The spin and pseudo-spin symmetries are analytically investigated by solving the three-dimensional Dirac equation for the Kratzer potential plus a ring-shaped potential. Relativistic Schrödinger-like wave equations coupled in energy are derived from Dirac equation. The energy eigenvalues and eigenfunctions are calculated by solving the coupled relativistic radial, and angular wave equations in the framework of asymptotic iteration method. Our numerical results revealed that the spin and pseudo-spin symmetries are relativistic symmetries of the Dirac Hamiltonian. Effects of the angle-dependent potential on the relativistic energy spectra are also investigated. In addition, we include illustrative tables to examine the solutions in detail.

中文翻译：

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原子核自旋和伪自旋对称性起源的研究

通过求解 Kratzer 势加上环形势的三维 Dirac 方程来分析研究自旋和赝自旋对称性。能量耦合的相对论薛定谔波动方程源自狄拉克方程。在渐近迭代法的框架下，通过求解耦合的相对论径向和角波方程来计算能量特征值和特征函数。我们的数值结果表明，自旋和赝自旋对称性是狄拉克哈密顿量的相对论对称性。还研究了角度相关势对相对论能谱的影响。此外，我们还包括说明性表格以详细检查解决方案。