2 - Physics of the electron

Abstract

This chapter focuses on the quantum theory of an electron. In the debut within a frame of a nonrelativistic consideration of a hydrogen atom with the aid of the method of factorization, we obtain Bohr’s famous formula for frequencies and touch on a question regarding the calculation of observable intensities of transitions. We then proceed to a relativistic consideration; according to Dirac’s original work in relation to the wave equation for an electron, we derive the values for spin and magnetic moment of an electron. The elegant devices of noncommutative algebra in combination with the developed method of factorization enable us to construct an exact solution for the energy levels of a relativistic hydrogen atom. We thus arrive at the expression for fine structure, bypassing conventional calculations with respect to a solution of a differential equation for an electron in a Coulombic potential. Moreover, we analyze in detail an elementary solution of the free Dirac equation in the form of plane waves and discuss qualitatively a question regarding the existence of positrons. In the endgame, we consider a problem regarding the motion of an electron in a magnetic field; in the case of a homogeneous magnetic field, an exact solution of an equation yields the well-known Landau levels, whereas for a weakly inhomogeneous magnetic field we resort to a solution according to perturbation theory and obtain the sought energy levels in the form of expansions with respect to the corresponding polynomials of quantum numbers.