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Student mathematical library

One-dimensional problems of quantum mechanics. A free one-dimensional particle

L. D. Faddeev, O. Yakubovskiĭ

Published Mar 9, 2009• 0 citations
Publisher
DOI
10.1090/stml/047/15

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Abstract This study aims to address the nature of state change, measurement, and probabilistic outcomes in non-relativistic quantum mechanics. We consider a pair of particles that interact in a one-dimensional setting via a delta-function potential. One of the particles is confined to a one-dimensional box, and the other particle is free. The free particle is incident from the left with specified energy, and it may cause changes in state of the confined particle before flying away to the left or to the right. We present a formulation and computational scheme that avoids the use of perturbation theory and determines the probability of any such outcome as a function of the initial state of the confined particle and the energy of the incident particle. As demonstrated by a direct comparison, this presented method holds multiple advantages over a standard perturbative method. The problem formulation and corresponding computational scheme may have applications in physical settings which admit one-dimensional scattering, e.g., in the study of quantum wires or quantum dots.
We investigate several Hamiltonians for a free particle in a one-dimensional box, in the context of supersymmetric quantum mechanics. Specifically, we study this problem with the Neumann boundary condition, the periodic and antiperiodic boundary condition, and some mixed and complex boundary conditions. This is achieved by using an approach recently proposed which expresses the factorization of the partner Hamiltonians in terms of the probability density and current for the ground-state eigenfunction of one of them.
The problems of a free classical particle inside a one-dimensional box: (i) with impenetrable walls and (ii) with penetrable walls, were considered. For each problem, the classical amplitude and mechanical frequency of the T -th harmonic of the motion of the particle were identified from the Fourier series of the position function. After using the Bohr-Sommerfeld-Wilson quantization rule, the respective quantized amplitudes and frequencies (i.e., as a function of the quantum label n ) were obtained. Finally, the classical-quantum results were compared to those obtained from modern quantum mechanics, and a clear correspondence was observed in the limit of n » τ.