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Brazilian Journal of Physics

Exact Solutions to Schrödinger Equation for a Charged Particle on a Torus in Uniform Electric and Magnetic Fields

Alexandre G. M. Schmidt

Published Jun 21, 2020• 10 citations
Publisher
DOI
10.1007/s13538-020-00764-9

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We calculate exact solutions of the Schrödinger equation for a particle constrained to move along a spherical surface and interacting with non-central potentials, namely, (i) Makarov, (ii) ring-shaped pseudo-harmonic oscillatory, and (iii) Kratzer potentials. We also study exact solutions of the Pauli equation in the same geometrical setting for a charged particle in the presence of a uniform magnetic field. In this case, the two-component spinor can adhere to the surface only if the magnetic field intensity has certain special values. The solutions of Schrödinger equations allow us to obtain exact Pauli spinors and their corresponding energy eigenvalues for the same non-central potentials.
We present an exact analytical solution for the quantum dynamics of a charged particle subjected to both a time-dependent electric field and a static magnetic field aligned along the zzz -direction. Using a systematic approach based on successive unitary transformations, we reduce the original three-dimensional problem to a two-dimensional system of decoupled, time-dependent harmonic oscillators. This technique produces free parameters that allow us to impose constrains to derive the exact solution of Schrödinger equation with the time-dependent Hamiltonian through the explicit derivation of the quantum propagator and shows the equivalence of this approach to established path integral methods for such systems. The developed framework provides new insights into quantum systems with time-dependent electromagnetic fields and offers analytical solutions.