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The Journal of Chemical Physics

A new uniform semiclassical wave function for single surface and multisurface scattering

Michael F. Herman

Published Sep 15, 1983• 19 citations
Publisher

Abstract

A procedure for the evaluation of one-dimensional uniform semiclassical wave functions is explored. This form of the approximate wave function is tested for the case of an exponential potential and is compared to the exact quantum mechanical and WKB wave functions. It is found that the uniform approximation provides a very accurate representation of the exact wave function, even at the turning points of the classical motion, where the WKB approximation diverges. It is demonstrated how correction terms can be constructed, leading to a formal expansion for the exact wave function. The individual terms of this expansion each involve a finite number of reflection points with uniform semiclassical propagation between these reflections. The analysis is extended to multisurface (nonadiabatic) problems, removing the turning point singularities which are generally present in semiclassical formulations of nonadiabatic problems.

DOI
10.1063/1.446096

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A procedure for the evaluation of one-dimensional uniform semiclassical wave functions is explored. This form of the approximate wave function is tested for the case of an exponential potential and is compared to the exact quantum mechanical and WKB wave functions. It is found that the uniform approximation provides a very accurate representation of the exact wave function, even at the turning points of the classical motion, where the WKB approximation diverges. It is demonstrated how correction terms can be constructed, leading to a formal expansion for the exact wave function. The individual terms of this expansion each involve a finite number of reflection points with uniform semiclassical propagation between these reflections. The analysis is extended to multisurface (nonadiabatic) problems, removing the turning point singularities which are generally present in semiclassical formulations of nonadiabatic problems.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash 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 turningpoint 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 semiclassical 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 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.
A long-standing problem and major shortcoming of semiclassical (WKB) bound-state wave functions is their divergence at the classical turning points. Presently available regularization schemes are accurate but rather complicated. It is shown how finite wave functions can be simply obtained by reorganizing the WKB perturbation expansion in powers of ℏ and retaining appropriate higher-order terms. The wave functions obtained by this method are compared with exactly known ones for the harmonic-oscillator and Morse potentials.
The semiclassical approximation for a quantum amplitude is given by the sum of contributions from intersections of the appropriate manifolds in classical phase space. The intersection overlaps are just the Van Vleck determinants multiplied by a phase given by a classical action. Here we consider two nonstandard instances of this semiclassical prescription which would appear to be on shaky ground, yet the corresponding physical situations are not unusual. The first case involves momentum-space WKB theory for scattering potentials; the second is a propagator for the whisker map that arises in generic two-dimensional systems. In the former case two manifolds become asymptotically tangent, and the semiclassical formula needs to be uniformized in order to give a meaningful wave function. We give a uniformization procedure. In the latter case, there are an infinite number of intersections in phase space within a zone with the area of Planck's constant (the limit of resolution for quantum mechanics), yet the semiclassical sum over all contributions is shown to be correct.
We demonstrate that the Bargmann representation of quantum mechanics is ideally suited for semiclassical analysis, using as an example the WKB method applied to the bound-state problem in a single well of one degree of freedom. While the WKB expansion formulas are basically the usual ones, in this representation they describe approximations that are uniform and nonsingular in the classically allowed region of phase space because no turning points appear there. The quantization of energy levels relies on a complex contour integral that tests the eigenfunction for analyticity. For the harmonic oscillator, this WKB method trivially gives the exact eigenfunctions in addition to the exact eigenvalues. For an anharmonic well, a self-consistent variational choice of the representation greatly improves the accuracy of the semiclassical ground state. Also, a simple change of scale illuminates the relationship of semiclassical versus linear perturbative expansions, allowing a variety of multidimensional extensions. All in all, the Bargmann representation appears to combine the advantages of a linear description and of a phase-space representation of the quantum state vectors.
In this thesis, we construct an approximate series solution of the Wigner equation in terms of Airy functions, which are semiclassically concentrated on certain Lagrangian curves in two-dimensional phase space. These curves are defined by the eigenvalues and the Hamiltonian function of the associated one-dimensional Schrödinger operator, and they play a crucial role in the quantum interference mechanism in phase space. We assume that the potential of the Schrödinger operator is a single potential well, such that the spectrum is discrete. The construction starts from an eigenfunction series expansion of the solution, which is derived here for first time in a systematic way, by combining the elementary technique of separation of variables with involved spectral results for the Moyal star exponential operator. The eigenfunctions of the Wigner equation are the Wigner transforms of the Schrödinger eigenfunctions, and they are approximated in terms of Airy functions by a uniform stationary phase approximation of the Wigner transforms of the WKB expansions of the Schrödinger eigenfunctions. The approximation of the eigenfunction series is an approximated solution of the Wigner equation, which by projection onto the configuration space provides an approximate wave amplitude, free of turning point singularities. It is generally expected that, the derived wave amplitude is bounded, and correctly scaled, even on caustics, since only finite terms of the approximate terms are significant for WKB initial wave functions with finite energy. The details of the calculations are presented for the simple potential of the harmonic oscillator, in order to be able to check our approximations analytically. But, the same construction can be applied to any potential well, which behaves like the harmonic oscillator near the bottom of the well.