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Physics

Classical Limit, Quantum Border and Energy

A.M. Kowalski, Angelo Plastino, Gaspar Gonzalez

Published Jul 26, 2023• 1 citation
Publisher

Abstract

We analyze the (dynamical) classic limit of a special semiclassical system. We describe the interaction of a quantum system with a classical one. This limit has been well studied before as a function of a constant of motion linked to the Heisenberg principle. In this paper, we investigate the existence of the mentioned limit, but with reference to the total energy of the system. Additionally, we find an attractive result regarding the border of the transition.

DOI
10.3390/physics5030053

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We analyze the (dynamical) classic limit of a special semiclassical system. We describe the interaction of a quantum system with a classical one. This limit has been well studied before as a function of a constant of motion linked to the Heisenberg principle. In this paper, we investigate the existence of the mentioned limit, but with reference to the total energy of the system. Additionally, we find an attractive result regarding the border of the transition.
Based on a quantum dynamical invariant of motion, I, we study the classical limit of a semiclassical Hamiltonian composed by a full quantum harmonic oscillator plus a classical particle plus a "semiclassical" coupling quartic term. The motion-invariant is closely related to the uncertainty principle. The classical limit (CL) is determined by the relationship between I and the total energy of the system, defining an adimensional invariant Er. We find that the CL coincides with the results of a purely classical treatment. Both invariants allow to follow the transit between quantum nonchaotic to the classical chaotic regime. Particularly, with Er we define the threshold above which chaos appears, and the interval during which both regimes co-exist.
In this paper, a reference to the semiclassical model, in which quantum degrees of freedom interact with classical ones, is considered. The classical limit of a maximum-entropy density matrix that describes the temporal evolution of such a system is analyzed. Here, it is analytically shown that, in the classical limit, it is possible to reproduce classical results. An example is classical chaos. This is done by means a pure-state density matrix, a rather unexpected result. It is shown that this is possible only if the quantum part of the system is in a special class of states.
We investigate the classical limit of the dynamics of a semiclassical system that represents the interaction between matter and a given field.On using as a quantifier the q-Complexity, we find that it describes appropriately the quantum-classical transition, detecting the most salient details of the changeover.Additionally the q-Complexity results a better quantifier of the problem than the q-entropy, in the sense that the q-range is enlarged, describing the q-Complexity, the most important characteristics of the transition for all q-value.
We investigate the classical limit of the dynamics of a semiclassical system that represents the interaction between matter and a given field. On using as a quantifier the q- Complexity, we find that it describes appropriately the quantum-classical transition, detecting the most salient details of the changeover. Additionally the q-Complexity results a better quantifier of the problem than the q-entropy, in the sense that the q-range is enlarged, describing the q-Complexity, the most important characteristics of the transition for all q-value.
We analyze the nonlinear dynamics of a quartic semiclassical system able to describe the interaction of matter with a field. We do it in both dissipative and conservative scenarios. In particular, we study the classical limit of both frameworks and compare the associated features. In the two environments, we heavily use a system's invariant, related to the Uncertainty Principle, that helps to determine how the dynamics tends to the pertinent classical limit. We exhibit the convergence to the classical limit and also verify that the Uncertainty Principle is complied with during the entire process, even in the presence of dissipation.
We set up a semiclassical approximation which helps us clarify by means of several simple examples the rich variety of time scale in the quantum domain. The underlying structure of quantum and classical mechanics is so completly different that it is naive to expect to reach a classical regime by counting powers of the quantum scale variant Planck's over 2pi. We show although it is possible to define a time scale for nonclassical phenomena, but it is impossible to characterize quantum dynamics through a unique time scale, such as Ehrenfest's time. We use simple systems to critically discuss and illustrate these features of the quantum-classical limit.