Generalized plane waves for Schrödinger and Dirac particles on the background of Bolyai–Lobachevsky geometry: simulating of a special medium

Abstract

Bolyai–Lobachevsky geometry substantially affects quantum-mechanical particles, simulating a medium with special reflecting properties of an ideal mirror. For Scrödinger particle the problem reduces to a second order differential equation which can be associated with one-dimensional Schrödinger problem for a particle in external potential field 2 U(z) U0e z . In quantum mechanics, curved geometry acts as an effective potential barrier with reflection coefficient R 1. Hyperbolic geometry simulates a medium that effectively acts as an ideal mirror. Penetration of the particle into the effective medium, depends on the parameters of quantum solutions 2 2 1 2   k  k , and the curvature radius  . Similar analysis is performed for the case of a Dirac spin 1/2 particle; additional to the quantum numbers 2 2 1 2   k  k for the spin 0 particle here is a quantum number related with an extended helicity operator. Key words: geometry Bolyai–Lobachevsky, plane wave, Schrödinger equation, Dirac equation, hypergeometric function.