Wave functions of higher dimensional hydrogen atom. Its algebraic derivation, graphic representation, and physical meaning

在我们以前的研究中，我们导出了n维氢原子波函数角部简并度的一般公式，它可以通过对著名的Pascal三角形的简单操作而递归地得到。我们的发现的本质在于它的代数和启发式方法，而不是求助于解决薛定谔方程的分析和复杂的技术，而是从现有的解析解之间的代数关系。在本研究项目中，1）我们成功地建立了动画技术，用于绘制投影到低维坐标系上的n维原子轨道横截面的各个方面。2)用这种方法，我们得到了一些s-、p-和d-型的普通三维原子轨道的截面，而这些截面在量子化学教科书中是没有的。3)考虑到这一经验，我们绘制了一些四维原子轨道的横截面，使我们能够容易地理解整个n维世界的原子轨道波函数的分析和几何结构。4)通过对新获得的关于n维原子轨道的信息的组合，揭示了这些轨道的简并度与非对称帕斯卡三角形之间的有趣关系的秘密。总体而言，本研究项目所获得的结果不仅可以为研究生，而且可以为化学和物理领域的活跃研究人员提供新的令人兴奋的信息和见解。

In our previous research we have derived the general formula of the degeneracy of the angular part of the n-dimensional atomic hydrogen wave functions, which can be obtained recursively by simple manipulation of the famous Pascal's triangle. The essence of our findings is in its algebraic and heuristic method without recourse to analytical and sophisticated techniques for solving the Schrodinger equation but from the algebraic relations among the existing analytical solutions.In this research project 1) we have succeeded in establishing the animation technique for drawing various facets of the cross sections of n-dimensional atomic orbitals projected on the lower dimensional coordinate systems. 2) With this technique a number of cross sections were obtained for ordinary 3-dimensional atomic orbtials of s-, p-, and d-types, which, however, have not been available in the quantum chemistry textbooks. 3) With this experience in mind a number of cross sections of 4-dimensional atomic orbitals were drawn, enabling easy understanding of the analytical and geometrical structure of the atomic orbital wavefunctions of the whole family of the n-dimensional world. 4) By assembling the newly obtained information on the n-dimensional atomic orbitals the secret of the interesting relation between the degeneracy of these orbitals and the asymmetrical Pascal's triangle.As a whole the obtained results in this research project could provide new and exciting information and insight not only for graduate students but also for active researchers in chemistry and physics.