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Banach Center Publications

A quantum route to Hamilton–Jacobi equation: comments and remarks

José F. Cariñena, Eduardo Martı́nez, Giuseppe Marmo, Xavier Gràcia, C. Muñoz-Lecanda

Published Jan 1, 2016• 6 citations
Publisher

Abstract

We consider the short-wave limit of evolutionary wave equations to derive the Hamilton–Jacobi equation. In particular we consider how to get wave mechanics from the abstract picture on Hilbert spaces. This ‘realisation’ contains an additional ingredient
DOI
10.4064/bc110-0-3

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In this article, we will answer a question posed in the book Classical Mechanics by H. Goldstein: ``"Is the Hamilton-Jacobi equation the short wavelength limit of the Schrödinger equation?" But, before that, we will identify an essential element that will take us from the Hamilton-Jacobi equation to the dynamic equation of non-relativistic quantum mechanics for a function ψ through an exact procedure. This element is the linear independence of the functions ψ and ψ* (their complex conjugate). Their independence is demonstrated for physical systems where the acting physical potential does not explicitly depend on time. Proceeding in reverse, from the Schrödinger equation, we obtain the Hamilton-Jacobi equation, exactly, without additional terms.