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arXiv:quant-ph/0106143

On Some Peculiarities of Solving Nonstationary Problems of Quantum Mechanics

V.E. Mitroshin

Published Jun 25, 2001• 0 citations
arXivHTMLPDF

Abstract

Exact solutions of several nonstationary problems of quantum mechanics are obtained. It is shown that if the initial conditions of the problem correspond to the localized-in-space particle, then it moves exactly along the classical trajectory, and the wave packet is not spread in time.
DOI
10.48550/arxiv.quant-ph/0106143

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This chapter considers an approximate method of solving the Schrodinger equation in the case when the wavelength of the particle is small by comparison with the distance over which the potential changes appreciably. Since the limit of short wavelength corresponds to the limit of classical mechanics, the quasiclassical approximation allows to explore the relation between classical and quantum mechanics. The chapter discusses how to match the quasiclassical wave function f in the region accessible to classical motion with the wave function in the forbidden region. Since the action function is a minimum along the classical trajectory, the pencil of trajectories adjacent to the classical one gives the minimum phase factor. The chapter examines the problem of the escape of a particle from a potential barrier; this problem reduces to the problem of the spreading of a wave packet which describes a particle which at the initial time was inside the barrier.
The behavior of a particle moving according to the laws of quantum mechanics in a field, which potential changes over time, is studied. The method of unitary transformations for solving the temporal Schrödinger equation is considered. The reduced Hamiltonian of the system of a quantum­-mechanical particle in time­-dependent potential is obtained, as well as the total operator of the evolution of such a system. We find a new version of the unitary transformation, which, compared to the known ones, simplifies solving the problem in analytical form. This transformation associates the linear potential quantum model with time-­dependent parameters and free­particle system. Based on the constructed evolution operator, we consider the wave packet dispersion. It is shown that the wave packet dispersion for this model is the same as that for a free particle. This conclusion is the general result of the behavior of a quantum­mechanical particle moving in the field of a time­dependent linear potential.