Coherent and semiclassical states of a free particle

Coherent states (CSs) 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 CSs 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 CSs regarded as examples of wave packets representing semiclassical motion. In this paper, we essentially follow the Malkin–Dodonov–Man'ko method to construct the CSs of a free nonrelativistic particle. We give a detailed discussion of the properties of the CSs 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 CSs can be considered semiclassical states.