Abstract. S. I. Vakarchuk. Supersymmetry of the electron in magnetic field

S. I. Vakarchuk. Supersymmetry of the electron in magnetic field.

A thesis for a Candidate of Sciences degree on the speciality 01.04.02 – theoretical physics, Ivan Franko National University of Lviv, 2009.

The thesis is devoted to investigation supersymmetry of the electron in the magnetic field. The non-relativistic case in the frame of the Pauli Hamiltonian and relativistic one in the frame of the Dirac Hamiltonian are considered. The examples of three dimensional magnetic fields for which supersymmetry with two, three and four supercharges exists are found. For instance, the supersymmetry with two supercharges is realized in magnetic field of magnetic moment, the supersymmetry with three supercharges — in magnetic field of straight current, the supersymmetry with three supercharges — in magnetic field of magnetic octuple. The superalgebra for these cases is extended using Witten parity operators.

We consider the motion of an electron in the plane perpendicular to the magnetic field which possesses supersymmetry with two supercharges. It is shown that stationary states as a consequence of supersymmetry can have entanglement between spin and coordinate variables of the electron. Squared concurrence for these states is equal to the sum of the squared mean value of supercharges divided by the energy of the electron. The eigenstates of supercharges are maximally entangled.

The motion of electron with position-dependent mass is studied. For three dimensional space with a spherically symmetric dependence of mass on coordinates there exists supersymmetry with two, three and four supercharges similarly to constant mass. In the two-dimensional case the supersymmetry with two supercharges exists for an arbitrary dependence of mass on position and an arbitrary magnetic field, which is perpendicular to the plane of electron motion. For the latter case and for an axially symmetric dependence of mass on coordinates exact wave functions for zero energy ground state are found. The number of zero modes for an electron with position dependent mass in magnetic field is found as well. This result is the generalization of the Aharonov–Casher theorem obtained for constant mass for the case of position-dependent mass.

The ground state of electron in the magnetic of straight current is studied. For this case the supersymmetry with three supercharges is realized. It is shown that ground state of the electron in the magnetic field of this configuration has non-zero energy and thus supersymmetry is broken. The supersymmetric method developed for construction quasi exactly solvable potentials with two known states in one dimensional case is generalized for the case of the two-dimensional Pauli equation. Explicit examples of quasi exactly solvable magnetic field are presented.