【作者】 李晓鹏；

【导师】 强稳朝；

【作者基本信息】 西安建筑科技大学 ， 计算数学， 2007， 硕士

【摘要】 狄拉克方程是相对论量子力学的基本方程。对于不同的势场，径向狄拉克方程是耦合的一阶微分方程组。但是只有氢原子和均匀磁场中的电子等少数问题求得了其精确解。在大多数情况下为了解释相对论现象我们必须用数值方法。本文针对库仑势场、谐振子势场、Morse势场和Hulthén势场，用B样条函数方法和B多项式函数方法求解径向狄拉克方程。第二章和第三章分别介绍了B样条函数方法和B多项式函数方法。为了检验这两种方法的正确性，首先将两种方法应用于解库仑势场下的径向狄拉克方程，其计算结果与精确解比较吻合，证明了这两种方法的正确性和有效性。然后应用这两种方法，去求解谐振子势场、Morse势场和Hulthén势场下的径向狄拉克方程能量本征值。这里使用了MATHEMATICA软件，并在实际计算中应用了多种技巧以提高计算精度。将我们计算的能量本征值和非相对论的能量本征值作比较，以此分析相对论效应。然后将两种方法得到的能量本征值作比较，分析了两者之间的误差。通过计算表明，这两种方法对于求解狄拉克方程是成功的。在第四章里，我们总结了主要计算结果，以及实际计算中的一些经验，并作了进一步的展望。最后，附录MATHEMATICA程序。更多还原

【Abstract】 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.更多还原