Physica A: Statistical Mechanics and its Applications
On a modified quantum statistical mechanics based on the Hartley information
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 and eigenvalue equation , where labels the eigenstates, . The number of particles of the system ( classes of particles) and the external parameters are described by the sets and , 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 is known, where 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 is assigned to the energy spectrum . It is assumed that an experiment would be capable of producing the eigenvalue with (theoretical) probability , 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 to the energy spectrum of the quantum system, taking into account that knowing is a conditio sine qua non for the application of the treatment proposed in Ref. [1]. For deducing 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 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 , denoted by and respectively, where is the entropic parameter of the Nonextensive Statistical Mechanics [2], and plays the role of the thermodynamic .
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 that will be assigned to the energy spectrum of the physical system described by the Hamiltonian . In the following, the notation will be simplified by dropping out the variables .
Given the eigenvalue equation , where 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 , with elements given by Eq. (5).
Quantum harmonic oscillators and blackbody radiation
Consider an oscillator of frequency , and the energy eigenvalues of the Hamiltonian, with . The whole spectrum of energy is known , and the partition function becomes For the calculation of thermodynamic potentials, one also requires
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 in the potential , where . The Hamiltonian has precisely one bound state associated with the eigenvalue , and unbounded eigenfunctions associated with the continuous spectrum of eigenvalues , labeled by the parameter : The normalization constant is chosen so that at large , where ,
Final remarks
In this article, an arbitrary quantum system was considered by assuming that its energy spectrum is known. Then a probability distribution was deduced theoretically (Eq. (5)), where emerged as a separation constant for a pair of equations. It was found that the probability for the -th state looks like the one for the canonical ensemble, where the transformation allows us to assign a “virtual temperature” to the system.
Afterwards, the procedure proposed in Ref. [1] was
Acknowledgments
Graphics and some integrals were done by using Mathematica [19].
References (19)
A thermodynamic-like approach for the study of probabilistic systems
Physica A
(2011)- et al.
Measurability of vacuum fluctuations and dark energy
Physica A
(2007) - et al.
Could dark energy be measured in the lab?
Phys. Lett. B
(2005) - et al.
Nonextensive thermostatistical investigation of the blackbody radiation
Chaos Solitons Fractals
(2002) Introduction to Nonextensive Statistical Mechanics
(2009)Quantum Mechanics
(2005)Fundamentals of Statistical and Thermal Physics
(2008)- et al.
Vorticity, Statistical Mechanics, and Monte Carlo Simulation
(2007) Probability Theory
(1970)
Cited by (1)
Macroscopic characterization of data sets by using the average absolute deviation
2014, Physica A: Statistical Mechanics and its Applications
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Member of the Colombian Academy of Sciences (ACCEFYN).