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随着石墨烯等二维材料的发现，相对论二维Dirac方程越来越受到研究者们的关注，准确求解电磁场中的Dirac方程是研究和调控Dirac电子量子状态的基础。通过将分区级数方法应用到Dirac方程中并对该方程在正则区、泰勒区和非正则区进行级数展开，Dirac电子束缚态的普适性判据被推导出来，束缚态的能级和波函数被计算出来。用该方法计算有质量Dirac电子在库仑电势下（相对论二维类氢原子）的能级和波函数并与解析解进行比较，结果表明该方法具有非常高的准确性。对均匀磁场和线性电势下Dirac电子态的计算结果表明该方法对于复杂电磁场中Dirac方程的求解具有普遍的适用性。用该方法研究了均匀磁场B和线性电势V=Fr下Dirac电子束缚态随着电势强度的变化，负能态的能级序列变化被观察到，在临界处F=0.5B正能态的束缚态仍然存在而只有能量超过0的少数负能态的束缚态才存在。该方法提供了求解Dirac方程的有效工具并能够丰富人们对相对论量子力学的认识。With the discovery of two-dimensional materials like graphene, the relativistic two-dimensional Dirac equation has garnered increasing attention from researchers. Accurately solving the Dirac equation in electromagnetic fields is the foundation for studying and manipulating quantum states of Dirac electrons. Sectioned Series Expansion method is successful and accurate in solving Schrodinger equation under complex electromagnetic fields. Dirac equation is a system of coupled first-order differential equations with undermined eigenvalues, more difficult to solve. By applying the sectioned series expansion principle to Dirac equation and conducting series expansions in regular, Taylor and irregular regions, we obtain an accurate method with wide applicability. With the method a universal criterion for bound states of Dirac electrons in electromagnetic fields has been derived and the energy levels and wave functions of bound states can be accurately calculated.The criterion provided by Equation (52) shows that the magnetic field and mass field help to confine Dirac electron while the electric field tends to deconfine it due to Klein tunneling. When the highest power of the electric potential is equal to that of the magnetic vector potential or the mass field, confined-deconfiend states depend on the comparison of their coefficients. We apply the method to two cases: one is massive Dirac electron in Coulomb electric potential (relativistic two-dimensional hydrogen-like atom) and the other is Dirac electron in uniform mangetic field (mangetic vector potenial is A=1/2Br) and linear electric potential V=Fr. The energy levels of the hydrogen-like atom are calculated and compared with analytical solutions, demonstrating the exceptional accuracy of the method. By solving Dirac equation under uniform magnetic field and linear electric potential, the method proves to be broadly applicable in the solutions of Dirac equation under complex electromagnetic fields. Under uniform magnetic field B and V=Fr, as the F increases, level orders of negative energy states change and at the critical point F=0.5B, the bound states of positive ones still exist while only certain negative ones can exist on the condition that their energies exceed zero. The sectioned series expansion method provides an effective computational framework for Dirac equation and it deepens our understanding of relativistic quantum mechanics.
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Keywords:
- Dirac equation /
- Sectioned series expansion method /
- Bound states /
- Relativistic two-dimensional hydrogen-like atom
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