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Journal of Mathematical Physics

A minimum principle for von Neumann’s equation

Sidney Golden

Published Jan 1, 1975• 7 citations

Abstract

A minimum principle is described which has von Neumann’s equation of motion of the statistical operator as its ultimate solution. When the statistical operator is initially restricted to be a projection of trace unity, the minimum principle has Schrödinger’s time−dependent equation as its ultimate solution. As a result, the present minimum principle is shown to be a generalization of one first introduced by McLachlan, with a modification of the Dirac−Frenkel variational principle as a necessary consequence. In physical terms, the quantity that is minimized is shown to be the mean−squared deviation of an effective time−dependent Hamiltonian, which can be attributed to a time−dependent statistical operator initially chosen to describe a system, from the actual time−dependent Hamiltonian of the system. Examples of approximate solutions to von Neumann’s equation and Schrödinger’s equation that are obtained by use of the minimum principle confirm this identification.

DOI: 10.1063/1.522409

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A minimum principle is described which has von Neumann’s equation of motion of the statistical operator as its ultimate solution. When the statistical operator is initially restricted to be a projection of trace unity, the minimum principle has Schrödinger’s time−dependent equation as its ultimate solution. As a result, the present minimum principle is shown to be a generalization of one first introduced by McLachlan, with a modification of the Dirac−Frenkel variational principle as a necessary consequence. In physical terms, the quantity that is minimized is shown to be the mean−squared deviation of an effective time−dependent Hamiltonian, which can be attributed to a time−dependent statistical operator initially chosen to describe a system, from the actual time−dependent Hamiltonian of the system. Examples of approximate solutions to von Neumann’s equation and Schrödinger’s equation that are obtained by use of the minimum principle confirm this identification.

Approximate solutions to the time-dependent Schr\"odinger equation can be sought in terms of time-dependent parameters. In this work, a reasonable definition of the "best" values for these parameters is given by requiring that the time integral of the deviation of the approximate solution from the exact solution be minimized. A procedure for specifically evaluating the parameters to optimize the approximate solution is given in terms of equations analogous to Hamilton's equations of classical mechanics, which produce a minimum-deviation integral in analogy to Hamilton's principle. A comparison of this technique with previous approaches is made, both in terms of general principles, and in terms of a simple hypothetical example of a system subjected to a time-dependent perturbation.

The Dirac-Frenkel-McLachlan variational principle is the basic tool for obtaining computationally accessible approximations in quantum molecular dynamics. It determines equations of motion for an approximate time-dependent wave function on an approximation manifold of reduced dimension. This paper gives a near-optimality result for variational approximations. It bounds the error in terms of the distance of the exact wave function to the approximation manifold and identifies the parameters that control the deviation of the variational approximation from the best approximation on the manifold.

The time dependent harmonic oscillator is solved explicitly for quantum mechanics by the operator method with an auxiliary condition as the classical solution. Two classical invariant quantities which determine whether or not the system is bound are derived by the classical equation of motion. We obtain the invariant operator from one classical invariant quantity. Its eigenfunction is related to the solution of Schrodinger equation of the system and its eigenvalue is related to another classical quantity. The wave function is evaluated exactly by the eigenfunction of the invariant operator but it is not the eigenfunction of the Hamiltonian of the system. The uncertainty which calculates with the wave function is not a minimum one. We will confirm that the function which holds minimum uncertainty is a eigenfunction of the Hamiltonian.

This paper attempts to explain both the basic concepts and the mathematical formalism of von Neumann's operator method in quantum statistics, in terms that should be understandable by anyone who has mastered first courses in quantum mechanics and statistical mechanics. A simple model is used throughout the paper to illustrate the various theorems and techniques. The discussion begins with the statistical interpretation of the Heisenberg uncertainty principle, the definition of mixed ensembles, and the meaning of measurement in quantum mechanics. The von Neumann projection operator is explained, and is used to set up the ensemble density matrix and the von Neumann entropy operator. In the second part of the paper, to be published later, the quantum-mechanical equation of change with time is applied to the von Neumann operators.

We investigate variational principles for the approximation of quantum dynamics that apply for approximation manifolds that do not have complex linear tangent spaces. The first one, dating back to McLachlan [Mol. Phys. 8, 39–44 (1964)], minimizes the residuum of the time-dependent Schrödinger equation, while the second one, originating from the lecture notes of Kramer and Saraceno [Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Lecture Notes in Physics Vol. 140 (Springer, Berlin, 1981)], imposes the stationarity of an action functional. We characterize both principles in terms of metric and symplectic orthogonality conditions, consider their conservation properties, and derive an elementary a posteriori error estimate. As an application, we revisit the time-dependent Hartree approximation and frozen Gaussian wave packets.