SOLVING SCHRÖDINGER EQUATION NUMERICALLY USING THE MATRIX FINITE DIFFERENCE METHOD

This paper applies the matrix finite-difference method to the numerical solution of the Schrödinger equation on a discrete spatial grid. The method enables controllable accuracy in calculating the bound-state energy eigenvalues and corresponding eigenfunctions for a prescribed potential. It can be implemented with only a few lines of code in common high-level programming languages, making it accessible and computationally efficient. More importantly, its logic and underlying principles are transparent, helping undergraduate students understand the essential idea of solving the Schrödinger equation numerically and establish a direct connection between differential equations and linear algebra. As examples, the energy eigenvalues and eigenfunctions of the one-dimensional harmonic oscillator are calculated, and the radial wave functions and energy levels of the hydrogen atom are obtained. The matrix finite-difference method is also compared with the widely used matrix Numerov method, with emphasis on their similarities and differences in derivation, accuracy, and applicability. This work aims to demystify the “black box” of numerical computation in quantum mechanics and provide students with a clear and intuitive understanding of the algorithm.