Analytical Study of Band Structure of Material Using Relativistic Concept

Abstract

In this paper, we present the study of band structure relativistically. Here, Dirac equation is formulated from Hamiltonian in which the formulation is found to contain a correction term known as spin-orbit coupling given as Theoretical Frame Work: Band Theorye that modifies the non-relativistic expression for the same formulation. This term leads to double spin-degeneracy within the first Brillioun zone which is a concept that is not found in other method of study of band structure of material.

Analytical Study of the Behavioral Trend of Klein-Gordon Equation in Different Potentials

“…Many authors have used one form potential out of different known potentials to solve Klein-Gordon or Dirac relativistic equation in order to study free particle behavior or to ascertain its energy Eigen value [1][2][3][4][5][6][7][8][9] the case of KGE, it describe so well spin-zero particle if potential is applied because it enable the reduction of complex KGE to solvable equation state such that the exact solution or the analytical solution can be obtained [10][11][12][13][14]. With this, it still requires the use of good mathematical methods such as the variational method, functional analysis, supers metric approach, Nikiforov-Uvarov (NU, the asymptotic iteration method and so on In recent period, work has been carried out on to study bound state of KGE for a number of special potentials [1,15] even in the case of equal vector and scalar potential [16], because it reduces KGE to a Schrodinger type of equation which could in turn be transformed into hypergeometric differential equation that has a known solution using [17][18][19][20] and this is more reason why KGE is receiving attention considerably in the literature when it comes to use of potentials [21,22]. In fact it has been shown that exact solution are possible with some certain central potentials [23][24][25][26] which has help in investigation of bound states of the KGE for that particular potential which on the other hand has invariably led to derivation of the exact expression of the energy eigenvalues and the corresponding normalized eigenfunctionns in terms of some special polynomials and hypergeometrical function [16,[27][28][29].…”

Analytical Study of the Behavioral Trend of Charged Particle Interacting with Electromagnetic Field: Klein-Gordon/Dirac Equation

“…Since the advent of quantum mechanics, the study of particle behaviour in different media especially the interaction of particle with electromagnetic has been of great interest in both non-relativistic and non-relativistic domain. This further led Gordon and Dirac to go ahead in formulation of their own equation respectively Akhiezer and Berestetskii (1965), Greiner (1987) which has revolutionalize the study of particle in a field that has brought about quantum electrodynamics thereby paving the ways for the use their equation for analysis of behavior of particle in E.M field (Simulk & Krivs ki, 2014; Frandkin et al, 1991;Ugwu & Echi, 2013). For instance, a new class of exact solutions of Klein-Gordon equation for a charge particle in electromagnetic wave medium using laser field as case study has been examined in which the analytic solutions of the particle interacting with field was considered (Fedorov, 1997).…”