A Numerical Approach to Bound-State Eigenvalues of the One-Dimensional Finite Square Quantum Well

The one-dimensional finite quantum well is one of the fundamental problems in quantum mechanics, illustrating the formation of bound states, the penetration of the wave function into classically forbidden regions, and the quantization of energy. Unlike the infinite potential well, the finite quantum well requires the solution of transcendental equations, which cannot be solved analytically. This study presents the derivation of the transcendental equations for a finite quantum well, and their numerical solution through an implementation in the Fortran programming language. The transcendental equations were derived by analyzing the Schrödinger equation inside and outside the potential well. These equations were then solved using the Newton–Raphson method, leading to the development of an algorithm for determining the energies of the bound states. The algorithm was implemented as a Fortran program and compiled to obtain the roots of the transcendental equations for a quantum well of depth and 2nm width, which were subsequently used to calculate the corresponding energy eigenvalues. The obtained results show excellent agreement with theoretical predictions. This efficient and portable implementation is suitable for educational and research applications in computational physics.