@MISC{Friedberg99solutionsof, author = {R. Friedberg and T. D. Lee and W. Q. Zhao}, title = {Solutions of the Hamilton-Jacobi Equation ∗}, year = {1999} }

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Abstract

We discuss a new relation between the low lying Schroedinger wave function of a particle in a one-dimentional potential V and the solution of the corresponding Hamilton-Jacobi equation with −V as its potential. The function V is ≥ 0, and can have several minina (V = 0). We assume the problem to be characterized by a small anhamornicity parameter g −1 and a much smaller quantum tunneling parameter ǫ between these different minima. Expanding either the wave function or its energy as a formal double power series in g −1 and ǫ, we show how the coefficients of g −m ǫ n in such an expansion can be expressed in terms of definite integrals, with leading order term determined by the classical solution of the Hamilton-Jacobi equation. A detailed analysis is given for the particular example of quartic potential V = 1 2 g2 (x 2 − a 2) 2.