物理学报2025，Vol.74Issue(14)：27-40,14.DOI:10.7498/aps.74.20250302
分区级数解法在二维Dirac方程中的应用
Solution of two-dimensional Dirac equation by sectioned series expansion method
摘要
Abstract
With the discovery of two-dimensional materials like graphene,the relativistic two-dimensional Dirac equation has received 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 Schrödinger equation under complex electromagnetic fields.Dirac equation is a system of coupled first-order differential equations with undermined eigenvalues,and it is 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 is derived and the energy levels and wave functions of bound states can be accurately calculated.The criterion given in the main text body shows that the magnetic field and mass field help to confine Dirac electrons while the electric field tends to deconfine them 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=l/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 to 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 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.关键词
Dirac方程/分区级数解法/束缚态/相对论二维类氢原子Key words
Dirac equation/sectioned series expansion method/bound states/relativistic two-dimensional hydrogen-like atom引用本文复制引用
孙松阳,朱嘉麟..分区级数解法在二维Dirac方程中的应用[J].物理学报,2025,74(14):27-40,14.基金项目
吉林省教育厅科学研究项目(批准号:JJKH20250474KJ)和吉林省高教科研重点项目(批准号:JGJX24B10)资助的课题. Project supported by the Scientific Research Foundation of the Education Department of Jilin Province,China(Grant No.JJKH20250474KJ)and the Scientific Research Program of the Higher Education Association of Jilin Province,China(Grant No.JGJX24B10). （批准号:JJKH20250474KJ）
