[Paper Review] Transfer matrices for piecewise constant potentials

G. F. Torres del Castillo, I. Rubalcava García|LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas)|Jun 7, 2007

Scientific Research and Discoveries3 citations

TL;DR

This paper presents a matrix exponential method to derive transfer matrices for one-dimensional piecewise constant potentials in quantum mechanics and electromagnetic wave propagation. By reformulating the Schrödinger equation and wave equation as first-order systems, the transfer matrix is computed exactly, enabling direct calculation of bound states and quasinormal modes through matrix eigenvalue analysis, with analogous results for layered dielectric media without using Fresnel coefficients.

ABSTRACT

By expressing the time-independent Schrodinger equation in one dimension as a system of two first-order differential equations, the transfer matrix for a rectangular potential barrier is obtained making use of the matrix exponential. It is shown that the transfer matrix allows one to find the bound states and the quasinormal modes. A similar treatment for the one-dimensional propagation of electromagnetic waves in a homogeneous medium is also presented.

Motivation & Objective

• To develop a systematic method for computing transfer matrices of piecewise constant potentials using matrix exponentials.
• To demonstrate that the transfer matrix formalism allows direct identification of bound states and quasinormal modes via matrix eigenvalue analysis.
• To extend the transfer matrix approach to one-dimensional electromagnetic wave propagation in layered dielectrics without relying on Fresnel coefficients.
• To unify the treatment of quantum scattering and electromagnetic wave propagation under a common mathematical framework based on first-order differential systems.

Proposed method

• Express the time-independent Schrödinger equation as a system of two coupled first-order differential equations.
• Apply the matrix exponential to integrate the system over regions of constant potential, yielding the transfer matrix for a rectangular barrier.
• Use the continuity of the wave function and its derivative at potential boundaries to construct the full transfer matrix.
• Derive the transfer matrix for electromagnetic waves by expressing the wave equation as a first-order system in terms of electric field and its derivative.
• Establish that the resulting transfer matrices belong to the SU(1,1) group, ensuring unit determinant and conservation of probability or power flow.
• Solve for bound states and quasinormal modes by setting the (1,1) entry of the transfer matrix to zero, leading to transcendental equations in the complex wave number.

Experimental results

Research questions

• RQ1 How can the transfer matrix for a rectangular potential barrier be derived using the matrix exponential method?
• RQ2 What conditions on the transfer matrix correspond to bound states and quasinormal modes in quantum systems?
• RQ3 Can the same matrix exponential approach be applied to electromagnetic wave propagation in layered media?
• RQ4 How does the transfer matrix formalism ensure conservation of probability current or power flow in both quantum and electromagnetic cases?
• RQ5 What is the mathematical structure of the transfer matrix in terms of Lie groups, and how does it constrain the physical solutions?

Key findings

• The transfer matrix for a rectangular potential barrier is derived as the matrix exponential of the system's coefficient matrix, providing an exact analytical solution for piecewise constant potentials.
• Bound states are identified by solving the condition that the (1,1) entry of the transfer matrix vanishes, leading to transcendental equations involving the wave number and potential parameters.
• Quasinormal modes appear as complex solutions of the same condition, corresponding to states with finite lifetime due to exponential decay in time.
• For electromagnetic waves in layered dielectrics, the transfer matrix is derived directly from the wave equation without using Fresnel coefficients, yielding a matrix in SU(1,1) with unit determinant.
• The method allows for the systematic computation of transmission and reflection amplitudes for any piecewise constant potential or refractive index profile through matrix multiplication.
• The formalism reveals that both quantum scattering and electromagnetic wave propagation are governed by the same underlying mathematical structure, unified through first-order differential systems and matrix exponentials.