Even-exponent potential wells solved by the finite difference method

Abstract This article examines a particle in a one-dimensional potential well V a ( x ) ∝ x a , where a ≥ 2 is an even integer. The system is referred to as a harmonic oscillator when a = 2 and as a particle in a box when a → ∞ . The finite difference method is applied to solve the Schrödinger equation, determining the ground-state energies and wave functions. Giving accurate description in the cases of a = 2 and a → ∞ , the numerical method allows us to explore the system for intermediate values of a , revealing how the energies and wave functions evolve between these two limits. The study includes discussions on the virial theorem, the uncertainty principle, excited states, and the limitation of the numerical method.