Two-dimensional Hydrogen-like Atom in a Constant Magnetic Field

TL;DR

This paper analyzes the two-dimensional hydrogen-like atom in a magnetic field, providing exact solutions for certain energy states and numerical methods for higher states, revealing how magnetic fields influence atomic behavior.

Contribution

It offers a detailed classification of the problem into six cases and derives exact solutions for specific energy states, advancing understanding of 2D atomic systems in magnetic fields.

Findings

Exact solutions for negative and positive shifted energy eigenvalues.

Numerical methods for higher order states.

Recursion relations for zero eigenvalue cases.

Abstract

The two-dimensional hydrogen-like atom in a constant magnetic field is considered. It is found that this is actually two separate problems. One for which the magnetic field causes an effective attraction between the nucleus and the electron and one for which it causes an effective repulsion. Each of the two problems has three separate cases depending on the sign of a shifted energy eigenvalue. For two of the six possibilities (shifted energy eigenvalue that is negative) it is shown that the first four solutions can be obtained exactly. For another two of the six possibilities (shifted energy eigenvalue that is positive) it is shown that the first eight solutions can be obtained exactly. For higher order states the energy eigenvalue is the root of a fifth or higher order polynomial, hence, the eigenvalue must be obtained numerically. Once the energy eigenvalue is known the solution to the radial wave equation is also known. Exact solutions for the radial wave equation, for the remaining two possibilities (shifted energy eigenvalue that is zero), are given to any desired order by means of a recursion relation.