Chapter 2. The Dynamic Variables in Quantum Mechanics

Eric Heller

A. O. Caldeira

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.

Alessandro Teta

Gabino Torres-Vega

There are many proposals for writing Classical and Quantum Mechanics in the same lan‐ guage. Some approaches use complex functions for classical probability densities [1] and other define functions of two variables from single variable quantum wave functions [2,3]. Our approach is to use the same concepts in both types of dynamics but in their own realms, not using foreign unnatural objects. In this chapter, we derive many inter relationships be‐ tween conjugate variables.

Valerio Magnasco

David K. Ferry

The introductory basics of quantum mechanics were said to arise from the changes from classical mechanics in the first chapter. In some sense, these changes are brought to an observable level by the smallness of some parameter, such as the size scale. In another sense, however, quantum mechanics introduces some fundamental changes, that are not at all present in classical mechanics. Perhaps the most important effect is the appearance of operators in place of dynamical variables and the non-commuting nature of these operators. While these properties are present in normal mathematics, they are not usually associated with discussions of classical mechanics.