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Applied Physics Research

Thermodynamic Properties of the Harmonic Oscillator and a Four Level System

Oladunjoye A. Awoga, Akpan N. Ikot, A. A. Essiett, Louis E. Akpabio

Published Apr 30, 2011• 4 citations
Publisher

Abstract

The thermodynamics properties of a quantum harmonic oscillator and four level oscillator systems are evaluated. These results lead to the exact value for the entropy of the system which corresponds to the second law of thermodynamics. We also show the numerical results for the harmonic and four level oscillators and it is in good agreement with the one obtained before in the literature.

DOI
10.5539/apr.v3n1p47

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The thermodynamics properties of a quantum harmonic oscillator and four level oscillator systems are evaluated. These results lead to the exact value for the entropy of the system which corresponds to the second law of thermodynamics. We also show the numerical results for the harmonic and four level oscillators and it is in good agreement with the one obtained before in the literature.
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We consider a harmonic oscillator (h.o.) in contact with a non-minimum uncertainty squeezed reservoir (but isolated from contact with other non-squeezed reservoirs). We calculate the h.o.’s density matrix and thermodynamic entropy. We interpret the derivatives of the entropy in terms of two temperatures, one for each quadrature of the reservoir. A change in the total (random) energy of the h.o. is shown to equal the sum of changes in the energies of each h.o. quadrature separately (a version of the First Law of thermodynamics). A change in the total entropy of the h.o. system is likewise shown to equal the sum of changes in the entropies of each h.o. quadrature separately (Second Law of thermodynamics). We also present equations that correspond to the so called “Fundamemental equation” and “Gibbs-Duhem equation” for the h.o system under consideration.
Boltzmann’s statistical interpretation of the second law of thermodynamics indicates that systems evolve from less probable to more probable states. In an isolated system, entropy reaches its maximum at equilibrium, corresponding to the greatest number of accessible microstates. The quantum uncertainty principle, a fundamental feature of quantum mechanics, reflects the intrinsic probabilistic nature of physical systems. This study proposes that the equilibrium state defined by maximal entropy and microstate accessibility also represents a condition of maximal intrinsic uncertainty. We examine the connection between Boltzmann’s entropy framework and the quantum mechanical uncertainty relation between position and momentum, focusing on localized harmonic oscillator systems. Properties of the uncertainty product are analyzed in detail. Both the uncertainty principle and the second law of thermodynamics are rooted in probabilistic foundations. Our findings suggest that the uncertainty principle provides a conceptual perspective on thermodynamic behavior.
We study the quantum entanglement of systems of coupled harmonic oscillators on the basis of thermo-field dynamics (TFD).For coupled harmonic oscillators at equilibrium, the extended entanglement entropy is derived using the TFD method, and it is demonstrated to be controlled by temperature and coupling parameters.For non-equilibrium systems, in addition to temperature and coupling parameters, the time dependence of the extended entanglement entropy is calculated in accordance with the dissipative von Neumann equation, and the dissipative dynamics of the systems of coupled harmonic oscillators is discussed.Consequently, based on TFD, the physical parametrization of the entanglement entropies is confirmed in both the equilibrium and nonequilibrium cases of harmonic oscillator systems by means of the laws of thermodynamics.
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We illustrate recent results concerning the validity of the work fluctuation theorem in open quantum systems [M. Campisi, P. Talkner, and P. Hänggi, Phys. Rev. Lett. {\bf 102}, 210401 (2009)], by applying them to a solvable model of an open quantum system. The central role played by the thermodynamic partition function of the open quantum system, -- a two level fluctuator with a strong quantum nondemolition coupling to a harmonic oscillator --, is elucidated. The corresponding quantum Hamiltonian of mean force is evaluated explicitly. We study the thermodynamic entropy and the corresponding specific heat of this open system as a function of temperature and coupling strength and show that both may assume negative values at nonzero low temperatures.