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Matemática Contemporânea

Solving the Schrödinger Equation by the Transfer-Matrix Method

Juliana Almeida, Tatiana Miguel Rodrigues de Souza, Alexys Bruno‐Alfonso

Published Jan 1, 2024• 0 citations
Publisher

Abstract

The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.

DOI
10.21711/231766362024/rmc598

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The transfer-matrix method was proposed in the 1940s to solve linear ordinary differential equations.It is broadly studied due to its efficiency and simplicity for solving one-dimensional physical problems.In this work, the method is applied to solve the timeindependent Schrödinger equation, in order to improve both the formalism and the computational treatment.Closed-form solutions are derived for quantum wells and periodic potentials of arbitrary profiles.Numerical results are given for symmetric finite-barrier quantum wells and sinusoidal potentials.
A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.
Abstract We discuss a general description of the solutions to the 1D time-independent Schrödinger equation that does not a priori distinguish between scattering states and bound states and emphasizes and reinforces their relationship and connection to each other. This manuscript also introduces the concept of transfer matrices, which it presents as a logical extension of the traditional approach to evaluating 1D potentials. Using the transfer matrix method and a finite step approximation allows for a simple and straight-forward numerical solution of arbitrary 1D potentials. It also separates the process of solving the Schrödinger equation from selecting physically relevant solutions, which is a critical skill in quantum mechanics and is at the core of physics problems in general.
A consistent approach to the description of a stationary motion of a quantum particle in an arbitrary one-dimensional potential has been developed. It has been proved that the wave function of an infinite motion can be expressed accurate to up two arbitrary constants with the use of one particular solution to the system of first-order linear differential equations. It has been shown that many well-known methods, such as the integral equation method, the transfer matrix method, the embedding method, and the method of combination of scattering parameters, are based on a general property of the solutions to the Schrödinger equation. Within the proposed approach, the relation between these methods becomes more transparent and their description can be well within a unified context.
Many solutions have been propounded for Schrodinger’s equation. The application of these solutions is restricted to a few aspects of physics. One of the methods used for solving the Schrodinger equation is the Finite Difference Time Domain (FDTD) method [1]. It was used to solve numerically the Schrodinger equation for ground state and excited state eigenvalues, and eigenfunctions for a single particle in one-, two-, and three-dimensional potential wells. In the past, analytical methods which have been used to solve the Schrodinger equation are: supersymmetry [1], Nikiforov Uvarov method (NU) [3], asymptotic iteration method (AIM) [4,5], hyperviria perturbation [6], exact quantization rule (EQR) [7], perturbative formalism [8], wave function ansatz method [9], and path integral [10]. In all these methods highlighted above, none has been incorporated with the Bloch NMR to solve nuclear spin among other problems in physics and chemistry. The solution of the Schrodinger equation using the mathematical modeling of Bloch NMR has wide range of applications in many fields of physics and chemistry. It can be used for checking and improving models developed to study quantum mechanical systems, improve on the effective analysis of biological systems, analyzing spin transport, nuclear spin dynamics, improvement of the numerical methods and so on. In Sec. 2, the solution of the Schrodinger is given. In Sec. 3, the solution was validated by applying it to the harmonic solution. The energy
The one-dimensional Schrödinger equation can be written as a first-order multicomponent equation by considering ψ and dψ/dx, or combinations thereof, as independent variables. A potential barrier is then represented by a matrix belonging to one of the homomorphic groups SU(1,1), SO(2,1), Sp(2,R), or SL(2,R). The relationship between these groups is clarified. In various applications, one of them may turn out more convenient than others. In particular, SO(2,1), which is obtained by using as a basis some bilinear combinations of ψ and dψ/dx, leads to remarkable results: The Schrödinger wavefunction is represented by a trajectory on a unit hyperboloid; a periodic potential corresponds to a pseudorotation around a fixed axis; a random potential gives a random walk on the hyperboloid. This method can also be used to calculate bound states (in potential wells) and may have many other interesting applications.