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Journal of Physics Conference Series

On the definition of entropy for quantum unstable states

Osvaldo Civitarese, Manuel Gadella

Published Jan 22, 2015• 0 citations
Publisher

Abstract

The concept of entropy is central to the formulation of the quantum statistical mechanics, and it is linked to the definition of the density operator and the associated probabilities of occupation of quantum states. The extension of this scheme to accommodate for quantum decaying states is conceptually difficult, because of the nature of these states. Here we present a way to treat quantum unstable states in the context of statistical mechanics. We focuss on the definition of the entropy and avoid the use of complex temperatures.

DOI
10.1088/1742-6596/578/1/012006

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For pt.I see ibid., vol.25, p.359 (1992). Regarding the discussion of Peres (1989) and Weinberg (1989) concerning a suitable definition of entropy in nonlinear quantum mechanics, two further observations are made. Firstly, by regarding the covariance matrix of the probability distribution over the phase space of wavefunctions as the nonlinear counterpart (p NL ) of the (linear) density matrix (p L ) and employing -Tr p NL In p NL , one obtains a limiting transition (as nonlinearities vanish), in which this entropy measure converges to the definition in ordinary quantum mechanics, -Tr p L In p L . Secondly, it is argued that Peres' contention that 'nonlinear variants of the Schrodinger equation violate the second law of thermodynamics' is flawed in that it relies upon the entropy of mixing of nonorthogonal states which as Dieks an van Dijk (1988) have indicated is an undefined concept. A proper approach to associating a quantum mechanical entropy with a mixture of a particle into two non-orthogonal states-by first estimating a suitable two-particle density matrix (p) and then employing -Tr p In p-is outlined.

Journal of Physics A Mathematical and General•Aug 7, 1992•2 citations
The concept of entropy is central to the formulation of the quantum statistical mechanics, and it is linked to the definition of the density operator and the associated probabilities of occupation of quantum states. The extension of this scheme to accommodate for quantum decaying states is conceptually difficult, because of the nature of these states. Here we present a way to treat quantum unstable states in the context of statistical mechanics. We focuss on the definition of the entropy and avoid the use of complex temperatures.
In this paper, we review the concept of entropy in connection with the description of quantum unstable systems. We revise the conventional definition of entropy due to Boltzmann and extend it so as to include the presence of complex-energy states. After introducing a generalized basis of states which includes resonances, and working with amplitudes instead of probabilities, we~found an expression for the entropy which exhibits real and imaginary components. We discuss the meaning of the imaginary part of the entropy on the basis of the similarities existing between thermal and time evolutions.
In this paper, we review the concept of entropy in connection with the description of quantum unstable systems. We revise the conventional definition of entropy due to Boltzmann and extend it so as to include the presence of complex-energy states. After introducing a generalized basis of states which includes resonances, and working with amplitudes instead of probabilities, we found an expression for the entropy which exhibits real and imaginary components. We discuss the meaning of the imaginary part of the entropy on the basis of the similarities existing between thermal and time evolutions.
Entropy is a fundamental concept which emerged along with other ideas during the development of thermodynamics and statistical mechanics Landau and Lifshitz (1978); Lieb and Yngvason (1999). Entropy has developed foremost out of phenomenological thermodynamical considerations such as the second law of thermodynamics in which it plays a prominent role Wehrl (1978). With the intense interest in the investigation of the physics of matter at the atomic and subatomic quantum levels, it maywell be askedwhether this concept can emerge out of the study of systems at a more fundamental level. In fact, it may be argued that a correct definition is only possible in the framework of quantum mechanics, whereas in classical mechanics, entropy can only be introduced in a rather limited and artificial way. Entropy relates macroscopic and microscopic aspects of nature, and ultimately determines the behavior of macroscopic systems. It is the intention here to present an introduction to this subject in a readable manner from the quantum point of view. There are many reasons for undertaking this. The intense interest in irreversible thermodynamics Grossing (2008), the statistical mechanics of astrophysical objects Padmanabhan (1990); Pathria (1977), quantum gravity and entropy of black holes Peres & al. (2004), testing quantum mechanics Ballentine (1970) and applications to condensed matter and quantum optics Haroche & al. (2006); Raimond & al. (2001) are just a few areas which are directly or indirectly touched on here. Let us begin by introducing the concept of entropy from the quantummechanical perspective, realizing that the purpose is to focus on quantum mechanics in particular. Quantum mechanics makes a clear distinction between observables and states. Observables such as position and momentum are mathematically described by self-adjoint operators in a Hilbert space. States, which are generally mixed, can be described by a density matrix, which is designated by ρ throughout. This operator ρ is Hermitean, has trace one and yields the expectation value of an observable A in the state ρ through the definition
This paper introduces a novel definition of non-classical information entropy, rooted in the principles of statistical physics, particularly quantum mechanics. The core premise is that traditional information entropy, derived from classical probability theory, is insufficient to capture the essence of information within non-classical systems exhibiting phenomena such as superposition and entanglement. We propose a definition based on the concept of accessible states and their associated probabilities, extending the notion of entropy beyond the realm of purely deterministic systems. The paper details the mathematical formulation of this non-classical information entropy, denoted as *Hnc*, and presents a method for its calculation. The method relies on identifying the set of accessible states for a given system and then applying a generalized form of Shannon's entropy formula, modified to account for the inherent uncertainties associated with quantum mechanics. The resulting *Hnc* provides a quantitative measure of the information content within non-classical systems, offering a potential framework for studying information processing and dynamics in quantum environments. The key contribution lies in bridging the gap between information theory and non-classical statistical physics, offering a new lens through which to understand information in quantum systems.
The proper definition of entropy is fundamental to the relationship between statistical mechanics and thermodynamics. It also plays a major role in the recent debate about the validity of the concept of negative temperature. In this paper, I analyze and calculate the thermodynamic entropy for large but finite quantum mechanical systems. A special feature of this analysis is that the thermodynamic energy of a quantum system is shown to be a continuous variable, rather than being associated with discrete energy eigenvalues. Calculations of the entropy as a function of energy can be carried out with a Legendre transform of thermodynamic potentials obtained from a canonical ensemble. The resultant expressions for the entropy are able to describe equilibrium between quantum systems having incommensurate energy-level spacings. This definition of entropy preserves all required thermodynamic properties, including satisfaction of all postulates and laws of thermodynamics. It demonstrates the consistency of the concept of negative temperature with the principles of thermodynamics.
For pt.I see ibid., vol.25, p.359 (1992). Regarding the discussion of Peres (1989) and Weinberg (1989) concerning a suitable definition of entropy in nonlinear quantum mechanics, two further observations are made. Firstly, by regarding the covariance matrix of the probability distribution over the phase space of wavefunctions as the nonlinear counterpart (p NL ) of the (linear) density matrix (p L ) and employing -Tr p NL In p NL , one obtains a limiting transition (as nonlinearities vanish), in which this entropy measure converges to the definition in ordinary quantum mechanics, -Tr p L In p L . Secondly, it is argued that Peres' contention that 'nonlinear variants of the Schrodinger equation violate the second law of thermodynamics' is flawed in that it relies upon the entropy of mixing of nonorthogonal states which as Dieks an van Dijk (1988) have indicated is an undefined concept. A proper approach to associating a quantum mechanical entropy with a mixture of a particle into two non-orthogonal states-by first estimating a suitable two-particle density matrix (p) and then employing -Tr p In p-is outlined.