Small Components of the Wave Function of Electron

Zhiyong Wang, Xiong Cai-Dong, Qiu Wei-gang +1 more

This paper shows that the moving or time-varying large components of four-component wavefunction of electron would induce small components, and vice versa. Then when a wave packet of electron is moving with high speeds or varies rapidly, or its size is sufficiently small, or in the presence of a strong electromagnetic field, its small components and the related effects cannot be ignored. Furthermore, the spin quantum states of both a moving electron and a motionless electron can be affected by some special electrostatic fields. This may open a new pathway for spintronics to the manipulation of electron spins in the absence of applied magnetic fields. In this paper, via an analogy between the wave functions of free electron and free electromagnetic fields, we study the relation between the large components and the small components of the wave function of electron, and show that in some cases the small components cannot be ignored. As an application, we will demonstrate that not only the spin quantum states of a moving electron but also those of a motionless electron can be affected by some special electrostatic fields. This may provide a new approach for enabling electrical control over electron spins. In this paper, we apply the natural units of measurement ( ) and take the space-time metric tensor as

John F. Ogilvie

The solution of Dirac's equation for the hydrogen atom according to relativistic wave mechanics yields for each state a vectorial amplitude function with four components, two large and two small. Each such component has its characteristic surface of constant amplitude, of which we plot several examples. For each state of the hydrogen atom there is both a density of electronic charge surrounding the atomic nucleus and an electronic current circulating about the polar axis; the latter generates a magnetic dipolar moment that agrees precisely with experiment.

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.