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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Most-general minimality-preserving Hamiltonian

David Stoler

Published May 15, 1975• 14 citations
Publisher

Abstract

This paper is concerned with Hamiltonians that preserve the minimum value of the uncertainty product. Using the method of time-dependent unitary transformations, we derive the most general such Hamiltonian.
DOI
10.1103/physrevd.11.3033

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We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classical equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The wave functions and the propagators are evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q and P, but the values of the functions of q and the kinetic momentum pk{\mathrm{p}}_{\mathrm{k}}pk​ are equal to those of the other Q and Pk{\mathrm{P}}_{\mathrm{k}}Pk​ in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.