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

Summary

We discuss a new relation between the low-lying Schrödinger wave function of a particle in a one-dimensional potential V andthe solution of the corresponding Hamilton-Jacobi equation with —V as its potential. The functionV is ≥ 0, andcan have several minima (V = 0). We assume the problem to be characterized by a small anharmonicity parameterg ^-1 anda much smaller quantum tunneling parameter ɛ between these different minima. Expanding either the wave function or its energy as a formal double power series ing ^-1 and ɛ, we show how the coefficients ofg ^-mɛn in such an expansion can be expressedin 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 potentialV =¹ ₂ g ²(x ² -a ²)².