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Time Evolution In Macroscopic Systems. I: Equations of Motion

Walter T. Grandy

Published Jan 1, 2003• 14 citations
OpenAlex

Abstract

Abstract. Time evolution of macroscopic systems is re-examined primarily through further analysis and extension of the equation of motion for the density matrix ρ(t). Because ρ contains both classical and quantum-mechanical probabilities it is necessary to account for changes in both in the presence of external influences, yet standard treatments tend to neglect the former. A model of time-dependent classical probabilities is presented to illustrate the required type of extension to the conventional time-evolution equation, and it is shown that such an extension is already contained in the definition of the density matrix. 1.

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•May 7, 2005•3 citations
Abstract. Time evolution of macroscopic systems is re-examined primarily through further analysis and extension of the equation of motion for the density matrix ρ(t). Because ρ contains both classical and quantum-mechanical probabilities it is necessary to account for changes in both in the presence of external influences, yet standard treatments tend to neglect the former. A model of time-dependent classical probabilities is presented to illustrate the required type of extension to the conventional time-evolution equation, and it is shown that such an extension is already contained in the definition of the density matrix. 1.
The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
The conventional quantum theory gives no real, that is, independent of the observer, natural description. If it is required that it must give an absolute reality of the natural phenomena at least in the macroscopic region, then the quantum theory must be changed. One possibility of that kind is discussed. A model of a quantum theory, which for small systems agrees with the usual theory, but for large systems deviates so much from them that the absolute reality of macroscopic size remains steady, is described. This was achieved by means of an additional term in the motion equation for the density matrix. This additional term plays no role for the small system. For macroscopic ion of the density matrix as if'' all macroscopic size of the system was observed perennially. (tr-auth)
Abstract We investigate quantum effects in the evolution of general systems. For studying such temporal quantum phenomena, it is paramount to have a rigorous concept and profound understanding of the classical dynamics in such a system in the first place. For this reason, we derive from first principles equations of motions that describe the classical propagation in quantum systems. A comparison of this classical model with the actual temporal quantum behavior enables us to identify quantum phenomena in the system’s dynamics and distinguish them from static quantum features at individual points in time. For instance, we show how Newton’s second law emerges as a special case of our general treatment, connecting it to a Schrödinger-type equation. As applications of our universal technique, we analyze nonlinear optical processes, semiclassical models, and the multipartite entanglement dynamics of macroscopic ensembles.
The theory of quantum statistical mechanics is reviewed in the light of the quantum probability theory developed in recent years by mathematicians. We show that—which is our main motivation here—the former is a special model in the framework of the latter, by extending the characteristic function in the classical probability theory to the quantum probability theory based on the requirements of quantum statistical mechanics. We find that due to the fact that fundamental dynamical variables of a quantum system must respect certain commutation relations, one has to reform the classical characteristic function defined on a Euclidean space to a normalized, nonnegative definite function defined on a group which is generated by a Lie algebra formed from these commutation relations. Our key finding is that the group-theoretical characteristic function could be adopted to replace the density matrix commonly used in quantum statistical mechanics for representing a state of the quantum ensemble, and more than that, the new representation has some significant advantages in applications. To show its advantages compared with the conventional density matrix scenario, we illustrate its applications in: solving the master equation that describes the irreversible evolution of a quantum open system; determining the pointer states of a quantum open system; and expanding the density matrix with the pointer states. In particular, thanks to the quantum characteristic function, we are able to write down the solutions of the quantum-optical master equation explicitly, obtain its steady solutions and verify that the pointer states are exactly coherent states. Besides, the Caldeira-Leggett master equation for the quantum Brownian motion and its extension to the Lindblad type are also studied.
Certain intriguing consequences of the discreteness of time on the time evolution of dynamical systems are discussed. In the discrete-time classical mechanics proposed here, there is an {\\it arrow of time} that follows from the fact that the replacement of the time derivative by the backward difference operator alone can preserve the non-negativity of the phase space density. It is seen that, even for free particles, all the degrees of freedom are {\\it correlated} in principle. The forward evolution of functions of phase space variables by a finite number of time steps, in this discrete-time mechanics, depends on the entire continuous-time history in the interval [0,infty][0, \\infty][0,infty]. In this sense, discrete time evolution is {\\it nonlocal} in time from a continuous-time point of view. A corresponding quantum mechanical treatment is possible {\\it via} the density matrix approach. The interference between non-degenerate quantum mechanical states decays exponentially. This {\\it decoherence} is present, in principle, for all systems; however, it is of practical importance only in macroscopic systems, or in processes involving large energy changes.