Shooting Method for Solving the Schrödinger Equation

In this paper, we employ MATLAB to solve the eigenvalues and eigenfunctions of the time-independent Schrödinger equation using the shooting method integrated with the Numerov algorithm. The implementation divides the energy range into discrete segments, applies the Numerov method to propagate wavefunctions segment by segment, and stitches solutions across segments via the shooting method's boundary condition matching. Key implementation steps include: 1) Initializing trial energy values with adaptive step size control 2) Implementing Numerov's finite-difference scheme for numerically stable integration 3) Applying root-finding algorithms (e.g., bisection or Newton-Raphson) to match logarithmic derivatives at boundary points Our findings indicate that increasing the number of energy segments significantly enhances solution accuracy. This methodology proves applicable to eigenvalue problems in diverse physical systems, including quantum wells and harmonic oscillators, with promising extensibility to multi-dimensional and relativistic wave equations. The MATLAB code structure features modular functions for potential definition, wavefunction propagation, and eigenvalue refinement, allowing straightforward adaptation to alternative potential profiles.