Basic Structures of Relativistic Wave Functions

Sihong Shao, Zhendong Li, Wenjian Liu

Z. Ficek

This chapter introduces the concepts of the relativistic theory to basic problems of quantum physics, in particular, the extension of the Schrodinger equation to a relativistic form. Starting from the fundamental laws of relativity, the chapter evaluates the Klein–Gordon equation and investigates if the equation can be considered a generalization of the non-relativistic Schrodinger equation to the case of the relativistic energy. The chapter shows that the wave function, which is a solution to the Klein–Gordon equation, cannot be connected with the probability wave function. For this reason, the Dirac equation is used to solve the problems faced by the Klein–Gordon equation. The Dirac equation also includes the spin. The advantage of the Dirac equation over the Schrodinger and Klein–Gordon equations is that it naturally includes the spin. In the Schrodinger and Klein–Gordon equations, the spin is not present and has to be added to the wave function manually whenever the wave function determines a system with the spin.

Alina V. Ivashkevich, Viktor M. Red'kov, Artur Mikhailovich Ishkhanyan

In the paper, we examine the non-relativistic approximation in the relativistic system of equations in Cartesian coordinates for 16-component wave functions with transformation properties of the vector-bispinor under the Lorentz group. When performing the non-relativistic approximation, for separating large and small components in the complete wave function we apply the method of projective operators. Accordingly, the complete wave function is presented as a sum of three parts: the large part depends on 6 variables, and the small ones depend on 14 variables. We have found two linear constraints on large components and two constraints on the small ones. After performing the procedure of the non-relativistic approximation we have derived 6 equations with a needed non-relativistic structure, which include only 4 large components. It is proved that only 4 equations are independent, so we have arrived at the generalized Pauli-like equation for the 4-component wave function. The analysis of transformation properties of the non-relativistic wave function permits us to generalize the structure of the derived equation to an arbitrary curved 3-space.

Paul E. Hussar

Konstantin K. Likharev

Amaury de Kertanguy

Starting with a very basic statement that any physical constants cannot be written with an infinite precision, it is shown how to introduce this uncertainty into the Hamiltonian of non-relativistic atomic (NR) physics and how to estimate errors on quantum operators (energy, frequency, momenta) when an uncertainty is assigned to . The Schrödinger equation is written and the kinetic energy term is transformed into a Laplacian: . This transformation leads (as known since 1926) to the wave equation, whose solutions are wave functions. The relativity correction to the kinetic energy term is introduced and its effect is discussed. (h constant has an uncertainty value taken from CODATA.)