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American Journal of Physics

Classical Limit and the WKB Approximation

Lowell S. Brown

Published Mar 1, 1972• 13 citations
Publisher

Abstract

We prove that a wave-packet solution in the WKB approximation to the Schrödinger equation follows the classical trajectory, and we discuss the nature of this solution.
DOI
10.1119/1.1986554

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Abstract In the context of nuclear scattering, we use the bipolar reduction of the Schrödinger equation to examine the effects of optical model absorption on incoming and outgoing scattering waves. We compare the exact solutions for these waves, obtained using a bipolar quantum trajectory-based formalism, with their approximate WKB counterparts. Aside from reducing the magnitudes of the incoming and outgoing waves, absorption smooths the variation of the potential at the turning point, reducing reflection in this region. This brings the incoming exact solution and WKB approximation into closer agreement, but tends to worsen the agreement between the outgoing solutions. Inside the turning point, the WKB approximation overestimates the inward decaying solution. The exact solution also possesses an outward going component, solely due to reflection, with no WKB counterpart.
We propose a semiclassical wave packet propagation method relying on classical trajectories in a complex phase space. It is based on the Schrödinger wave equation and the usual expansion with respect to ℏ, except that the amplitude of the wave packet is taken into account at the very zeroth order, unlike in the usual WKB method where it is treated as a corrective or first order term. Formally, it amounts to making both the wavelength and the width of the wave packet tend to zero with ℏ. The action and consequently the classical trajectories derived are complex. This method is tested successfully in many cases, analytically or numerically, including the bounce and even the splitting of the wave packet. Our method appears to be much more accurate than the WKB method while less computationally demanding than the Van-Vleck formula. Moreover, it has a particularly interesting property: the singularities (caustics) of the usual semiclassical theories do not appear in this formalism in all cases tested.