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arXiv:1004.2620

Coherent states for free particles

A. de la Torre, Dardo Goyeneche

Published Apr 15, 2010• 3 citations
arXivHTMLPDF

Abstract

The coherent states are reviewed with particular application to the free particle system. The didactic advantages of the formalism is emphasized. Several interesting features, like the relation of the coherent states with the Galilei group and with the Husimi distribution are presented. Keywords: quantum mechanics, coherent states, free particle.

DOI
10.48550/arxiv.1004.2620

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The coherent states are reviewed with particular application to the free particle system. The didactic advantages of the formalism is emphasized. Several interesting features, like the relation of the coherent states with the Galilei group and with the Husimi distribution are presented. Keywords: quantum mechanics, coherent states, free particle.
Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian. However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion. In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle. We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density. We describe the physical conditions under which free-particle CS can be considered semiclassical states.
Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian.However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion.In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle.We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density.We describe the physical conditions under which free-particle CS can be considered semiclassical states.
Despite the didactic importance of a free particle in quantum mechanics, its coherent state analysis has long been untouched. It is only recently that it has been noticed and studied in the semiclassical domain. While the previously known solutions, reported by Bagrov et al. for a free particle, are described using the linear non-Hermitian invariant operator, we show in this work that the general solution of the Schrödinger equation can also be naturally derived using a simpler method based on an Hermitian linear invariant operator. According to this, an exact Gaussian wave function that corresponds to a coherent state solution is obtained. An interpretation for such general quantum solution designated within the Lewis-Riesenfeld framework is provided and, further, quantum-classical correspondence principle for the system is reexamined.
We construct the coherent states of a free particle by implementing a coordinate transformation in the extended configuration space, which establishes a correspondence between the solutions of the Schrödinger equation for a harmonic oscillator and those for a free particle. Using this framework, we derive analytical expressions for the coherent states of a free particle, avoiding the complexities of non-normalizable fiducial states, integrals of motion, or group-theoretic approaches. Our method provides a systematic way to characterize these states at any instant while ensuring that they satisfy the Robertson-Schrödinger uncertainty relation.
In this work, we review two formalisms of coherent states for the case of a particle in a magnetic field. We focus our revision on both pioneering (Feldman and Kahn 1970 Phys. Rev. B 1 4584) and recent (Kowalski and Rembieliński 2005 J. Phys. A: Math. Gen. 38 8247 ) formulations of coherent states for this problem. We introduce a general application, the Husimi function, which takes into account collective and environmental effects, and which can be derived in a variety of ways. However, we calculate this function as the expected value of the density operator in a basis of coherent states.
In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the coherent states are isomorphic to a coset space of group geometrical space. Thus the topological and algebraic structure of the coherent states as well as the associated dynamical system can be extensively discussed. In addition, a quantum-mechanical phase-space representation is constructed via the coherent-state theory. Several useful methods for employing the coherent states to study the physical phenomena of quantum-dynamic systems, such as the path integral, variational principle, classical limit, and thermodynamic limit of quantum mechanics, are described.