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CRC Press eBooks

Time Evolution of Quantum Systems

Nandita Rudra

Published Aug 19, 2019• 0 citations
Publisher

Abstract

Our ultimate objective is to know the dynamics of quantum system and obtain the equation of motion. We emphasize that time in non-relativistic mechanics is not a dynamical variable and we have treated it like a parameter.
DOI
10.1201/9780429325762-4

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In the two preceding chapters of this book we have analyzed many interesting physical phenomena in magnetic and superconducting systems which could adequately be described by phenomenological dynamical equations in terms of collective classical variables. One unavoidable consequence of this approach is that, as we are always dealing with variables that describe only part of the whole system, the interaction with the remaining degrees of freedom shows up through the presence of non-conservative terms which describe the relaxation of those variables to equilibrium. Those phenomenological equations are able to describe a very rich diversity of physical phenomena, in particular, those which can be studied in the context of quantum mechanics. Since these are genuine dynamical equations, there is no reason why they should be restricted to classical physics. However, as we do not yet know how to treat dissipative effects in quantum mechanics, we have deliberately neglected those terms when trying to describe quantum mechanical effects of our collective variables. In this chapter we will describe the general approach to dealing with dissipation in quantum mechanics. However, before we embark on this enterprise we should spend some time learning a little bit about the classical behavior of dissipative systems. In this way we can develop some intuition on how systems evolve during a dissipative process and, hopefully, this will be useful later on when we deal with quantum mechanical systems. The immediate problem we have to face concerns the choice of dissipative system to be studied.
Abstract Our conception of time changed from the lofty absolute coordinate of the classical paradigm to merely a connection between local reference frames, thanks to the Special Theory of Relativity, a connection that depends on knowing the relative velocity between frames, and that entails knowing local velocities. We measure local velocities using a clock that cannot give us precise time because knowing time depends solely on knowing the process that we use to generate time. In the case of both macroscopic and microscopic clocks, an intrinsic uncertainty blocks us from knowing both the dynamic and static states of the variables involved. By focusing on the conceptual process that we use to describe physical systems, we can approach a common epistemology of both microscopic and macroscopic dynamical systems. We find that both systems of description require that we inser t time implicitly into the formalism to account for the conservation of identity through our own psychological experience of time. Although we cannot define exactly the nature of the conserved identity, we can describe its evolution with pairs of conjugate variables, one tracking the static state and one the dynamic state. Uncertainty extends from our conceiving system into our descriptive theories. An epistemology that emphasizes the balance between the conjugate variables as they serve to describe the evolution of the identity provides a common ground on which to discuss both the Heisenberg uncertainty principle of quantum mechanics and the bounded randomness of a deterministic macroscopic system. We have seen time go from a global to a local phenomenon, and perhaps now we must allow time to become an epiphenomenon even at the local level, one that we ourselves insert to account for the continuity of our own conscious experience.
This paper presents a novel approach to understanding the role of harmonic dynamics and gaining a deeper appreciation for its impact within and outside of quantum mechanics. This includes consequences of harmonic dynamics and the uncertainty principle for anomalous diffusion and for the time-frequency analysis of chirp signals. In this approach, we consider a contact transformation to view a system of canonical variables with coordinate xxx and momentum pxp_xpx​ in the context of a new system of "generalized" coordinates and momentum. This new system is first studied in the context of non-relativistic quantum mechanics. The classical analog is then explored by use of the Poisson bracket equation. From this, new implications are demonstrated in classical phenomena. One is for a new model of Anomalous and Normal Diffusion. In another, we introduce the concept of the "Mixed Fourier Transform" which explores a new Gaussian Fourier Transform kernel in terms of the generalized variables. This has the ultimate objective of "harmonizing" chirp signals or producing a harmonic signal from an otherwise non-harmonic chirp.