Solution of the Feshbach–Villars equation for the step potential

Abstract

We solve the one dimensional Feshbach-Villars equation for spin-1/2 particle subjected to a scalar smooth potential. The eight component wave function is given in terms of the hypergeometric functions and via a limiting procedure, the wave functions of the step potential are deduced. These wave functions are used to test the validity of the boundary conditions deduced from the Feshbach-Villars transformation. The creation of pairs is predicted from the boundary condition of the charge density.

Statistical properties of the two dimensional Feshbach–Villars oscillator (FVO) in the rotating cosmic string space–time

“…ξ denotes a real dimensionless coupling constant, and R is the Ricci scalar curvature given by R = g µν R µν , where R µν is the Ricci curvature tensor. The inverse metric tensor is g µν , and g = det (g µν ) [45][46][47][48][49][50].…”

“…In recent years, a significant number of studies have been devoted to exploring the dynamical properties of single particles in a relativistic context and solving their wave equations using the FV approach. For example (see [41][42][43][44][45][46][47]), Bouzenada et al [48] investigated the Feshbach-Villars oscillator (FVO) case in a spinning cosmic string space-time and reported their findings on the thermal properties and density of the system.…”

Statistical proerties of the two-dimensional Feshbach-illars oscillator (FVO) in the rotating cosmic string

“…In recent decades, a number of papers have been produced with the aim of exploring the relativistic dynamical properties of single particles and solving their wave equations by adopting the FV scheme (e.g, Refs. [34][35][36][37][38][39][40] and other related references cited therein).…”

Matrix Continued Fraction Solution to the Relativistic Spin-0 Feshbach–Villars Equations

“…They split the Klein-Gordon wave function into two components and for the components vector they arrived at a Schrödinger-like equation with first order in time derivative. Although the Feshbach-Villars formalism appear in some advanced quantum mechanics books [9,10,11,12,13,14,15], and they were utilized in gaining deeper insight into relativistic physics of Klein paradox pair production, [16,17,18,19,20,21,22,23], in exotic atoms [24,25,26], used in theoretical consideraions [27,28,29,31], study relativistic scattering [32,32] and optics [33] or demosntrate PT symmetry [34,35,36,37], they were hardly used as a computational tool. The equations look like ordinary coupled differential equations, but the components are coupled by the kinetic energy operator, which makes them very hard to solve.…”