Research On Analytic Solution For The Non-relativistic Schr(?)dinger Equation And Relativistic Dirac Equation

In this paper, we have sketched the theory of supersymmetric quantum mechanics, discussed the analytic solution of Bohr equation in several special cases, and investigated the pseudospin symmetry based on the framework of relativistic mean field theory. More details are summarizedi) In the first section, we have briefly introduced the supersymmetric quantum mechanics, and presented a set of methods to find exact solution for shape invariant potential. And then we have solved the Schrodinger-Woods Saxon problem by mapping Woods Saxon potential into Poschl-Teller I potential under point canonical transformations. Considering that Woods-Saxon potential is interesting in studying the properties of atomic nuclei, we have presented the exact solutions for bound states with the energy spectra and Eigen functions obtained.ii) In the second section, we have solved Bohr Hamiltonian for Eckart potential by adopting N-U method to Bohr equation HBψ= Eψ. With this technique, we have obtained the energy spectrum expression and the corresponding wave functions for the bound states with arbitrary angular momentuml. In the case ofγ-unstable andγ≈0, the analytic solution of Bohr Hamiltonian are presented with the approximation taken into account for centrifugal term.iii) In the third section, we have researched the pseudospin symmetry and checked the conditions of exact pseudo-spin symmetry in the framework of RMF theory. We have discussed the Dirac Woods-Saxon problem, and obtained the energy Eigen values and the corresponding Dirac spinors under the condition of exact pseudospin symmetry.iv) In the fourth section, we have solved Dirac equation for an electron moving in the presence of a varying magnetic field, and presented the corresponding exact solutions based on the same scheme as.