Dynamics of a quantum particle in a static modulated magnetic field B=(0,0,B/cosh ((x−x ₀)/δ))

Abstract

Quantum motion of a particle with its massm and its chargee>0 in thex−y plane under the influence of the static unidirectionally modulated magnetic fieldB=(0,0,B/cosh²((x−x ₀)/δ)) is studied in this paper. We solved the single-particle problem exactly. Expressions for eigenenergies and eigenfunctions are found. Several physical phenomena are described: the energy spectrum separates into two parts which we call a discrete part and a continuous part—the discrete part of the spectrum corresponds with those states states which describe localized behavior in the direction of the field modulation; these states are extended in the perpendicular direction; the effective mass of this quasi-one-dimensional motion is found to be negative and dependent on the discrete quantum numbern—the states corresponding to the continuous part of the energy spectrum are found to be of two types: the reflection states, which are extended only in that part of thex−y plane where the effective potential barrier is small, and the states describing overbarrier motion in the whole plane; there exists a minimum of the energy for the states from this part of the spectrum which corresponds with a nonzero momentum of the particle motion in they-direction and almost zero momentum for thex-direction motion; there exists twofold degeneracy for those states from the continuous part of the spectrum for which their energy is lower than a certain value.