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Journal : BULETIN FISIKA

Solution of Time-Independent Schrodinger Equation (TISE) by Using Finite Difference Approach Widagda, I Gusti Agung; Artawan, Nengah; Kasmawan, I Gde Antha; Suharta, I Wayan Gede; Nurmalasari, Ni Putu Yuni
BULETIN FISIKA Vol. 26 No. 1 (2025): BULETIN FISIKA
Publisher : Departement of Physics Faculty of Mathematics and Natural Sciences, and Institute of Research and Community Services Udayana University, Kampus Bukit Jimbaran Badung Bali

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24843/BF.2025.v26.i01.p02

Abstract

This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (β1), regression coefficient (β0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.
Co-Authors Adnyana, I Gusti Agung Putra Anandika Putra, I Made Raditya Anastasia Jessing Gamalita Mberato Artawan, I Nengah Artawan, Nengah Baskoro, Winardi Tjahyo Cory Amelia Dali Sinaga, Ade Nisa Daniel Happy Putra Dedy Pratama, I Putu Djoko Hadi Prajitno Efrensiana Ana Melti Gracia Paula Fahik Gusti Ngurah Sutapa Hadi Apriandi I G. A. Widagda I Gusti Agung Ayu Ratnawati I Gusti Agung Ngurah Ari Kusuma Putra I Ketut Putra I Ketut putra, I Ketut I Made Satriya Wibawa I Made Yuliara I Nengah Sandi I Nengah Sandi I Nengah Simpen I NYOMAN WIJAYA I Putu Wahyu Pranata Kusuma Jaya I Wayan Supardi I. M. Yuliara I. M. Yuliara I.M. Yuliara Ida Ayu Dwi Giriantari Ida Bagus Made Suryatika Ida Bagus Made Suryatika Ida Bagus Putu Priastana Bawa Irhas, Rozi Kelo, Febriyanti Jia M. Zully Amrul Hadi Made Sumadiyasa Made Suryatika, Ida Bagus Mas Janudinata, Gde Ari Murzid, Jihan Qothrunada N. L. P. Trisnawati Ni Kadek Nova Anggarani Ni Kadek Nova Anggarani Ni Luh Putu Trisnawati Ni Made Wedayani Ni Nyoman Rupiasih Ni Nyoman, Ratini Ni Putu Yuni Nurmalasari Nurmalasari, Ni Putu Yuni Nyoman Wendri Pranditayana, I Nyoman Prawistya Putra, I Made Gede Mas Prayuda, I Made Ardi Putra Adnyana, I Gusti Agung Rosvita Hilbrida Sulastri Rozi Irhas Sari, Kamelin Diah Satriya Wibawa, I Made Setiawan, Yohanes Agus Silalahi, Dio Carpry Gunawan Silalahi, Johanes J. Sismanto Spardi, I Wayan Suandayani, Ni Komang Tri Sukarasa, I Ketut Sutapa Gusti Ngurah Tri Jaya Maghuna, Komang Trisnawati, Ni Luh Putu Umi Hariyani Wayan Balik Sudarsana Wayan Gede Suharta Widagda, I Gusti Agung Widagda, I. G. A Widiana, I Komang Widianingsih, Ni Kadek Indah Winardi Tjahjo Baskoro Winardi Tjahyo Baskoro Windaryoto Windaryoto Winer Johan Bora Yuliara, I Made Yuliara, Made