In this work,we employ Gaussian functions as basis to numerically solve for the eigenvalues and eigenfunctions of the one-dimensional quantum harmonic oscillator. The numerical results are compared with analytical solutions, demonstrating good agreement.Furthermore, we compare this method with the fast Fourier transform (FFT) approach, showing that the Gaussian basis method achieves superior computational accuracy in determining the energy eigenvalues of the harmonic oscillator. Additionally, we investigate the influence of the number of basis functions and Gaussian parameters on both computational precision and efficiency.This paper helps deepen physics majors' understanding of fundamental concepts in quantum mechanics. Specifically, by applying numerical methods to solve the energy eigenvalue problem of a one-dimensional harmonic oscillator, students can better understand practical computational techniques beyond analytical solutions and grasp the basic ideas of numerical methods such as basis function expansion.