Abstract: This study numerically solves the Schrdinger equation for a threedimensional isotropic harmonic oscillator using Gaussian basis expansion and Fourier transform methods. Wave functions are constructed via Gaussian basis functions, with optimization of variational parameters to enhance the accuracy of energy level calculations. Additionally, the Fourier transform method is utilized to discretize the Schrdinger equation in momentum space, employing the fast Fourier transform algorithm to improve computational efficiency. Results are benchmarked against analytical solutions, and the applicability, accuracy, and computational efficiency of both methods are evaluated. This work serves as a reference for numerical computation teaching in quantum mechanics.

Key words: Gaussian expansion, Fourier transform, threedimensional isotropic harmonic oscillator