本文采用高斯函数作为基函数,通过数值方法求解了一维谐振子的本征值和本征波函数,并将结果与解析解进行了对比.我们将该方法与快速傅里叶变换方法进行比较,验证了其在求解一维谐振子能量本征值问题中具有更优的计算精度.此外,本文还探讨了基函数数量及高斯参数对计算精度和效率的影响.本文有助于加深物理专业学生对量子力学基本概念的理解.具体而言,本文通过应用数值方法求解一维谐振子的能量本征值问题,可以帮助学生理解解析解之外的实际计算手段和基函数展开等数值方法的基本思想.

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.