On a modified quantum statistical mechanics based on the Hartley information

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Abstract

The theoretical method described by Campos [D. Campos, A thermodynamic-like approach for the study of probabilistic systems, Physica A 390 (2011) 214. http://dx.doi.org/10.1016/j.physa.2010.09.023] is used for the study of quantum systems giving rise to a modified quantum statistical mechanics. The core of this approach includes a nonlinear relationship between Hartley information (pseudo-energy) and energy eigenvalues, and the use of thermodynamic-like functions parameterized by the continuous entropic parameter q (q[0,)).

The method is applied to a system with an infinite number of independent oscillators (e.g., blackbody radiation), and it is found that the pseudo-energy remains finite in contrast to the zero-point energy that becomes infinite. Since the quantum energy catastrophe problem (infinite energy) can be circumvented by focusing the physical analysis on the average Hartley information, the renormalization of the energy is avoided and the pseudo-energy emerges as a fundamental concept more basic than energy. As a second example, the method is applied to an analytical solvable system that has a discrete and continuous energy spectrum.

Highlights

► Quantum statistical mechanics is reformulated on the basis of Hartley information. ► Quantum energy catastrophe of the blackbody radiation is circumvented. ► Hartley information emerges as a fundamental concept more basic than energy.

Introduction

Consider a quantum mechanical system with Hamiltonian Hˆ(a,ν) and eigenvalue equation Hˆ(a,ν)|ϕn(a,ν)=En(a,ν)|ϕn(a,ν), where n labels the eigenstates, n=1,2,. The number of particles of the system (N classes of particles) and the external parameters are described by the sets ν{ν1,ν2,,νN} and a{a1,a2,,ag}, respectively.

The quantum system is characterized by using the thermodynamic-like approach for the study of probabilistic systems proposed in Ref. [1], and it is assumed that the energy spectrum E(a,ν){E1(a,ν),E2(a,ν),,EN(a,ν)} is known, where N is the number of states taken into account (finite or infinite).

Since the quantum system is in contact with the rest of the universe (the environment of the system) and the energy is a measurable or observable property, a probability set P{P1,P2,,PN} is assigned to the energy spectrum E(a,ν). It is assumed that an experiment would be capable of producing the eigenvalue En with (theoretical) probability Pn, if we were able to perform an infinite sequence of experiments.

The structure of the paper is the following. Section 2 describes the method for assigning a probability distribution P to the energy spectrum of the quantum system, taking into account that knowing P is a conditio sine qua non for the application of the treatment proposed in Ref. [1]. For deducing P theoretically, first an ensemble of systems is introduced and then a hypothetical compound system arranged with two non-interacting copies of the physical system is considered, so that one can take advantage of the product rule of probabilities for independent events.

In Section 3, the method proposed in Ref. [1] for the study of probabilistic systems is summarized, and it is applied directly to the probability distribution for N states deduced in Section 2. A special expression linking the Hartley information (also called surprise or pseudo-energy) with the energy eigenvalues of the system is obtained (see Eq. (7)), and thermodynamic-like functions are derived allowing a macro-description of the quantum system. This description is mainly based on the average pseudo-energy of the system and the Shannon entropy for the escort probability set of P, denoted by U(P(β),q) and S(P(β),q) respectively, where q(q[0,)) is the entropic parameter q of the Nonextensive Statistical Mechanics [2], and β plays the role of the thermodynamic β1/(kBT).

In Section 4, the method is applied to a system of quantum harmonic oscillators. For the case of the blackbody radiation that is described by an infinite number of oscillators, it follows that the average Hartley information remains finite in contrast to the zero-point energy that becomes infinite. In this way the quantum energy catastrophe problem, due to the zero-point energy, can be circumvented and the renormalization of the energy becomes unnecessary.

Assuming that the standard quantum statistical mechanics breaks down, because the energy of the blackbody radiation becomes infinite, the modified quantum statistical mechanics described in this paper emerges as an alternative for the macro-description of quantum systems. The essence of this approach is the concept of average pseudo-energy (Hartley information), instead of the concept of average energy. Along this line of thinking it seems that the Hartley information could be more basic than energy, a conclusion derived from Eq. (7), which links both concepts.

In Section 5, another analytically solvable example is considered, where the Hamiltonian has a spectrum with discrete and continuous parts. Since the system has a gap in the energy spectrum, the pseudo-heat capacity shows a Schottky behavior. Finally, Section 6 concludes with some remarks.

Section snippets

An ensemble of systems

In this section, a suitable ensemble of systems is introduced for deducing the probability distribution P that will be assigned to the energy spectrum E(a,ν) of the physical system described by the Hamiltonian Hˆ(a,ν). In the following, the notation will be simplified by dropping out the variables (a,ν).

Given the eigenvalue equation Hˆ|ϕn=En|ϕn, where n labels the eigenstates, an ensemble of systems is introduced as a collection of many copies of the system, all characterized by the same

Macro-description of a quantum system

In this section, the thermodynamic-like method described in Ref. [1] is applied to the probability distribution P(β), with elements given by Eq. (5).

Quantum harmonic oscillators and blackbody radiation

Consider an oscillator of frequency ωe, and the energy eigenvalues En=(n+1/2)ħωe of the Hamiltonian, with n=0,1,2,. The whole spectrum of energy is known (ωN(P(β))=1), and the partition function becomes Z(β)n=0exp(β(n+1/2)ħωe)=exp(βħωe/2)1exp(βħωe). For the calculation of thermodynamic potentials, one also requires ln(Z(qβ))=[12qβħωe+ln(1exp(qβħωe))].

In order to provide a visual exhibit of some results, consider the numerical values shown in Table 1, taken from Refs. [8], [9], [10].

An analytical solvable example

Consider a particle of mass m in the potential V(x)=2V0sech2(p0x/ħ), where V0p02/(2m)>0. The Hamiltonian has precisely one bound state ϕ0(x) associated with the eigenvalue E0=V0, and unbounded eigenfunctions ϕ(x,p1) associated with the continuous spectrum of eigenvalues E(p1)=p12/(2m)=(p1/p0)2V0, labeled by the parameter 0<p1<: ϕ0(x)=p02ħsech(p0xħ),ϕ(x,p1)=|p0|p02+p12exp(ip1xħ)(tanh(p0xħ)ip1p0). The normalization constant is chosen so that at large |x|, where tanh(p0x/ħ)=sign(p0x/ħ)=±1,

Final remarks

In this article, an arbitrary quantum system was considered by assuming that its energy spectrum is known. Then a probability distribution P(β) was deduced theoretically (Eq. (5)), where β emerged as a separation constant for a pair of equations. It was found that the probability Pn(β) for the n-th state looks like the one for the canonical ensemble, where the transformation β=1/(kBT) allows us to assign a “virtual temperature” T to the system.

Afterwards, the procedure proposed in Ref. [1] was

Acknowledgments

Graphics and some integrals were done by using Mathematica [19].

References (19)

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