Relations between low-lying quantum wave functions and solutions of the Hamilton-Jacobi equation
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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 =12g2(x2 -a2)2.
PACS 11.10.EfLagrangian andHamiltonian approach
PACS 03.65.GeSolutions of wave equations: boundstates
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