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arXiv:2312.08412 · Physica Scripta

One-dimensional quantum scattering from multiple Dirac δ − potentials: a Python-based solution

Erfan Keshavarz, S. Habib Mazharimousavi

Published May 30, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract In this research, we present a Python-based solution designed to simulate a one-dimensional quantum system that incorporates multiple Dirac δ − potentials. The primary aim of this research is to investigate the scattering problem within such a system. By developing this program, we can generate wave functions throughout the system and compute transmission and reflection amplitudes analytically and numerically for an infinite range of combinations involving potential strengths, potential separations, and the number of potential centers in the form of the Dirac δ − functions. Furthermore, by modifying the code, we investigate the so-called ‘transmission resonances’ which yields the energy of the quantum particles undergoing a perfect transmission. Subsequently, our research is extended by considering impurities in the system.

DOI
10.1088/1402-4896/ad5234

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Abstract In this research, we present a Python-based solution designed to simulate a one-dimensional quantum system that incorporates multiple Dirac δ − potentials. The primary aim of this research is to investigate the scattering problem within such a system. By developing this program, we can generate wave functions throughout the system and compute transmission and reflection amplitudes analytically and numerically for an infinite range of combinations involving potential strengths, potential separations, and the number of potential centers in the form of the Dirac δ − functions. Furthermore, by modifying the code, we investigate the so-called ‘transmission resonances’ which yields the energy of the quantum particles undergoing a perfect transmission. Subsequently, our research is extended by considering impurities in the system.
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We present an analytical framework for studying quantum tunneling through multiple Dirac delta potential barriers in one dimension. Using the transfer matrix method, we derive a closed-form expression for the total transfer matrix of a system composed of NNN equally spaced delta barriers. In a systematic manner, a compact expression is obtained for the first element of the transfer matrix, based on triangular numbers. This, in turn, allows us to compute the transmission coefficient exactly as a function of the number of barriers. The proposed method successfully reproduces well-known results for one and two barriers and efficiently captures complex interference effects for larger values of NNN, such as N=4N=4N=4.
This paper discussed the digitization of some analytical solutions of the fundamental equation of non-relativistic quantum mechanics, the Schrödinger equation, in the Python programming environment. The main objective of the work is to automate the solution of several known analytically solvable problems of quantum mechanics and ensure their use in more complex problems. For this purpose, the Python programming environment was chosen, which, due to the large number of libraries and flexibility, is currently widely used in solving physical and mathematical problems. In the program we propose, which is available on the GitHub platform, it is solved easily. The problem of determining the probability of the spatial distribution of an electron in a hydrogen-like atom is discussed. The Schrödinger equation is presented in the spherical coordinate system. The wave function, describing the electron's state, quantized energy values, and probability density, is derived analytically. A recursive function was written in Python to handle the series coefficients defining the wave function. Utilizing relevant libraries, the resulting program constructs electron spatial distribution functions for different excitation levels. The paper then examines two one-dimensional tunneling scenarios and provides the tunneling coefficients in analytical form. A Python program was developed to plot the dependence of these coefficients on the particle's coordinate and energy. All the code is available on GitHub.
In one-dimensional scattering, the portion of particles that is transmitted in general vanishes as the kinetic energy of the incident particles approaches zero. Based upon published results from the literature, the present work shows that for a potential consisting of two Dirac δ functions of arbitrary strength, a finite portion of the incident particles is transmitted at threshold for certain choices of the set of parameters defining the potential. The underlying reasons are shown to be related to the structure of the bound levels of the potential. It is shown that for a potential with a finite range but of general shape the reflection coefficient in general tends to unity near threshold, which is what is usually observed. However, in the presence of a bound level at the onset of the continuum, a finite portion of the incident particles is transmitted at threshold for a potential with a general shape and, furthermore, it can be shown that if the potential V is symmetric with V(x)=V(−x) then resonance transmission is observed, i.e., the reflection coefficient vanishes at threshold.
Regarding the quantum mechanical transmission–reflection problem in one dimension with a potential of the form of the derivative of the Dirac delta function δ'(x) = dδ(x)/dx, Christiansen et al recently found that, depending on how δ'(x) is interpreted, there can be a resonance which leads to partial transmission. This is in contrast to the earlier consensus that such a potential allows no transmission. The δ'(x) can be regarded as the narrow-width limit of a certain function Δ'(x) of a finite range. Christiansen et al assumed a rectangular function for Δ'(x). We examine various other forms and how the resonance depends on the shape of Δ'(x). We also present some general observations related to the 'threshold anomaly'.