Abstract
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.
Highlights
Despite the didactic importance of a free particle in quantum mechanics, its coherent state analysis has long been untouched
It has been shown that the group-theoretic approach to the time evolution of quantum states is equivalent to the corresponding Lewis-Riesenfeld approach and the free-particle dynamics can be reproduced from the dynamics of the time-dependent harmonic oscillator by letting ω(t) → 0 for t →∞12
The difference between them is just a multiplication by a phase factor exp[iαλ(t)]. This leads to a localized Schrödinger solution that corresponds to the coherent state
Summary
Despite the didactic importance of a free particle in quantum mechanics, its coherent state analysis has long been untouched. A class of the well known coherent states is those of the simple harmonic oscillator[1,2,3,4]; they were originally obtained by Schrödinger[1] as specific quantum states, where the expectation values of the position and momentum operators in these states were the same as the corresponding classical solutions These states have a number of other interesting properties including the followings: (a) They are eigenstates of the destruction operator; (b) They are created from the ground state by a unitary operator; (c) They minimize the uncertainty relations and do not spread over time; (d) They are (over)complete and normalized, but not orthogonal. It is only recently that quantum dynamics of a free particle has been noticed and the time behavior of its semiclassical wave packets has been studied[13,14,15,16]
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