We study the quantum relativistic wave equations (Klein–Gordon and Dirac) for the non-pure dipole potential $$V(r)=-Ze/r+D\cos \theta /r^{2}$$ V ( r ) = - Z e / r + D cos θ / r 2 , in the case of two-dimensional systems. We consider either spin symmetry or anti-spin symmetry cases in our computations. We give the analytical expressions of the eigenfunctions, compute the exact values of the energies and study their dependence according to the dipole moment D . Our study generalizes the energies of the Kratzer potential as well as the magnetic quantum number m , which is replaced with the Mathieu characteristic values obtained during the resolution of the angular equations. For each magnetic quantum number, we demonstrate the existence of a critical value for the dipole moment, beyond which the corresponding bound state can no longer exist. We find that the critical value is null when $$m=0$$ m = 0 ; this means that these s -states cannot exist for this system and this is in agreement with non-relativistic studies.

中文翻译：

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二维系统中非纯偶极势的 Klein-Gordon 和 Dirac 方程的解

我们研究了非纯偶极子势 $$V(r)=-Ze/r+D\cos \theta /r^{2}$$ V ( r ) 的量子相对论波动方程（Klein-Gordon 和 Dirac） = - Z e / r + D cos θ / r 2 ，在二维系统的情况下。我们在计算中考虑自旋对称或反自旋对称情况。我们给出本征函数的解析表达式，计算能量的精确值并根据偶极矩 D 研究它们的依赖性。我们的研究概括了 Kratzer 势的能量以及磁量子数 m ，它被在角方程求解过程中获得的 Mathieu 特征值所取代。对于每个磁量子数，我们证明了偶极矩的临界值的存在，超过该临界值，相应的束缚态将不再存在。我们发现当 $$m=0$$m = 0 时临界值为空；这意味着该系统不能存在这些 s 状态，这与非相对论研究一致。