14 - The Klein–Gordon and Dirac Equations

Abstract

Covered in this chapter are the Klein–Gordon equation and the Dirac wave equation for spin 1/2 particles. The general Dirac equation is solved first for a free electron and then solved for a bound electron bound to a proton. The resulting Hamiltonian is evaluated to yield the LS coupling term (coupling of the total electron orbital L and spin S angular momentums) of magnetic neutron scatter, and the hyperfine interaction terms for the atomic spectra of hydrogen-like atoms. Also obtained are the equations of motion of the bound electron spin, orbital, and total angular momentum vectors. Next, we derive the Dirac equation for hydrogen and hydrogen-like atoms as a pair of coupled, Dirac eigenequations, which are solved to yield asymptotic and then regular solutions. From the regular solution, we obtain the electron energy levels formula for the hydrogen atom (and hydrogen-like atoms) and the quantum number relationships. Finally, we derive the nuclear electric quadrupole potential, which is obtained from the electric potential energy term of the Dirac Hamiltonian.