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Canadian Mathematical Bulletin

A Correspondence Principle for Momentum Operators

Jerrold Eldon Marsden

Published Jun 1, 1967• 10 citations
Publisher

Abstract

The purpose of this note is to give a precise formulation of the correspondence principle between classical and quantum mechanics, for the case of momentum operators. (See e. g. I. Segal, Journ. Math. Phys. (1960), 475).
DOI
10.4153/cmb-1967-023-x

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In a previous article in this series [E. M. Corson, J. Math. Phys. 4, 42(1963); herein referred to as I], we showed that a new variational principle, of Lagrangian structure, includes Hamilton's principle as a unique derived consequent and encompasses both the classical and quantal definitions of ``state'' and the corresponding Lagrangian and Schrödinger equations of motion, respectively. In this theory, the essential difference between classical and quantal domains was shown to arise solely from the superadded postulates of determinism vs indeterminism; other than this, the two basic postulates may be reduced to (a) Newton's equations and (b) the new variation principle. It was shown that in the quantal domain, Schrödinger's equations obtain for the transition amplitudes (propagators) between pure eigenstates of homologous variables in configuration space, at different times, thus clearly implying all the results of conventional Hamiltonian-operator quantum formalism. In the present article we show that our variational formulation, under the familiar classical canonical transformation to momentum space, also automatically leads to the corresponding Schrödinger equations for the transition amplitudes between pure eigenstates of homologous variables in momentum space, at different times, the unitarity of the formalism and the reciprocal Fourier integral relations between coordinate and momentum representations. Thus we bridge a long-standing gap in theory and derive, uniquely and apparently for the first time, the basic quantum commutators (P.B.'s) which are postulated in the more familiar Hamiltonian theory.
Semiclassical bound state theory is considered from the viewpoint of the phase space representations of quantum mechanics. In the small h/ limit, within a representation defined by the ordering chosen for the noncommuting position and momentum operators, we obtain: (1) equations of motion, which need not be classical and (2) correspondence principles in action-angle variables, regarded as those initial conditions on the equations of motion which dominate in the small h/ limit. The Weyl ordering is seen to generate the classical equations of motion and the normal correspondence result. Possible evaluation of semiclassical bound state energies is discussed by means of the eigenvalue approach to classical mechanics used by Prigogine and co-workers.