Generalized plane waves for particles of Schrödinger and Dirac particles on the background of Bolyi-Lobachevsky geometry: simulating of a special medium
DOI:
https://doi.org/10.24144/2415-8038.2012.32.133-140Keywords:
Geometry Bolyai–Lobachevsky Plane wave, Schrödinger equation, Dirac equation, Hypergeometric functionAbstract
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 U(z)=U0e2z. 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 e, k12+k22, and the curvature radius r. Similar analysis is performed for the case of a Dirac spin 1/2 particle; additional to the quantum numbers e, k12+k22 for the spin 0 particle here is a quantum number related with an extended helicity operator.References
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