A Simple Solution of the Relativistic Dirac Equation for a Particle in a Central Potential and an Example

Abstract

In previous studies, a simple procedure for the general solution of the radial Schrodinger equation has been found for spherical symmetric potentials without making any approximation. The wave functions are always periodic. Two difficulties may be encountered: one is to solve the equation, E=U(r), where E and U(r) are the total and effective potential energies, respectively; and the other is to calculate the integral, ∫▒〖√(U(r))  dr 〗. If these calculations cannot be made analytically, it should then be performed by numerical methods. To find the energy of the ground state (minimum energy), there is no need to calculate this integral, it is sufficient to find the classical turning points, that is to solve the equation, E=U(r). We have applied this simple procedure to a lot of non-relativistic cases.  In this short article, by using this simple procedure, the relativistic Dirac equation for a particle in central potential well of any form was solved without making any approximation. It has been applied to the hydrogen-like atom and the results have been compared with the other results