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arXiv:1404.0506

A note on the short-time quantum propagator

Sanayei, Ali

Published Apr 2, 2014• 1 citation
arXivHTMLPDF

Abstract

In the Feynman formalism of quantum mechanics one encounters a postulate, namely, that the propagator in an infinitesimal time-interval is the classical wave function. This postulate, which was later studied thoroughly by Holland, was recently highlighted by using the improved Makri-Miller propagator. The present note, whose conclusion is in agreement with that recent achievement, demonstrates that the Heisenberg picture of quantum mechanics invariably includes the Feynman postulate and is able to yield a proof for it. In other words, by starting out from the Heisenberg picture, it is proved that when terms of second-order in time can be neglected, the dynamics of a system is classical.

DOI
10.48550/arxiv.1404.0506

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In the Feynman formalism of quantum mechanics one encounters a postulate, namely, that the propagator in an infinitesimal time-interval is the classical wave function. This postulate, which was later studied thoroughly by Holland, was recently highlighted by using the improved Makri-Miller propagator. The present note, whose conclusion is in agreement with that recent achievement, demonstrates that the Heisenberg picture of quantum mechanics invariably includes the Feynman postulate and is able to yield a proof for it. In other words, by starting out from the Heisenberg picture, it is proved that when terms of second-order in time can be neglected, the dynamics of a system is classical.
We have shown in previous work that the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics requires the use of the Born and Jordan quantization rules. In the present work we give further evidence that the Born-Jordan rule is the correct quantization scheme for quantum mechanics. For this purpose we use correct short-time approximations to the action functional, initially due to Makri and Miller, and show that these lead to the desired quantization of the classical Hamiltonian.
In non-relativistic quantum mechanics, there exist several different ways of describing the time dependence of dynamics of physical systems. Of particular importance for quantum scattering theory are the Schrodinger, Heisenberg and Dirac pictures. For establishing a connection between quantum mechanical formalisms and classical mechanics, as well as for investigating relativistic solutions, the Heisenberg picture appears to be the most convenient. This picture of quantum mechanics is particularly adapted for the examination of free fields. The customary wave-mechanical version of quantum theory is obtained if we choose, e.g., in the framework of the Schrodinger picture to work with the coordinate representation. Possible results of measurements on a quantum system are called eigenvalues. To each of the latter values corresponds one or more eigenstates. An eigenstate is degenerate when more than one eigenstate correspond to a single eigenvalue.
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.