基于MATLAB对球谐函数及其原子轨道的可视化研究

Visualization research of the spherical harmonic function and atomic orbital based on MATLAB software

摘要:

球谐函数是拉普拉斯方程的球坐标系形式解的角度部分，在经典场论、量子力学等领域广泛应用.基于量子力学中的角动量算符和连带勒让德方程，设定广义函数的条件下，利用分离变量法推导出球谐函数的一般方程式，进而解出n、l分别为0，1，2，3，4，5时的球谐函数.另外，根据球坐标的自变量定义域，设定自变量θ、φ的条件下，得到了不同状态的球谐函数及其原子轨道，并对其进行了可视化研究.最后，利用 MATLAB软件对不同的n、l所对应的球谐函数和原子轨道进行仿真模拟，得到了简明扼要、直观清晰的可视化结果.这种可视化的研究思路为研究球谐函数及其原子轨道的其他特性提供了一条便利的途径.

Abstract:

The spherical harmonic function is the angular part of the form solution of the spherical coordinate system of the Laplace equation. It is widely used in classical field theory, quantum mechanics and other fields. Based on the angular momentum operator and associated Legendre equation in quantum mechanics, the general equation of spheric harmonic function is deduced using the separation variable method under the generalized function condition, and then the spherical harmonic function is obtained when n and l are 0, 1, 2, 3, 4 and 5, respectively. According to the independent variable domain of spherical coordinate, under the independent variables θ, φ condition, spheric harmonic function and its atomic orbitals at different states are studied, and the visualization research is made. Finally, using MATLAB software, we simulate spheric harmonic function and its atomic orbitals at different n, l, then obtain concise, readable and clear visualization results. The visualization study provides a feasible way to investigate other characteristics for spheric harmonic function and its atomic orbitals.