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28 March 2024
 
  » arxiv » quant-ph/9910047

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Relations Between Low-lying Quantum Wave Functions and Solutions of the Hamilton-Jacobi Equation
R. Friedberg ; T. D. Lee ; W. Q. Zhao ;
Date 12 Oct 1999
Subject quant-ph
AbstractWe 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 $geq 0$, 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 $epsilon$ between these different minima. Expanding either the wave function or its energy as a formal double power series in $g^{-1}$ and $epsilon$, we show how the coefficients of $g^{-m}epsilon^n$ 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 $V={1/2}g^2(x^2-a^2)^2$.
Source arXiv, quant-ph/9910047
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