Harmonic Oscillator And Hydrogen Atom And Its Teaching Research

In this paper, the current states of research about the basic theory of harmonic oscillator and hydrogen atom in quantum mechanics are firstly reviewed. On base of it, and the relationships between harmonic oscillator and hydrogen atom in quantum mechanics are studied. By use of the relationships between the hermite polynomial and the laguerre polynomial, the eigenequations of one-dimensional harmonic oscillator and hydrogen atom are conversed into the same equations in form. Therefore the relationships between energy levels and wave functions of one-dimensional harmonic oscillator and hydrogen atom are found. Through the coordinates transform, the relationships between energy levels and wave functions of two-dimensional harmonic oscillator and hydrogen atom are found. At the same time in terms of SU(1, 1) algebra, the eigenequations of two and three-dimensional harmonic oscillator and hydrogen atom with inverse square potential are conversed into the same equations in form. Then the relationships between energy levels and wave functions of them are found. At last, about how to improve teaching harmonic oscillator and hydrogen atom in quantum mechanics is briefly reviewed on base of modern teaching theory and self-study experience.