Solution and numerical analysis of two-dimensional time-independent Schrödinger equation based on finite difference method

Research Article
Open access

Solution and numerical analysis of two-dimensional time-independent Schrödinger equation based on finite difference method

Yuqi Peng 1*
  • 1 University of Colorado Denver    
  • *corresponding author yuqipeng0726@gmail.com
Published on 17 November 2023 | https://doi.org/10.54254/2753-8818/11/20230388
TNS Vol.11
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-83558-133-9
ISBN (Online): 978-1-83558-134-6

Abstract

With the continuous development and widespread use of quantum mechanics, solving the Schrödinger equation has become a hot research topic. The finite difference method has the advantages of simple calculation and high accuracy, which means that it has high potential in solving the numerical solutions of the Schrödinger equation. In this paper, we deeply explore the problem of using the finite difference method to solve the numerical solution of the time-independent Schrödinger equation, propose a solution method based on the finite difference method, and evaluate its performance under different conditions. Firstly, by analyzing the principles and characteristics of the finite difference method, we construct a difference format for the time-independent Schrödinger equation. Then, by converting the difference format of the numerical solutions of the equation into a matrix, the numerical calculation problem is transformed into a matrix eigenvalue and eigenvector problem. Finally, for different physical scenarios, the established model is numerically solved and its performance is analyzed. This study found that the constructed numerical solution method exhibits high accuracy and stability in solving the numerical solutions of the time-independent Schrödinger equation. In different physical scenarios, this method can provide satisfactory results, thus verifying the feasibility of applying the finite difference method to this problem.

Keywords:

finite difference method, Schrödinger equation, numerical solution, eigenvalue.

Peng,Y. (2023). Solution and numerical analysis of two-dimensional time-independent Schrödinger equation based on finite difference method. Theoretical and Natural Science,11,112-120.
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References

[1]. Hazewinkel, M. (1994). Encyclopaedia of Mathematics (1st ed.). Springer Dordrecht.

[2]. Morton, K. W., & Mayers, D. F. (2005). Numerical Solution of Partial Differential Equations, An Introduction. Cambridge University Press.

[3]. Fan G., & Liu W. (2023). Analysis of thermal insulation of double-layer glass windows based on differential implicit method. Journal of Ludong University: Natural Science Edition, 39(1), 56-62.

[4]. Ge M., & Xu D. (2011). Numerical solution of the inverse time fractional diffusion problems. Journal of Zhejiang Normal University: Natural Science Edition, 34(1), 5.

[5]. Zhang J., Luan S., Han H., & Liang B. (2022). Finite difference method for the heat conduction equation with nonlinear convection term. Journal of Dalian Jiaotong University, 43(5), 115-117.

[6]. Romao E. C., & Assis L. (2018). Numerical simulation of 1d unsteady heat conduction-convection in spherical and cylindrical coordinates by fourth-order fdm. Engineering, Technology and Applied Science Research, 8(1), 2389-2392.

[7]. Si X., & Chen D. (2022). Computational simulation of one-dimensional wave equation. Journal of Huaibei Normal University: Natural Science Edition, 043.

[8]. Zhang Q. (2022). Calculation of wave equation solution based on five-point central difference algorithm. Journal of Chengdu Technological University, 025.

[9]. Sun S., & Wang B. (2017). Numerical solution of a nonlinear parabolic equation in MEMS. Journal of Henan University: Natural Science Edition, 47(6), 6.

[10]. Qiao P., Fang J. & Niu Z. (2019). Finite difference method for solving Schrodinger equation. Journal of Guizhou Normal College, 35(12), 5.

Cite this article

Peng,Y. (2023). Solution and numerical analysis of two-dimensional time-independent Schrödinger equation based on finite difference method. Theoretical and Natural Science,11,112-120.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

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About volume

Volume title: Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation

ISBN：978-1-83558-133-9(Print) / 978-1-83558-134-6(Online)
Editor：Roman Bauer
Conference website: https://www.confmpcs.org/
Conference date: 12 August 2023
Series: Theoretical and Natural Science
Volume number: Vol.11
ISSN：2753-8818(Print) / 2753-8826(Online)

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References

[1]. Hazewinkel, M. (1994). Encyclopaedia of Mathematics (1st ed.). Springer Dordrecht.

[2]. Morton, K. W., & Mayers, D. F. (2005). Numerical Solution of Partial Differential Equations, An Introduction. Cambridge University Press.

[3]. Fan G., & Liu W. (2023). Analysis of thermal insulation of double-layer glass windows based on differential implicit method. Journal of Ludong University: Natural Science Edition, 39(1), 56-62.

[4]. Ge M., & Xu D. (2011). Numerical solution of the inverse time fractional diffusion problems. Journal of Zhejiang Normal University: Natural Science Edition, 34(1), 5.

[5]. Zhang J., Luan S., Han H., & Liang B. (2022). Finite difference method for the heat conduction equation with nonlinear convection term. Journal of Dalian Jiaotong University, 43(5), 115-117.

[6]. Romao E. C., & Assis L. (2018). Numerical simulation of 1d unsteady heat conduction-convection in spherical and cylindrical coordinates by fourth-order fdm. Engineering, Technology and Applied Science Research, 8(1), 2389-2392.

[7]. Si X., & Chen D. (2022). Computational simulation of one-dimensional wave equation. Journal of Huaibei Normal University: Natural Science Edition, 043.

[8]. Zhang Q. (2022). Calculation of wave equation solution based on five-point central difference algorithm. Journal of Chengdu Technological University, 025.

[9]. Sun S., & Wang B. (2017). Numerical solution of a nonlinear parabolic equation in MEMS. Journal of Henan University: Natural Science Edition, 47(6), 6.

[10]. Qiao P., Fang J. & Niu Z. (2019). Finite difference method for solving Schrodinger equation. Journal of Guizhou Normal College, 35(12), 5.

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