薛定谔方程的矩阵有限差分数值解法（英文）SOLVING SCHR??DINGER EQUATION NUMERICALLY USING THE MATRIX FINITE DIFFERENCE METHOD
冯逸轩,隆笑,郑华,张文超,孙辉,王新刚
摘要(Abstract)：
本文采用矩阵有限差分法,在离散网格上对薛定谔方程进行数值求解。该方法在计算给定势场中束缚态的本征能量及相应本征函数时,能够实现数值精度的有效调控。该算法仅需少量代码即可在任何高级编程语言中实现,具有极高的易用性和计算效率。更为重要的是,该数值方法的逻辑与底层原理简明易懂,有助于本科生掌握数值求解薛定谔方程的核心要领,同时在微分方程与线性代数之间建立起直观联系。为展示其应用,本文计算了一维谐振子的本征能量与本征函数,以及氢原子的径向波函数与能级。此外,将矩阵有限差分法与广为人知的矩阵Numerov方法进行了对比,阐述了两者之间的异同。本文旨在揭开量子力学数值计算的“黑箱”面纱,为学生提供对数值算法清晰且直观的理解。
关键词(KeyWords)： 薛定谔方程;矩阵Numerov方法;矩阵有限差分法;一维谐振子;氢原子
基金项目(Foundation): 国家自然科学基金(11905120);; 中央高校基本科研业务费专项(GK202202003)
作者(Author): 冯逸轩,隆笑,郑华,张文超,孙辉,王新刚
DOI: 10.27024/j.wlygc.2026.02.26.01
参考文献(References)：
- [1] GRIFFITHS D J,SCHROETER D F.Introduction to quantum mechanics[M].Cambridge University Press,2018.
- [2] BRANSDEN B H,JOACHAIN C J.Quantum mechanics[M].Pearson Prentice Hall,2000.
- [3] DIRAC P A M.The principles of quantum mechanics[M].Clarendon Press,1930.
- [4] 郑海荣，BOYE D,邢永忠，等.量子力学[M].北京：科学出版社，2014.ZHENG H R,BOYE D,XING Y Z,et al.Quantum mechanics[M].Science Press,2014.
- [5] MATTHES E.Python crash course[M].No Starch Press,2023.
- [6] ATTAWAY D C.MATLAB:a practical introduction to programming and problem-solving[M].Butterworth-Heinemann,2022.
- [7] PILLAI M,GOGLIO J,WALKER T G.Matrix Numerov method for solving Schr??dinger's equation[J].American Journal of Physics,2012,80(11):1017-1019.
- [8] NUMEROV B.Note on the numerical integration of ■.Astronomische Nachrichten,1927,230(19):359-364.
- [9] BLATT J M.Practical points concerning the solution of the Schr??dinger equation[J].Journal of Computational Physics,1967,1(3):382-396.
- [10] CHOW P C.Computer solutions to the Schr??dinger equation[J].American Journal of Physics,1972,40(5):730-734.
- [11] LANDAU R H,PAEZ M J,BORDEIANU C C.A survey of computational physics[M].Princeton University Press,2008.
- [12] DIJK W.On numerical solutions of the time-dependent Schr??dinger's equation[J].American Journal of Physics,2023,91(10):826-839.