High-Accuracy Numerical Solution of the One-Dimensional Schrödinger Equation Using the Numerov–Shooting Method
DOI:
10.29303/jpft.v12i1.10794Published:
2026-05-30Downloads
Abstract
The Time-Independent Schrödinger Equation (TISE) plays a crucial role in quantum mechanics for obtaining the eigenenergies and wave functions of a system. This research numerically solves the TISE for the one-dimensional infinite potential well model by applying the Numerov method combined with the shooting technique. The Numerov method was chosen because it has an order of accuracy of O, making it effective for solving second-order differential equations with a very small truncation error. Numerov is used to solve the second order differential equation with a high degree of accuracy, while the shooting technique is applied to determine the eigenenergies that satisfy the physical boundary conditions of the system. Wave function normalization is performed using the Simpson integral so that the total probability is. Numerical validation is carried out by comparing the computed eigenenergies to the analytical solution in dimensionless units for up to . The results show that the numerical energies have a relatif error in the range of order to , indicating that the Numerov method is capable of producing very accurate and consistent solutions. Based on these findings, the Numerov method can be used effectively in solving quantum mechanics problems that do not have analytical solutions.
Keywords:
Eigen energy Numerov Method Shooting Method Time-Independent Schrödinger Equation Infinite Potential WellReferences
Al-Ani, L., & K. Abid, R. (2019). Solving Schrödinger Equation for Finite Potential Well Using the Iterative Method. Al-Nahrain Journal of Science, 22(4), 52–58. https://doi.org/10.22401/ANJS.22.4.07
Akhsan, H., Putra, G. S., & Ariska, M. (2023). Newtonian Yoyo (Lato-Lato) Phenomenon in Indonesia: An Innovative Resource for IGCSE Physics Teaching and Learning? Jurnal Penelitian & Pengembangan Pendidikan Fisika, 9(1), 119–126. https://doi.org/10.21009/1.09111
Angraini, L. M., & Sudiarta, I. W. (2018). Non-Standard and Numerov Finite Difference Schemes for Finite Difference Time Domain Method to Solve One-Dimensional Schrödinger Equation. Journal of Physics: Theories and Applications, 2(1), 27. https://doi.org/10.20961/jphystheor-appl.v2i1.26352
Ariska, M., Akhsan, H., & Muslim, M. (2022). Impact profile of ENSO and dipole mode on rainfall as anticipation of hydrometeorological disasters in the province of South Sumatra. Spektra: Jurnal Fisika Dan Aplikasinya, 7(3), 127–140. https://doi.org/10.21009/SPEKTRA.073.02
Budak, H., Hezenci, F., Kara, H., & Sarikaya, M. Z. (2023). Bounds for the Error in Approximating a Fractional Integral by Simpson’s Rule. Mathematics, 11(10), 2282. https://doi.org/10.3390/math11102282
Caruso, F., Oguri, V., & Silveira, F. (2022). Applications of the Numerov method to simple quantum systems using Python. Revista Brasileira de Ensino de Física, 44. https://doi.org/10.1590/1806-9126-rbef-2022-0098
Chowdhury, T. A., Islam, T., Mujahid, A. A., & Ahmed, Md. B. (2021). The Periodicity of the Accuracy of Numerical Integration Methods for the Solution of Different Engineering Problems. Journal of Engineering Advancements. https://doi.org/10.38032/jea.2021.04.006
Esposito, G., & Santorelli, P. (2022). Numerov and phase-integral methods for charmonium. The European Physical Journal Plus, 137(5), 642. https://doi.org/10.1140/epjp/s13360-022-02849-7
Gaigalas, G., & Fritzsche, S. (2021). Angular coefficients for symmetry-adapted configuration states in jj-coupling. Computer Physics Communications, 267, 108086. https://doi.org/10.1016/j.cpc.2021.108086
Gamper, J., Kluibenschedl, F., Weiss, A. K. H., & Hofer, T. S. (2023). Accessing position space wave functions in band structure calculations of periodic systems. The Journal of Physical Chemistry Letters, 14(33), 7395–7403. https://doi.org/10.1021/acs.jpclett.3c01707
Hagino, K. (2024). Barrier penetration with a finite mesh method. Physical Review C, 109(3), 034611. https://doi.org/10.1103/PhysRevC.109.034611
Hammad, A., Park, M., Ramos, R., & Saha, P. (2023). Exploration of parameter spaces assisted by machine learning. Computer Physics Communications, 293, 108902. https://doi.org/10.1016/j.cpc.2023.108902
Khan, A., Ahsan, M., Bonyah, E., Jan, R., Nisar, M., Abdel-Aty, A.-H., & Yahia, I. S. (2022). Numerical Solution of Schrödinger Equation by Crank–Nicolson Method. Mathematical Problems in Engineering, 2022, 1–11. https://doi.org/10.1155/2022/6991067
Liu, C.-S., Chang, J.-R., Shen, J.-H., & Chen, Y.-W. (2022). A boundary shape function method for computing eigenvalues and eigenfunctions of Sturm–Liouville problems. Mathematics, 10(19), 3689. https://doi.org/10.3390/math10193689
Ljungberg Rydin, Y., Mattsson, K., Werpers, J., & Sjöqvist, E. (2021). High-order finite difference method for the Schrödinger equation on deforming domains. Journal of Computational Physics, 443, 110530. https://doi.org/10.1016/j.jcp.2021.110530
Mughal, M. Z., & Khan, F. (2025). A numerical solution of Schrödinger equation for the dynamics of early universe. Astronomy and Computing, 50, 100894. https://doi.org/10.1016/j.ascom.2024.100894
Mushtaq, A., Noreen, A., & Olaussen, K. (2020). Numerical solutions of quantum mechanical eigenvalue problems. Frontiers in Physics, 8. https://doi.org/10.3389/fphy.2020.00390
Saad, D. Y., & Ikraiam, F. A. (2025). Modeling time-independent Schrödinger equation for an infinite potential well using Numerov and matrix methods. Libyan Journal of Science &Technology, 15(1), 218–222. https://doi.org/10.37376/ljst.v15i1.7226
Schürger, P., & Engel, V. (2023). On the relation between nodal structures in quantum wave functions and particle correlation. AIP Advances, 13(12). https://doi.org/10.1063/5.0180004
Wu, Y., & Chen, W. (2023). On strongly indefinite Schrödinger equations with non-periodic potential. Journal of Applied Analysis & Computation, 13(1), 1–10. https://doi.org/10.11948/20210036
License
Copyright (c) 2026 Sopia Sopia, Dea Amalia, Edo Anugrah, Hamdi Akhsan, Melly Ariska
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Authors who publish with Jurnal Pendidikan Fisika dan Teknologi (JPFT) agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License 4.0 International License (CC-BY-SA License). This license allows authors to use all articles, data sets, graphics, and appendices in data mining applications, search engines, web sites, blogs, and other platforms by providing an appropriate reference. The journal allows the author(s) to hold the copyright without restrictions and will retain publishing rights without restrictions.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in Jurnal Pendidikan Fisika dan Teknologi (JPFT).
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
Downloads
How to Cite
