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# Physics & Astronomy MSc thesis defense by Yann Audin

## July 31 @ 2:00 pm

In the present thesis, we consider an electron under an external electromagnetic field composed of a linear electric field parallel to a uniform magnetic field above the z = 0 plane and antiparallel below the plane. Classically, the electron undergoes a cyclotron motion in the z = 0 plane and harmonic oscillations along the z-axis. With SchrÃ¶dingerâ€™s equation, we find the energies of the electron to be functions of the cyclotron frequency w_{c}, a second frequency w_{z}, and two quantum numbers n and nz. When the ratio of frequencies w = w_{c}/w_{z} is a rational number, there sometimes can be more than one pair (n, n_{z}) yielding the same energy, leading to jumps in degeneracy. This is radically different from the regular Landau levels (an electron under a uniform magnetic field), where each energy level has the same degeneracy. We plot the energies as functions of w to point out that the energies have an extra degeneracy where lines intersect, and find analytical formulas for the degeneracy. For concreteness, we consider the electric field generated in the vicinity of the centre of a ring of uniformly distributed charge in the z = 0 plane. Such an electric field is linear in z, but adds an electric field in x and y which we treat with perturbation theory. The Landau level degeneracy is lifted by the x-y field and replaced by energy bands of tightly spaced levels. Finally, we use the Dirac equation to compute the first and second order relativistic corrections to the energies of the electron. The first order correction lowers the energy and includes cross-terms that split the extra degeneracy found in the non-relativistic case. We show that an extra degeneracy still occurs in the relativistic case, but at different ratios w.