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Physical Review Letters

Hamilton-Jacobi Theory and the Quantum Action Variable

Robert A. Leacock, Michael J. Padgett

Published Jan 3, 1983• 148 citations
Publisher

Abstract

A quantum action variable is defined in the context of a quantum Hamilton-Jacobi theory. The action variable can be used to find the exact bound-state energy levels of a quantum system without solving an equation of motion for the system wave functions.
DOI
10.1103/physrevlett.50.3

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A quantum mechanics is constructed which is patterned on classical Hamilton-Jacobi theory. The dynamical basis of the theory is a quantum Hamilton-Jacobi equation with accompanying physical boundary conditions. Basic objects of the formulation are quantum Hamilton's principal and characteristic functions. As an application of the formalism a theory of quantum action-angle variables is set up. The central feature of this theory is the definition of the quantum action variable which permits the determination of the bound-state energy levels without solving the dynamical equation.
It is well known in classical mechanics that the frequencies of a periodic system can be obtained rather easily through the action variable, without completely solving the equation of motion. Analogously, the equivalent quantum action variable appearing in the quantum Hamilton–Jacobi formalism can provide the energy eigenvalues of a bound state problem, without the necessity of solving the corresponding Schrödinger equation explicitly. This elegant and useful method is elucidated here in the context of some known and not so well-known solvable potentials. It is also shown how this method provides an understanding as to why approximate quantization schemes such as ordinary and supersymmetric WKB can give exact answers for certain potentials.
Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.
In this article, we develop quantum mechanics upon the framework of the quantum mechanical Hamilton-Jacobi theory. We will show, that the Schroedinger point of view and the Hamilton-Jacobi point of view are fully equivalent in their description of physical systems, but differ in their descriptive manner. As a main result, a wave function in Hamilton-Jacobi theory can be decomposed into travelling waves in any point in space, not only asymptotically. The well known WKB-theory will be a special result of the more general theory, we will develop below. By the example of the linear potential and the harmonic oscillator, we will discuss quantum mechanics from the Hamilton-Jacobi point of view. Soft boundary value problems as the connection problem can be solved exactely. Quantizised energies and Maslov-indices can be calculated directely without orthonormalizing wave-functions. Also, we will focus on trajectory themes, which, in contrast to the Schroedinger point of view, follow naturally from the quantum mechanical action function.
A few years ago, one of the former Editors of this journal launched “a call to action” (E. F. Taylor, Am. J. Phys. 71, 423–425 (2003)) for a revision of teaching methods in physics in order to emphasize the importance of the principle of least action. In response, we suggest the use of Hamilton's principle of stationary action to introduce the Schrödinger equation. When considering the geometric interpretation of the Hamilton–Jacobi theory, the real part of the action S defines the phase of the wave function exp iS/ℏ, and requiring the Hamilton–Jacobi wave function to obey wave-front propagation (i.e., Re(S) is a constant of the motion) yields the Schrödinger equation.