Coherent states of an accelerated particle

TL;DR

This work advances the theory of coherent states for unbounded quantum motion by constructing generalized coherent states for a massive accelerated particle. Using a non-commutative integration (integrals of motion) framework, it builds a complete nonstationary basis parameterized by $η$ and connects it to Airy-function stationary states, then derives time-dependent GCS with Gaussian wave packets following classical accelerated trajectories. The authors establish completeness, orthogonality, and uncertainty-minimization properties, and identify a one-parameter CS family governed by the initial coordinate spread $\sigma_q$ that remains semiclassical over time for acceleration. The results provide a rigorous, tractable framework for CS in driven, unbounded systems and offer pedagogical insight into quantum-classical correspondence in continuous-spectrum scenarios.

Abstract

We construct generalized coherent states (GCS) of a massive accelerated particle. This example is an important step in studying coherent states (CS) for systems with an unbounded motion and a continuous spectrum. First, we represent quantum states of the accelerated particle both known and new ones obtained by us using the method of non-commutative integration of linear differential equations. A complete set of non-stationary states for the accelerated particle is obtained. This set is expressed via elementary functions and is characterized by a continuous real parameter $η$, which corresponds to the initial momentum of the particle. A connection is obtained between these solutions and stationary states, which are determined by the Airy function. We solved the problem of constructing GCS, in particular, semiclassical states describing the accelerated particle, within the framework of the consistent method of integrals of motion. We have found different representations, coordinate one and in a Fock space, analyzing in detail all the parameters entering in these representations.

Figures (2)

• Figure 1: The shape of the wave packet at the initial moment of time for $\sigma _{q}=0.2$ and $q_{0}=0$.
• Figure 2: Evolution of the probability density for $q_{0}=0$, $p_{0}=1$, $\sigma_{q}=0.4$. Blue color shows the particle distribution density at the initial moment of time with initial conditions $q_{0}=0$, $p_{0}=1$. The standard deviation at the initial time is chosen to be $\sigma _{q}=0.4$. Yellow color shows the evolution of the distribution density of an accelerated particle, which is acted upon by a force $F=2$ and which is located at point $q=1$ and has momentum $p=2.23$. Green color shows the evolution of the distribution density of an accelerated particle at the same point $q=1$ with momentum $p=3.61$, which is acted upon by a force $F=6$.