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The effect of the geometric potential and an external magnetic field on a charged particle on a helicoid

We perform an analysis of the combined effects of geometry and a magnetic field for the case of a charged particle on a helicoid. The effective quantum potentials for a charged spinless particle confined on a helicoid for two simple magnetic field configurations are derived. These potentials depend nontrivially on the surface curvature and the external magnetic field. We find that the qualitative behavior of the effective potential can be altered by changing the strength of the applied magnetic field. The application of a magnetic field results in effective potentials that are either repulsive or attractive, depending on the magnitude of the magnetic field and the angular momentum of the particle. Finally, for the case of effective potentials that have a minimum, we also obtain approximate expressions for the energy levels valid when the particle is near a minimum, and these are found to be similar in form to the energy levels of a particle in a harmonic oscillator potential.