A beginner’s guide to solving the Schrödinger equation

Bingyi DU, Yan Xu

Quantum tunneling is an important model in elementary quantum mechanics. However, since solution of its wave function involves transcendental equation, many elementary quantum mechanics textbooks often focus on the explanation of transmission coefficient, and rarely mention the evolution of wave function. Although this process is mentioned in some textbooks, it is often solved by graphic method, which is not conducive to beginners' understanding of the process. In this paper, a numerical calculation method for beginners is proposed to solve this problem. According to Dirac's vectorization idea, the state vectors and operators in the Schrodinger equation are discretized in the form of vectors and matrices, respectively, and the time-dependent evolution of wave function tunneling under several potential barriers is solved by Julia numerically.

M. Roncadelli

Daniel A. Fleisch

Quantum mechanics is a hugely important topic in science and engineering, but many students struggle to understand the abstract mathematical techniques used to solve the Schrödinger equation and to analyze the resulting wave functions. Retaining the popular approach used in Fleisch's other Student's Guides, this friendly resource uses plain language to provide detailed explanations of the fundamental concepts and mathematical techniques underlying the Schrödinger equation in quantum mechanics. It addresses in a clear and intuitive way the problems students find most troublesome. Each chapter includes several homework problems with fully worked solutions. A companion website hosts additional resources, including a helpful glossary, Matlab code for creating key simulations, revision quizzes and a series of videos in which the author explains the most important concepts from each section of the book.

Nilesh P. Barde, Sandeep D. Patil, Pravin Manoharrao Kokne +1 more

The concept of time dependent Schrödinger equation (TDSE) illustrated in literature and even during class room teaching is mostly either complex or meant for advanced learners. This article is intended to enlighten the concept to the beginners in the field and further to improve knowledge about detailed steps for abstract mathematical formulation used which helps in understanding to derive TDSE using various tools and in more comprehensible manner. It is shown that TDSE may be derived using wave mechanics, time independent equation, classical & Hamilton-Jacobi’s equations. Similar attempts have been done earlier by some researchers. However, this article provides a comprehensive, lucid and well derived derivation, derived using various approaches, which would make this article unique.

Doug Jensen