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arXiv:2001.06298 · Journal of Physics Communications

Relativistic J-matrix method of scattering in 1 + 1 space-time I. Theory *

Abdulaziz D Alhaidari

Published Sep 1, 2020• 0 citations
arXivHTMLPDF

Abstract

Abstract We make a relativistic extension of the one-dimensional J-matrix method of scattering. The relativistic potential matrix is a combination of vector, scalar, and pseudo-scalar components. These are non-singular short-range potential functions (not necessarily analytic like a square well) such that they are well represented by their matrix elements in a finite subset of a square integrable basis set. This set is chosen to support a tridiagonal symmetric matrix representation for the free Dirac operator. We derive the transmission and reflection coefficients. This work will be followed by another where we apply the theory to obtain scattering information for different potential coupling modes including those with evidence of Klein tunneling and of supercritical resonances.

DOI
10.1088/2399-6528/abb4b4

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Abstract We make a relativistic extension of the one-dimensional J-matrix method of scattering. The relativistic potential matrix is a combination of vector, scalar, and pseudo-scalar components. These are non-singular short-range potential functions (not necessarily analytic like a square well) such that they are well represented by their matrix elements in a finite subset of a square integrable basis set. This set is chosen to support a tridiagonal symmetric matrix representation for the free Dirac operator. We derive the transmission and reflection coefficients. This work will be followed by another where we apply the theory to obtain scattering information for different potential coupling modes including those with evidence of Klein tunneling and of supercritical resonances.
We propose the quantum simulation of the Dirac equation with potentials, allowing the study of relativistic scattering and Klein tunneling. This quantum relativistic effect permits a positive-energy Dirac particle to propagate through a repulsive potential via the population transfer to negative-energy components. We show how to engineer scalar, pseudoscalar, and other potentials in the 1+11+11+1 Dirac equation by manipulating two trapped ions. The Dirac spinor is represented by the internal states of one ion, while its position and momentum are described by those of a collective motional mode. The second ion is used to build the desired potentials with high spatial resolution.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.
The covariant single-time equations of the quantum field theory are formulated in the relativistic configurational representation. The explicit formulae for the Green functions corresponding to the scattering states are calculated in this representation. Using the derived nonhomogeneous equations the scattering problem is solved exactly for certain potentials (combinations of zero-range potentials). The equations and their solutions are studied in the non-relativistic limit. The conditions of total reflection, available for such potentials, are investigated.
The Dirac wave function is represented in a form where all its components have obvious geometrical and physical interpretations. Six components compose a Lorentz transformation determining the electron velocity are spin directions. This provides the basis for a rigorous connection between relativistic rigid body dynamics and the time evolution of the wave function. The scattering matrix is given a new form as a spinor-valued operator rather than a complex function. The approach reveals a geometric structure of the scattering matrix and simplifies scattering calculations. This claim is supported by an explicit calculation of the differential crosssection and polarization change in Coulomb scattering. Implications for the structure and interpretation of relativistic quantum theory are discussed.
This is the first in a series of papers where we succeed in enlarging the class of exactly solvable potentials in one and three dimensions by obtaining solutions for new relativistic and nonrelativistic problems. This is accomplished by constructing a matrix representation of the wave operator in a complete square integrable basis that makes it tridiagonal. Expanding the wave function in this basis makes the wave equation equivalent to a three-term recursion relation for the expansion coefficients. Consequently, finding solutions of the recursion relation is equivalent to solving the original problem. Doing so results in a larger class of solvable potentials. The usual diagonal representation constraint results in a reduction from the larger class to the conventional class of solvable potentials, giving the well-known energy spectra and the corresponding wave functions. Moreover, some of the new solvable problems show evidence of a Klauder-like phenomenon. In the present work, we give an exact solution for the infinite potential well with a bottom that has a sinusoidal shape.