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THE QUANTUM-MECHANICAL INTERPRETATION OF TIME

Syed Arif Kamal

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

If time is substituted in the equation of motion in the Heisenberg picture the results obtained may lead to confusions if the equations are not interpreted correctly. Some examples are given.

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We clarify certain confusions in the literature of the density operator in quantum mechanics, and demonstrate that the quantum Liouville theorem has the same form in both the Schrodinger and the Heisenberg pictures.Our starting point is to treat the density operator as an observable which has its specific time dependence in each of the two pictures.It is further shown that such a formulation will provide the exact correspondence between classical and quantum statistical mechanics with the Liouville theorem being interpreted as a conservation law, which is derivable from the equation of motion only in the quantum case.
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Open quantum system theory is fundamental for a proper description of quantum systems in different contexts such as chemistry, condensed matter physics, bio-physics and opto-mechamics. Quantum brownian motion which is analytically described by the Hu-Paz-Zhang (HPZ) model represents one of the most important examples of an open quantum system. In this paper we propose a novel derivation of its master equation, based on the Heisenberg picture. We provide an exact and analytic equation both for the operators as well as for the states. The result, equivalent to the one derived originally by HPZ, is expressed in terms of the spectral density, regardless the strength of the coupling between the system and the environment and allows to compute the time evolution of physically relevant quantities in a much easier way, since it is directly expressed in terms of the spectral density. An example is explicitly studied.