High-precision Calculation Of Relativistic Corrections For Hydrogen-like Ions With Screened Coulomb Potentials

In recent yeas,studies on the abundant physical properties of hydrogen atom or hydrogen-like ions embedded in various environments have continuously been a hot topics in many fields of physics,such as atomic and molecular physics,plasma physics and condensed matter physic-s.Among them,Debye shielding effect in plasma has always been the focus of researches.Shielding effect is a phenomenon in which the attraction of an electron is weakened by the repulsion of other electrons.Debye shielding effect refers to the shielding effect in plasma,which is caused by introducing electric field into the plasma,after a period of time,electrons and ions in the plasma will move to shield the electric field.The thickness of the shield be-comes the Debye length,denoised as D.In this case,if we take some simple approximations,we can fomulize the potential generated by hydrogen-like ions in the plasma as follows(?) where ?=1/D refer to screening parameter.This potential is the Debye-Hiickel potential,which is the most representative screened Coulomb potential and is of great value interest for present work.The other two potentials we study are the exponential cosine screened Coulomb potential and the Hulthen potential,which also play an important role in many fields of physic-s.The exponential cosine screended Coulomb potential can be used to effectively simulate the strongly coupled dense quantum plasma environment,while the Hulthen potential is weak-er than the other two potentials,and is often used as an approximation of the Debye-Hiickel potential under weak screened conditions.We mainly study the eigenenergies of hydrogen and hydrogen-like ions under the three screened Coulomb potentials mentioned above.In the previous literature,the non-relativistic energies of bound states are obtained by solving the Schrodinger equation by some numerical approximation methods.In the framework of direct perturbation theory,we calculate the first-order relativistic corrections to the non-relativistic energy embedded in three screened environ-ments.The relativistic corrections include the relativistic mass correction,Darwin term and spin-orbit coupling term.utilizing the general pseudospectral method to solve the Schrodinger equation with high precision,and combining with the extrapolation method proposed by Zhu Ling et al.,the precision of our first-order relativistic correction results and the precision of non-relativistic energy can reach the same level.For different bound states,including the ground state and some excited states,the numerical results of three relativistic corrections with differ-ent values of screened parameters are systematically sorted out and tabulated.By comparing our results with corresponding results in other literatures,we found that the mass velocity term and the spin-orbit coupling term calculated by us are in good agreement with the results given in a literature,while significant discrepancy and even opposite trend is found for the Darwin term.In order to determine the problem,we compare it with another literature which uses fully relativistic calculations,and the total relativistic corrections of system energy given by present work are in good agreement with the results obtained in that literature.Then,we combine the data of these several literatures with our own data,and make reasonable speculation and discus-sion on the possible reasons for the existence of this discrepancy.We finally present the scaling law of the first-order relativistic corrections and discuss the validity of the direct perturbation theory with respect to both the nuclear charge and the screening parameter.Finally,we utilize the general pseudospectral method to solve the Dirac equation.First,we test our calculation accuracy by using the Coulomb potential.Then,we tabulate the relativistic energies of some bound states of hydrogen atoms in three types of screened environments.