arXiv:cond-mat/9808287 · Europhysics Letters (EPL)

Thermodynamics of the harmonic oscillator using coherent states

Jürgen Schnack

Published Mar 15, 1999• 26 citations

Abstract

The ongoing discussion whether thermodynamic properties can be extracted from a (possibly approximate) quantum-mechanical time evolution using time averages is fed with an instructive example. It is shown for the harmonic oscillator how the Hilbert space or an appropriately defined phase space must be populated in terms of coherent states in order to obtain the quantum result, respectively the classical one.

DOI: 10.1209/epl/i1999-00216-4

Related papers

Three-dimensional harmonic oscillator and time evolution in quantum mechanics

Pavel Kundrát, M. V. Lokajíček

The problem of defining a time (or phase) operator for a three-dimensional harmonic oscillator has been analyzed. A formula for this operator has been derived. The results have been used to demonstrate a possibility of representing quantum-mechanical time evolution in the framework of an extended Hilbert-space structure. The physical interpretation of the extended structure has been discussed briefly, too.

arXiv:quant-ph/0211026Physical Review AJan 13, 20035 citations

A phase in a coherent-state waveform - it is always irrelevant?

Waldemar Berej, P. Rozmej, Robert Arvieu

We point out that harmonic oscillator coherent states, in coordinate representation, require particular phase factor, in order to represent classical time evolution properly. The presence of such a phase is clearly stated only in a minority of scientific sources discussing properties of coherent states.

European Journal of PhysicsJan 1, 19990 citations

Coherent states

J. C. Garrison, R. Y. Chiao

The elementary quantum theory of quasi-classical states for mechanical harmonic oscillators is applied to the radiation oscillators describing light, and thus used to define quasi-classical states of light. These states are called “coherent”, because the expectation value of any normally ordered product of creation and annihilation operators factorizes for them. A theoretical discussion of possible sources for coherent states is followed by experimental evidence for their existence. The mathematical properties of coherent states, e.g., nonorthogonality and overcompleteness, are explained in order to use them to represent states and operators. These representations provide a suitable replacement for the classical phase space description of light, which is forbidden by the uncertainty principle. The quantum phase space approach leads to the definition of the Wigner distribution, the Q-function, the P-function, and Gaussian mixed states.

Oxford University Press eBooksJun 5, 20080 citations

Complex Coherent States and Non Equilibrium Quantum Thermodynamics

E. Celeghini

A possible thermodynamic evolution to equilibrium of a quantum statistical system is discussed. The description is entirely algebraic and it is done by means of complex coherent states built in the Universal Enveloping Algebra of the complex algebra https://www.w3.org/1998/Math/MathML" display="inline"> A 1 . Finite dimensional representations of https://www.w3.org/1998/Math/MathML" display="inline"> A 1 give fermion distributions while infinite dimensional ones describe bosons. The scheme is thermodynamic in the sense that fluctuations are not considered and this allows us to resort to an evolution operator that realizes, for t → ∞, a projector on the state of equilibrium. The shrinking of the Hilbert space is related to non unitary evolution that change the normalization of the state vectors. The theory is invariant under time reversal: vectors at t → ±∞ differ only in phases and both correspond to the equilibrium probability distribution. Equilibrium and non-equilibrium thermodynamic quantities can be calculated at microscopic level in a framework that is in between thermodynamic and statistical mechanics. Our approach is a physically founded example of non-linear quantum theory.

CRC Press eBooksNov 30, 20030 citations

Coherent States for General Potentials

Michael Martin Nieto, L. M. Simmons

We define coherent states for general potentials, requiring that they have the physically interesting properties of the harmonic-oscillator coherent states. We exhibit these states for several solvable examples and show that they obey a quantum approximation to the classical motion.

Physical Review LettersJul 24, 1978232 citations

Generalized coherent states and classical limits in quantum mechanics

M. G. A. Crawford

Coherent states in the harmonic oscillator have long been thought of as a bridge from quantum mechanics to classical mechanics. Many of their important properties (most notably those involving time evolution) are only valid in the harmonic oscillator and are at best approximations elsewhere. This thesis is an investigation into several means of generalizing coherent states for other systems and examines the resultant states for some systems. One system of interest is the harmonic oscillator with centripetal barrier for which coherent states of several disparate definitions coincide. Also studied is this spherical rotator, a system which is particularly amenable to defining annihilation operator coherent states. A third system under investigation is the hydrogen atom. This system serves as an arena for the development of an extension to a generalization due to Klauder [J. Phys. A. 29(12):L293-L298. 1996]. Klauder's construction is only applicable to systems without energy degeneracies and must be extended for application where degeneracies are present. The author provides a means for this extension and applies the complete construction to the hydrogen atom problem. As a demonstration of how this construction may be adapted, the author constructs Rydberg wave packets which are initially localized and exhibit full and fractional revivals in the long time evolution.

Jan 1, 20000 citations