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Physical Review A

Approximate solutions of the Schrödinger equation in the semiclassical limit via generalized Fourier integrals

Ira B. Bernstein

Published Jul 1, 1985• 10 citations
Publisher

Abstract

It is shown in the limit \ensuremath{\Elzxh}\ensuremath{\rightarrow}0 that an approximate solution of the time-dependent Schr\"odinger equation can be obtained by the use of generalized Fourier integrals. These are constructed by the continuous superposition of eikonals, each generated by solving a problem in classical mechanics. The solution thus generated is valid even in the presence of caustics, provided that they are not too dense. The method applies to localized wave functions and provides a short-time solution. The nature of the solution and its relation to an exact solution is demonstrated in the case of the one-dimensional problem of the surface-state electron. The Wigner distribution f arises in a natural way, leading to the interpretation of the semiclassical limit as classical kinetic theory.

DOI
10.1103/physreva.32.1

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It is shown in the limit \ensuremath{\Elzxh}\ensuremath{\rightarrow}0 that an approximate solution of the time-dependent Schr"odinger equation can be obtained by the use of generalized Fourier integrals. These are constructed by the continuous superposition of eikonals, each generated by solving a problem in classical mechanics. The solution thus generated is valid even in the presence of caustics, provided that they are not too dense. The method applies to localized wave functions and provides a short-time solution. The nature of the solution and its relation to an exact solution is demonstrated in the case of the one-dimensional problem of the surface-state electron. The Wigner distribution f arises in a natural way, leading to the interpretation of the semiclassical limit as classical kinetic theory.
A new method is proposed for constructing approximate solutions to the Schrödinger equation. In place of the wave function, its Gaussian-windowed Fourier transform is used as the fundamental entity. This allows an intuitively attractive connection to be made with a family of classical trajectories and, at all times, the wave function is inferred from the present state of these trajectories. The fact that the connection between the wave function and the classical trajectories is consistently constructed in phase space allows this method to be free of the limitations of other methods.
A symplectically covariant quantum-mechanical uncertainty relation more accurate than previously known ones is derived for multidimensional systems. It is shown that the quantum-mechanical description of a linear Hamiltonian system in terms of squeezed states is completely equivalent to its description in terms of a phase-space distribution function. A new approach to the semiclassical limit is proposed, based on the use of squeezed states. By analyzing explicit formulas for squeezed states, a semiclassical asymptotic form of the solution to the Cauchy problem for a multidimensional Schrödinger equation is found in the limit of ħ → 0. The behavior of the semiclassical solution in the neighborhood of a caustic is analyzed in the one-dimensional case, and the phase shift across the caustic is determined. General properties and examples of squeezed states are discussed that point to the fundamental importance of squeezed states for developing a nonrelativistic quantum-mechanical description of a system of charged particles in an electromagnetic field in the dipole approximation.
We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.
A mapping that relates the Wigner phase-space distribution function of a given stationary quantum mechanical wave function, a solution of the Schr\"odinger equation, to a specific solution of the Liouville equation, both subject to the same potential, is studied. By making this mapping, bound states are described by semiclassical distribution functions still depending on Planck's constant, whereas for elastic scattering of a particle by a potential they do not depend on it, the classical limit being reached in this case. Following this method, the mapped distributions of a particle bound in the P\"oschl-Teller potential and also in a modified oscillator potential are obtained.
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.