Coherent and semiclassical states of a free particle
a Tomsk State University, prosp. Lenina 36, Tomsk, 634050, Russian Federation
b Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prosp. 53, Moscow, 119991, Russian Federation
c Universidade de São Paulo, Instituto de Física, São Paulo, Brazil
d Universidade de São Paulo, R. da Reitoria 109, São Paulo, 05508-900, Brazil
Coherent states (CS) were first introduced and studied in detail for bound motion, discrete spectrum system like the harmonic oscillator and similar systems with a quadratic Hamiltonian. However, the problem of constructing CS has not yet received detailed investigation 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 where CSs can be considered as examples of wave packets representing semiclassical motion. In this paper we follow essentially 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 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 as semiclassical states.