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Journal of Physics A Mathematical and General

Geometrical approach to scattering in one dimension

Donald W. L. Sprung, G. V. Morozov, Joan Martorell

Published Jan 19, 2004• 16 citations
Publisher

Abstract

The transfer matrix technique is much used to study one-dimensional scattering, such as electrons in semiconductor superlattices. We elaborate on the geometrical interpretation of the transfer matrix, in particular its relation to conformal mapping in the unit disc, and the analogy to Lorentz transformations on a Dirac spinor in 2 + 1 dimensions.

DOI
10.1088/0305-4470/37/5/026

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The transfer matrix technique is much used to study one-dimensional scattering, such as electrons in semiconductor superlattices. We elaborate on the geometrical interpretation of the transfer matrix, in particular its relation to conformal mapping in the unit disc, and the analogy to Lorentz transformations on a Dirac spinor in 2 + 1 dimensions.
The Generalized Method of Transfer Matrix (MMT) appears as an alternative to quantum phenomena research in semiconductor materials and optoelectronic components. In this paper the MMT is applied to the propagation phenomena of a particle through a well or a potential barrier to the quantum case. The relations scattering amplitudes are deduced using the coefficients of the transfer matrix (transmission and reflection). The results are compared with related literature, to determine the accuracy and scope of the MMT, namely which can be applied to study many other problems of wave propagation and electrons, even in layered media as photonic crystals.
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.
We offer a consistent dynamical formulation of stationary scattering in two and three dimensions (2D and 3D) that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear operator acting in an infinite-dimensional function space which we can represent as a $2\ifmmode\times\else\texttimes\fi{}2$ matrix with operator entries. This operator encodes the information about the scattering properties of the potential and enjoys an analog of the composition property of its one-dimensional ancestor. Our results improve an earlier attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the role of the evanescent waves. We show that a proper formulation of this approach requires the introduction of a pair of intertwined transfer matrices, each related to the time-evolution operator for an effective nonunitary quantum system. We study the application of our findings in the treatment of the scattering problem for $\ensuremath{\delta}$-function potentials in 2D and 3D and clarify its implicit regularization property which circumvents the singular terms appearing in the standard treatments of these potentials. We also discuss the utility of our approach in characterizing invisible (scattering-free) potentials and potentials for which the first Born approximation provides the exact expression for the scattering amplitude.
This paper aims at explaining that the key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R 4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R 3. In the three-dimensional rotation group the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding acquired to derive the free-space Dirac equation from scratch proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine's statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.
Abstract Having been linked to numerous applications, the one-dimensional Lenz–Ising model in a non-zero magnetic field is well-known in graduate- and senior-level statistical physics. The transfer matrix method is the standard way to solve the model. In this line, we reformulate the transfer matrix by subjecting the usual Pauli matrices to rotation. The transfer matrix is then shown as a ‘spin’-representative of a two-dimensional rotation just like the special case when the magnetic field is zero. Consequently, eigenvalues in exponential form are obtained, and the zeroes of the partition function lie on a unit circle in agreement with the Lee–Yang Circle Theorem. The zeroes pinching the positive side of the real axis give the known trivial transition temperature, T = 0 K. Finally, we discuss the significance of the results in teaching the model.