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Revista Principia - Divulgação Científica e Tecnológica do IFPB

Soluções variacionais e numéricas da Equação de Schrödinger 1D submetida ao potencial de Pöschl-Teller

Lucas Carvalho Pereira, João Vítor Batista Ferreira, Valdir Aragão do Nascimento

Published Mar 30, 2020• 2 citations

Abstract

This paper presents the numerical and variational solutions of the 1D Schrödinger Equation submitted to the Pöschl-Teller potential. The methods used were the Variational Method and the Finite Difference Method. They were presented in a didactic and detailed way with the purpose of instructing both undergraduate and graduate students, about the applicability and effectiveness of the aforementioned methods. We use the Pöschl-Teller potential due to the fact that it is little explored in the books of Quantum Mechanics used in undergraduation courses and also because of its diverse applications, such as in Bose-Einstein condensates, waveguides, topological defects in field theory and so on. We conclude this paper comparing the variational and numerical solutions with the analytical solution and present the advantages of each method.

DOI: 10.18265/1517-03062015v1n48p156-168

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This work is from a coursework for Part I of my 10 credit Computational Physics module. This is not in a typical report style, instead being tailored to a set of four tasks approached with the various methods explored in this part of the course. The report serves to store an online portfolio of my work so far.The specific tasks addressed in this work were:Illustrating de-dimensionalisation of the one-dimensional Schrödinger equation subject to a Pöschl-Teller potential well.Finding numerically all localised eigenstates of this system.Implementation of a numerical method to simulate the dynamics of the given system, subject to specified initial conditions.Modifying the code from 3. to simulate a similar system, in which the potential has a small time-dependence.I was happy to achieve a mark of 100% for this report and its accompanying code, which can be found on my GitHub.

Condensação de Bose-Einstein é um fenômeno quântico que pode ser observado macroscopicamente. Para a sua obtenção são necessários aprisionamentos externos, porém a presença desses leva ao colapso da função de onda. As interações de longo alcance são propostas como uma forma alternativa ao confinamento externo, um vez que podem prevenir o colapso da função de onda. Neste trabalho será apresentada uma revisão sobre os estudos de condensados de Bose-Einstein. Também, será buscada a solução aproximada da equação de Schrödinger não linear e não-local, a qual descreve condensados de Bose-Einstein com auto-interações de longo alcance. Para isso, será suposta uma forma espacial da função de onda, permitindo o tratamento analítico do sistema dinâmico resultante. Ao fim, por meio do método variacional dependente do tempo, será demonstrado que existem soluções estáveis para a função de onda sujeito a interações de longo alcance na forma gaussiana e gravitacional.

An important technique for teaching introductory wave mechanics is provided by the numerical calculation of stationary state solutions of the Schrödinger equation. Realistic problems in wave mechanics that do not require sophisticated computer techniques are very valuable for the instruction of undergraduates. In the central-field model for atoms, the calculation of electron states reduces to the solution of a one-dimensional differential equation whose numerical solutions are no more difficult to obtain than those for a finite square well. A method used by our students is discussed in this paper for the calculation of the electron wavefunctions in a central-field potential. By working this problem, students obtain meaningful results that give them insight into the theory of atomic structure. Results of their experience are included, and possibilities of extending this example are discussed.