# Exact Energy Levels and Eigenfunctions of an Electron on a Nanosphere Under the Radial Magnetic Field

## TL;DR

This paper derives exact energy levels and wave functions for an electron on a sphere under a radial magnetic field, using Jacobi polynomials, and explores behavior in strong magnetic fields, comparing with existing results.

## Contribution

It provides a precise analytical solution for the electron's energy levels and wave functions on a spherical surface under radial magnetic fields, including the strong field limit.

## Key findings

- Wave functions expressed in Jacobi polynomials with orthogonality properties.
- Energy levels analyzed for very large magnetic fields, resembling Landau levels.
- Comparison with previous literature confirms the validity of the results.

## Abstract

The exact energy levels and wave functions of an electron free to move on a sphere under the radial magnetic field is found. Wave functions are expressed in terms of Jacobi polynomials which were well-defined and have orthogonality relation, recurrence relations, series expansions etc. We have also discussed the the wave functions and energy levels in case of very large magnetic field. Landau energy levels are shown for strong constant magnetic field occurring on two-dimensional surfaces, if the radius is very large. The results compared with those previously found in the literature.