WKB Approximation in Three Dimensions

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Abstract
An extension to three dimensions of the WKB approximation method for the quasiclassical wave function is discussed. The extended method is applicable to problems which possess an axis of symmetry, but for which the potential need not be a separable function of the coordinates. The essential difference from the one-dimensional WKB approximation lies in the effect of the curvature of the wave fronts, which plays a central role in the three-dimensional case, and which has a direct physical interpretation. Formulas are derived for the wave function on the axis in both the allowed and the forbidden zones, as well as for the three-dimensional connection formulas. Application of the new method is made to the case of a pure Coulomb wave function and to the case of a particular nonseparable potential which is of interest in the theory of nuclear reaction rates at high density.

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