正则化映射在二维和三维量子力学系统中的应用
Application of regularization maps to quantum mechanical systems in 2 and 3 dimensions
摘要：
我们扩展了Levi-Civita（L-C）和Kustaanheimo-Stiefel（K-S）正则化方法，该方法将经典系统映射为具有倒六次势的谐振子，其中粒子在$\frac{1}{r}$和$r^2$势的联合影响下运动，并与相应的量子力学对应物相互作用，无论是在2维还是3维。利用后一类系统的薛定谔方程的微扰解，我们导出了附加谐振势下氢原子的本征谱。我们还利用Bohlin-Sundman变换得到了在量子态下，在移动的简谐势中运动的粒子到H原子的映射。利用这一等价性，由后者的解得到前者的薛定谔方程的解。
英文摘要：
We extend the Levi-Civita (L-C) and Kustaanheimo-Stiefel (K-S) regularization methods that maps the classical system where a particle moves under the combined influence of $\frac{1}{r}$ and $r^2$ potentials to a harmonic oscillator with inverted sextic potential and interactions to corresponding quantum mechanical counterparts, both in 2 and 3 dimensions. Using the perturbative solutions of the Schrödinger equation of the later systems, we derive the eigen spectrum of the Hydrogen atom in presence of an additional harmonic potential. We have also obtained the mapping of a particle moving in the shifted harmonic potential to H-atom using Bohlin-Sundman transformation, for quantum regime. Exploiting this equivalence, the solution to the Schrödinger equation of the former is obtained from the solutions of the later.
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