannot legitimately be applied. In fact the quantum-classical transition is a singular perturbation problem analogous to, but simpler than, the transition from viscous to frictionless flow. This arises because the Schrodinger equation which, for a particle of mass m and energy E moving in a potential field V(r)is given by k2V2#(r)+2m(E- V(r))#(r)= 0, (1.1) suffers a reduction of order on setting F, equal to zero. The resulting formula, which is not a differential equation at all and does not give the classical limit correctly, implies that #(r) is zero except at the classical turning points, where E- V(r)is zero, and this is an even more drastic approximation than simply taking the classical limit would be, because it is at the turning points that classical quantities themselves frequently become infinite. However, the procedure does at least serve to direct attention to the turning points, which play a crucial role as we shall see. It is however often possible to obtain perturbation series in ascending (not always integral) powers of 6, by starting not from the classical limit but from what we propose to call the semiclassical limit, which takes full account of the various types of singularity at k = 0 (a simple example is discussed in $2.1). In order to keep this article down to a reasonable length we have virtually ignored the problems of finding the correction terms and of analysing the nature of the resulting series; strictly these are matters of asymptotic theory, and the simpler cases can be studied in appropriate texts (eg Copson 1965, Dingle to be published). Instead, we have concentrated on finding the limiting semiclassical forms-the first terms-for a variety of problems, The resulting formulae are often analytically quite com- plicated, but they have the great merit of describing almost all the physics (see Semiclassical mechanics 317 eg the resonance formulae in $3.3). As numerical approximations they are often astonishingly accurate (see $ 6.3, particularly figure 20) ; this is important, because it is precisely in the semiclassical limit that many of the standard calculational methods of wave mechanics-for instance, eigenfunction expansions-converge very slowly. The difficulty of solving a given problem in ‘semiclassical mechanics’ is, fairly obviously, directly related to the complexity of the pattern of classical paths. In particular, we shall find time and again that it is the topology of the orbits that affects the form of the semiclassical expressions. Furthermore, in all cases that have been fully worked out, it is found that the formulae involve only 7i in combina- tion with purely classical quantities, suitably analytically continued (see $9 6.3, 7.3). Semiclassical methods are as old as quantum theory itself, and the literature is correspondingly enormous. Nevertheless, we have frequently found that these techniques are not as widely applied as they might be, and it is the purpose of this article to present them as simply as possible, without losing sight of the complica- tions which are likely to occur in practice. Almost all the results derived here have appeared before, and are ‘well known’ to workers in various different fields. What we have tried to do is to gather together in one place a variety of methods which share the property of being applicable in the semiclassical limit; thus our treatment does not seriously overlap with that given in the books by Heading (1962) and Froman and Froman (1965), which concentrate on the complex method for one- dimensional problems (see our $3). Nevertheless, we have left out a number of topics, the most important being the motion of wave packets (Furry 1963), the semiclassical approximation of relativistic wave equations (eg Rubinow and Keller 1963, Rosen and Yennie 1964), motion in magnetic fields (Furry 1963, Pippard 1969, Lifshitz and Kaganov 1960, 1962), and inelastic collision theory (Percival 1971, Crothers 1971, Child 1971). We have designed the work to be read as a whole, although it is possible to read each section (but not subsection, except $07.1, 7.4) separately without a great deal of reference to the others. The plan is inductive, starting in $2 with the simplest one-dimensional problems not involving real classical turning points, and dealing with progressively more complicated cases until in 3 7 we consider the semiclassical limit of very general situations. We conclude in 0 8 by suggesting some areas where further research is particularly needed.
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