Applications of Numerical Calculation in Teaching and Learning on Harmonic Oscillator

This paper uses one-dimensional and two-dimensional harmonic oscillators as examples for a learning mode that combines numerical computation with understanding quantum phenomena. It first applies the factorization method to derive analytical results for the one-dimensional harmonic oscillator’s eigen energy and eigenfunctions. Then, it discusses solving one-dimensional problems numerically with the finite difference method. Next, it extends these analytical and finite difference approaches to two-dimensional cases and proposes using one-dimensional Hamiltonian matrix calculations for efficient two-dimensional problem solving. Finally, it introduces the finite element method as an alternative for two-dimensional harmonic oscillators. To ensure accurate results with the finite difference and finite element methods, it stresses choosing a large enough simulation area to meet boundary conditions.