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arXiv:1105.1220 · Journal of Modern Physics

Temperature Dependent Motion of a Massive Quantum Particle

Jian-Ping Peng

Published Jan 1, 2012• 2 citations
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Abstract

We report model calculations of the time-dependent internal energy and entropy for a single quasi-free massive quantum particle at a constant temperature. We show that the whole process started from a fully coherent quantum state to thermodynamic equilibrium can be understood, based on statistics of diffracted matter waves. As a result of thermal interaction between the particle and its surroundings, the motion of the particle shows new feature.

DOI
10.4236/jmp.2012.37083

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We report model calculations of the time-dependent internal energy and entropy for a single quasi-free massive quantum particle at a constant temperature. We show that the whole process started from a fully coherent quantum state to thermodynamic equilibrium can be understood, based on statistics of diffracted matter waves. As a result of thermal interaction between the particle and its surroundings, the motion of the particle shows new feature.
Abstract Quantum thermodynamics describes dynamic processes by means of the operators of entropy production P and time t . P and t do not commute. It exists the non-vanishing t–P commutator [ t , P ]=ik. Here the Boltzmann constant k has the physical meaning of a quantum of entropy. The t–P commutator immediately leads us to the t–P uncertainty relation ΔtΔP≥k/2. Hence the observables t and P are not sharply defined simultaneously. Similar uncertainty relations can also be expected for other pairs of conjugate variables with products of the physical meaning of an entropy. The free energy F and the reciprocal temperature (1/T) are the respective conjugate variables of an isolated system of many particles, which leads us to the F–(1/T) uncertainty relation ǀΔFǁΔ(1/T)ǀ≥k/2. It can be traced back to the t–P uncertainty relation mentioned above. In this way the Helmholtz free energy F and the temperatur T are introduced into quantum thermodynamics. The uncertainties ǀΔFǀ→0 and ǀΔTǀ→0 are negligible at low temperatures T→0, and quantum thermodynamics turns into the time-independent classical thermodynamics. Against this the uncertainties ǀΔFǀ→∞ and ǀΔTǀ→∞ grow unlimited at high temperatures T→∞, and classical thermodynamics loses its sense. In the limit of one particle the uncertainties cannot be neglected even at low temperatures. However a detailed discussion shows that the free energy f of a single particle vanishes within the whole range of temperatures T. This defines the particle entropy σ=ε/T=ak. The dimensionless entropy number a connects the particle energy ε=akT with the temperature T. The entropy number a of a single (s) independent particle can be calculated with the extended, temperature-dependent Schrödinger equation A s φ=aφ. Here A s=−(Λ 2/4π)∇2 means the dimensionless entropy operator describing the entropy number a and thus the particle entropy σ=ak. Λ is the thermal de Broglie wave length. Finally we calculate by means of quantized particle entropies σ the internal energy E, the Helmholtz free energy F, the entropy S, the chemical potential μ, and the equation of state of an ideal gas of N monatomic free particles in full agreement with classical thermodynamics. We also calculate the partition function q=V/Λ 3 of a single free particle within the volume V. Here Λ 3 is a small volume element taking into account the wave–particle dualism of a single free particle of mass m at temperature T. Extension to a system of N free particles leads us to a simple geometrical model and to the conclusion that an ideal gas of independent particles becomes instable below a critical temperature T C. T C corresponds to the critical temperature T BE of Bose–Einstein condensation.
In this paper the detailed investigation of one of the most interesting models in the non relativistic quantum mechanics of one massive particle i.e., introduced by G. Poeschl and E. Teller in 1933 is continued; the starting point of dynamic analysis for potentials, wave functions and energy spectrum was carried on by the same authors in the paper. The generalization of these results on the case of nonzero temperature includes the approximate calculation of the partition function of the model and then also the Massieu Planck thermodynamic potential and its most important derivatives, i.e. internal energy, heat capacity and pressure. The analysis of the results is carried on separately for low and high temperature regions because of the lack of exact unifying expressions of the potential at all temperatures.
A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.