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

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Abstract
It is shown in the limit ħ→0 that an approximate solution of the time-dependent Schrödinger 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.

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