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American Journal of Physics

Use of operator wave functions to construct a refined correspondence principle via the quantum mechanics of Wigner and Moyal

John Snygg

Published Nov 1, 1980• 8 citations
Publisher

Abstract

We represent the momentum operator p by p+(h//2i)(∂/∂p) and the position operator q by q−(h//2i)(∂/∂p). We then apply each resulting operator that appears in the nonrelativistic quantum mechanics of Schrödinger and Heisenberg to the constant 1. What results is a Wigner–Moyal theory which is as complete as the usual Schrödinger–Heisenberg theory. Replacing Planck’s constant h/ by zero in the Wigner–Moyal theory results in the classical physics of Newton. We thereby obtain a refined correspondence principle.

DOI
10.1119/1.12197

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We represent the momentum operator p by p+(h//2i)(∂/∂p) and the position operator q by q−(h//2i)(∂/∂p). We then apply each resulting operator that appears in the nonrelativistic quantum mechanics of Schrödinger and Heisenberg to the constant 1. What results is a Wigner–Moyal theory which is as complete as the usual Schrödinger–Heisenberg theory. Replacing Planck’s constant h/ by zero in the Wigner–Moyal theory results in the classical physics of Newton. We thereby obtain a refined correspondence principle.
Although it was in fact the study of the Wigner distribution function that suggested to the author the general problem discussed in Part I, a consideration of the question of the formation of quantum mechanical operators corresponding to dynamical variables, being of fundamental importance to quantum theory, should properly precede a discussion of the Wigner distribution function. In both Parts I and II spin and relativistic effects will be ignored.\nA word about notation is in order. A symbol such as A will denote the quantum mechanical operator corresponding to the dynamical quantity A. For example, H is the Hamiltonian operator; q is the coordinate operator and p the conjugate momentum operator. Whenever there is a possibility of confusion, O(A) is used to represent the operator corresponding to the dynamical quantity A. For example, O(pq) is the operator corresponding to the product pq, and O(H2) is the operator corresponding to the square of the classical Hamiltonian.\nA subscript appearing on a differential operator, such as in 66PH , indicates the function on which the operator acts. Unless otherwise indicated all integrations are to be taken from -infinity to infinity. As usual, h denotes Planck's constant divided by 2pi.\nSince any physical investigation is directly dependent on the quality of the tools used in the investigation, mathematical rigor is much to be desired in physical theory. That rigor has not been achieved in this work is all too obvious, but it was nonetheless a constant goal.
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
This is a review paper on the formulation of nonrelativistic Quantum Mechanics on Phase Space. In an overlook to the traditional approach, we present the basic concepts including the Weyl mapping, the Wigner function, the Moyal propagator and the Spectral projections. We discuss the advantages and the disadvantages of this formulation. We apply it to the study of quadratic Hamiltonians. The Stratonovich-Weyl correspondence is presented as the natural framework in which the theory can be extended to include spin, relativity and constraints. The theory of identical particles is also considered.
The fundamental problem considered is that of the existence of a joint probability distribution for momentum and position at a given instant. The philosophical interest of this problem is that for the potential energy functions (or Hamiltonians) corresponding to many simple experimental situations, the joint “distribution” derived by the methods of Wigner and Moyal is not a genuine probability distribution at all. The implications of these results for the interpretation of the Heisenberg uncertainty principle are analyzed. The final section consists of some observations concerning the axiomatic foundations of quantum mechanics.
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.
The momentum operator may be generalized for the correspondence of the limit of quantum commutators of observables with the classical commutators. The substitution of the momentum operator into the nonrelativistic equation for the conservation of energy yields a generalization of the stationary wave equation. Replacing also the energy operator, a generalization of the time-dependent Schrödinger equations, there exists a special class of wavefunctions that satisfies the nonrelativistic wave equation in quantum theory.