基于超对称量子力学的能级及波函数的研究

【作者】 卫高峰；

【导师】 龙超云；

【作者基本信息】 贵州大学 ， 理论物理， 2008， 硕士

【摘要】 波函数的精确解在量子力学中具有非常重要的作用,因为它几乎包含了所有需要的量子信息,如:氢原子和谐振子薛定谔方程的精确解在量子力学的诞生初期起到了重要的基石作用,对于量子理论正确性的证明提供了强有力的证据。因此,对于精确解的寻求一直是一项非常有意义的工作。然而,能够解析或精确求解的只是极少数的简单量子系统,如谐振子和氢原子。因此,对于精确解的寻求也是一项非常有挑战性的工作。本文正是在这一背景下,试图采用超对称及其它方法来求解几个典型模型的波函数及能级。首先,本文对超对称量子力学进行了简单的介绍,说明了超对称方法在量子力学波函数及能级求解中的应用。然后,我们采用超对称方法研究了二维各向同性变频率谐振子的超对称性及其不变量的超对称量子力学精确解,并且对不变量的超对称性进行了讨论。最后,我们还采用合适的指数近似对两个典型的指数分子Eckart和Manning-Rosen模型进行了研究,求解了它们的散射态解析近似解,为我们进一步结合超对称方法及指数近似方法给出任意L波情况下这两个模型的超对称量子力学解析解做好铺垫。更多还原

【Abstract】 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.更多还原