Numerical Analysis of Bound States in the One-Dimensional Finite Potential Well

The 1D finite square well is a textbook quantum system with only a finite number of bound states, qualitatively different from the infinite well. We numerically solve the time-independent Schrödinger equation for the bound state energies and wavefunctions in a symmetric finite well. The second-order equation is discretized with a finite difference scheme (central difference) and reformulated as a matrix eigenvalue problem, which can be solved using the ARPACK eigensolver through SciPy. We demonstrate the method by convergence tests of increasing grid resolution and benchmarking with analytical results like spectra in the case of the infinite well in the large-depth limit. We find that with increasing well depth, the bound state energies tend to those of an infinite well and that further states result from a deeper well. The calculated wavefunctions are sinusoidal within the well and have exponential decay outside the well, showing tunneling behavior into regions classically forbidden. These results demonstrate the accuracy and versatility of this numerical scheme, its compatibility with theory, and its educational value. Possible generalizations to more complicated or multi-dimensional quantum wells are also mentioned.