The Application Of B-spline And B-polynomial In Solving The Dirac Equation

The Dirac equation plays a fundamental role in relativistic quantum mechanics. The radial Dirac equations are pairs of differential equation of first order, which can be exactly solved only for a few cases such as hydrogen atom and an electron in a uniform magnetic field. In most cases in order to explain relativistic quantum phenomena one must appeal to numerical method. This thesis devotes to the numerical solutions of the Dirac equation for some typical potential such as Coulomb potential, harmonic potential and Morse potential as well as Hulthen potential by using B spline and B polynomial methods.We first give a brief outline of B spline and B polynomial methods in chapter 2 and 3 respectively. To check the correctness of two methods, we apply these methods to the Dirac equation with Coulomb potential. The good agreement of our results with exact solution demonstrates the effectiveness and correctness of two methods. Then we apply two approach to find bound state solutions of Dirac equation with harmonic potential and Morse potential as well as Hulthen potential by using powerful mathematical software MATHEMATICA. In our calculation some skills and techniques have been used to improve the accuracy of the result. We obtained the energy eigenvalues for each potential and compared our data with that of the same potential in non-relativistic quantum mechanics to probe the relativistic effect. The energy eigenvalues calculated by two methods are also compared to each other. The calculation errors were analyzed. Our calculation shows that B spline and B polynomial methods can be successfully used to find the numerical solutions of Dirac equation with varies potentials.In chapter 4, we summarized the main results and some experience come from our practical calculation. A promising prospect for the succedent research work is also outlined.All MATHEMATICA codes are given in the appendix.