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Physical Review A

Equivalence of time-dependent variational descriptions of quantum systems and Hamilton's mechanics

Kit-Keung Kan

Published Nov 1, 1981• 35 citations
Publisher

Abstract

It is shown that any time-dependent description of quantum systems derived from the variational principle is equivalent to Hamilton's description of a classical system. This is done by establishing the fact that the Euler's equations from the variation of the wave function in any parametrization can be transformed to a system of Hamilton's equations. The problem of obtaining collective dynamical variables in quantum many-body systems is discussed in the light of this equivalence.

DOI
10.1103/physreva.24.2831

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It is shown that any time-dependent description of quantum systems derived from the variational principle is equivalent to Hamilton's description of a classical system. This is done by establishing the fact that the Euler's equations from the variation of the wave function in any parametrization can be transformed to a system of Hamilton's equations. The problem of obtaining collective dynamical variables in quantum many-body systems is discussed in the light of this equivalence.
The quantum mechanical, time-dependent wave equation in generalized coordinates is obtained by a slight modification of Schrödinger's original derivation. If the classical Hamiltonian for a system is known, the variational technique may be applied to the Hamilton-Jacobi equation to yield an Euler-Lagrange equation. This latter is then the wave equation for the system. The method, which is generally easy to apply, is illustrated by applications to a single particle in curvilinear coordinates and to a symmetric top molecule.
Abstract The transition from classical mechanics to quantum mechanics has been formulated in various ways. But all these ways have the common feature that they use as starting point mechanics not in its Newtonian form, but in the form which it was given by Lagrange and Hamilton. The essence of this form consists in the fact that the differential equations are represented as Euler equations of a variational principle for v functions, the generalized coordinates, of one variable, the time. In its purely mathematical aspect the quantization method may, therefore, be regarded as a mode of taking certain notions and certain relations arising from a variational principle for several dependent and one independent variable and of giving them a new meaning.
It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.
In the first days of quantum mechanics Dirac pointed out an analogy between the time-dependent coefficients of an expansion of the Schr\"odinger equation and the classical position and momentum variables solving Hamilton's equations. Here it is shown that the analogy can be made an equivalence in that, in principle, systems of classical oscillators can be constructed whose position and momenta variables form time-dependent amplitudes which are identical to the complex quantum amplitudes of the coupled wave function of an NNN-level quantum system with real coupling matrix elements. Hence classical motion can reproduce quantum coherence.