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SOLUTION OF SCHRÖDINGER'S EQUATION USING THE TRANSFER MATRIX METHOD FOR ONE-DIMENSIONAL POTENTIALS

Jhon H. Vargas, Raúl E. Castiblanco, Maicol Cárdenas, John Morales

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

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.

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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.
Based on the similarity between time-independent Schrodinger equation and Helmholtz equation, the analytical transfer matrix method was utilized to analyze macroscopic quantum tunneling effect, and the potential filed and eigenvalue were corresponded to the refractive index and propagation constant, respectively.Without solving the Schrodinger equation and any approximation,the exact analytic expressions of reflection coefficient and reflection probability of the particle were obtained.Initial results show the equations obtained here are exact and have clear physical meanings.It is expected to be applied in the researches of many basic quantum phenomena,such as quantum tunneling effect,quantum reflection, quantum particles and tunneling resonance.
We show that application of the immersing and transfer-matrix methods to one-dimensional problems of particles scattering leads to the system of two linear equations for the functions F and Φ expressed by means of the transmission and reflection amplitudes. The expressions of these functions are derived. The offered method is illustrated by the finding of transmission and reflection coefficients for the potential barrier with a constant height. The developed method can be applied in solving the quasi-one-dimensional and two-dimensional problems of scattering.
We present a theoretical study of the correspondence between quantum mechanics (QM) and electromagnetism (EM) using the transfer matrix approach (TMA). This analogy has been studied previously [1] , [2] , [3] ; here we adapt it to electromagnetic wave propagation through an isotropic metamaterial for comparison with quantum tunneling through a one-dimensional barrier. Wigner trajectory calculations for quantum mechanical tunneling have been developed previously [4] , [5] to present a method for modeling electron emission at an atomistic level and to allow for coupling to larger-scale particle-based simulations. To provide a means for computational verification and macroscopic experimental validation of these microscale simulations, we use the TMA to calculate both the QM and EM transmission and reflection coefficients, which enables direct comparison of the respective outputs.
An alternative approach to the one-dimensional scattering problem is presented. We show that the reflection and the transmission coefficients of any number of quantum wells or barriers can directly be obtained by using two mathematical methods. The first method arises from a generalization of the so-called elementary symmetric functions used in the mathematical theory of polynomials; the second one consists of a complex generalization of Einstein's addition law for parallel velocities. These laws for the reflection and transmission amplitude coefficients permit a rapid evaluation of the transmittivity of any kind of potential barrier. Some examples are considered. Results are compared with the ones obtained with the WKB method. We also present a numerical study to test the stability of the method.