In this paper, an algebraic solution for the bound states of the relativistic hydrogen atom is presented. The method discussed here adds an operator associated with the phase of the energy eigenstates to the set of variables of the problem. In terms of this set, appropriate ladder operators are constructed in order to express the full solution of the Dirac hydrogen equation. These ladder operators are used to form the Lie algebra of a su (1, 1) group, in the same way as that applied to angular momentum algebra and the ladder operators L ± . The elements of the vector space associated with the representation of this algebra are related to a generalization of the Laguerre polynomials of the non integer index, also known as Sonine polynomials. In addition, we find that the eigenvalues of the operator constructed with the sum of the square of the three su (1, 1) generators gives precisely the relativistic energy spectrum of the hydrogen atom, including i...

中文翻译：

---

使用相作为额外变量的狄拉克氢原子

本文提出了相对论氢原子的束缚态的代数解。这里讨论的方法将与能量本征态的相位相关联的算子添加到问题的变量集中。根据该集合，构造了适当的梯形算子以表示Dirac氢方程的完整解。这些阶梯算子用于形成su（1，1）群的李代数，与应用于角动量代数和阶梯算子L±的方法相同。与该代数表示相关的向量空间的元素与非整数索引的Laguerre多项式的泛化有关，也称为Sonine多项式。此外，