Based On The Energy Levels And Wave Functions Of The Supersymmetric Quantum Mechanics

It is well known that exact solutions play an important role in quantum mechanics since they contain all the necessary information regarding the quantum system under consideration. For example, the exact solutions of the Schrodinger equation for a hydrogen atom and for a harmonic oscillator in three dimensions are an important milestone at the beginning stage of quantum mechanics, which provided a strong evidence for supporting the correctness of the quantum theory. So finding exact or analytical solutions is an important and significant work all the time. However, only a few simple quantum systems can be analytically solved such as the harmonic oscillator and the hydrogen atom. Thus finding exact or analytical solutions is also a very challenging work. Based on this background, this work is attempted to solve the wave functions and energy for several typical models by supersymmetric method and the others. First, we briefly introduce the supersymmetric quantum mechanics and give some applied explanation in solving wave functions and energy by supersymmetric method. Second, we investigate the supersymmetry of two dimensional isotropic harmonic oscillator with time dependent frequency and present supersymmetric quantum mechanics exact solutions of the invariant for this model, meanwhile, discuss the supersymmetry of the invariant for this model. Finally, we also investigate two typical exponential models (namely, Eckart and Manning-Rosen models) by a proper exponential approximation and present their approximately analytic scattering solutions, which would provide a basis for further solving these two typical models in the case of arbitrary 1-wave by supersymmetric method and that exponential approximation method.