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Journal of Physics A Mathematical and General

A shooting approximation to wavefunctions with harmonic amplitude

F. Riordan

Published Nov 11, 1987• 0 citations
Publisher

Abstract

States which in the Schrodinger representation have harmonic amplitudes or phases or both are discussed. The variational problem for the quantum mechanical action functional reduces to the solution of classical equations of motion by shooting methods, when the variation is restricted to wavefunctions whose phase satisfies the classical Hamilton-Jacobi equation. These wavefunctions have harmonic amplitude.

DOI
10.1088/0305-4470/20/16/032

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States which in the Schrodinger representation have harmonic amplitudes or phases or both are discussed. The variational problem for the quantum mechanical action functional reduces to the solution of classical equations of motion by shooting methods, when the variation is restricted to wavefunctions whose phase satisfies the classical Hamilton-Jacobi equation. These wavefunctions have harmonic amplitude.
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.
An ordinary unambiguous integral representation for the finite propagator of a quantum system is found by starting of a privileged skeletonization of the functional action in phase space, provided by the complete solution of the Hamilton-Jacobi equation. This representation allows to regard the propagator as the sum of the contributions coming from paths where the momenta generated by the complete solution of the Hamilton-Jacobi equation are conserved -as it does happen on the classical trajectory-, but are not restricted to having the classical values associated with the boundary conditions for the original coordinates.
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.
The problem of the reduction of quantum mechanics to classical mechanics is considered. It is demonstrated that the customary derivation of classical mechanics from the Schrödinger equation via the Hamilton-Jacobi action equation is incomplete. An improved derivation, involving the expectation value of the action, rather than the action itself, is given. Under this formulation the Hamilton-Jacobi description of classical mechanics is obtained from quantum mechanics, for suitable wave packets as h goes to zero.