Relations between low-lying quantum wave functions and solutions of the Hamilton-Jacobi equation

We discuss a new relation between the low-lying Schrodinger wave function o f a particle in a one-dimensional potential V and the solution of the corre sponding Hamilton-Jacobi equation with -V as its potential. The function V is greater than or equal to 0, and can have several minima (V = 0). We assu me the problem to be characterized by a small anharmonicity parameter g(-1) and a much smaller quantum tunneling parameter epsilon between these diffe rent minima. Expanding either the wave function or its energy as a formal d ouble 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 int egrals, 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 = 1/2g(2)(x(2) - a(2))(2).