Coherent and semiclassical states of a free particle

TL;DR

This paper constructs and analyzes coherent states for a free particle, exploring their properties, semiclassical conditions, and potential as wave packets representing classical-like motion in quantum mechanics.

Contribution

It provides a detailed construction of free-particle coherent states using the Malkin-Dodonov-Man'ko method, filling a gap in the understanding of CS for unbound systems.

Findings

Coherent states for free particles are explicitly constructed.

Properties such as completeness and uncertainty minimization are analyzed.

Conditions for semiclassical behavior of free-particle CS are identified.

Abstract

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.