2019/07/01 by Yann Audin, Audin, Yann, Ariel Edery +1
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Noncommutative and Quantum Gravity Theories #Quantum Electrodynamics and Casimir Effect #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1907.00769
openalex publication_date 2019/07/01 · openalex created_date 2019/07/12 · openalex updated_date 2026/07/28
We consider an electron moving under a constant magnetic field (in the z-direction) and a linear electric field parallel to the magnetic field above the z=0 plane and anti-parallel below the plane. Two frequencies characterize the system: the cyclotron frequency ωc corresponding to motion along the x-y plane and associated with the usual Landau levels, and a second frequency ωz corresponding to motion along the z-direction. In previous work, the non-relativistic energies of this system were obtained, and it was shown that an extra degeneracy (beyond the Landau degeneracy) occurs when the ratio w=ωc/ωz is rational. In this paper, we use Dirac's equation to obtain compact formulas for the first and second order relativistic corrections to this system via perturbation theory. The formulas are expressed in terms of the two frequencies ωc and ωz, and two quantum numbers, n and nz, both of which are non-negative integers. The first order correction is negative and lowers the original energies. We plot the energy (zeroth plus first order) versus the ratio w and there are degeneracies at all points where lines intersect. However, the degeneracy does not occur at the same w as before. To illustrate this, we show how the first order correction splits the energy levels for the case ωc=ωz.