Hydrogen Atom in Electric and Magnetic Fields: Dynamical Symmetries, Superintegrable and Integrable Systems, Exact Solutions

TL;DR

This paper explores the symmetries, integrability, and exact solutions of the hydrogen atom under electric and magnetic fields, highlighting differences in solvability and chaos, and providing analytical solutions for complex cases.

Contribution

It offers a comprehensive analysis of the hydrogen atom's symmetries and integrability in various fields, including new exact solutions in magnetic fields despite non-separability.

Findings

Hydrogen atom exhibits SO(4) symmetry and superintegrability in free space.

In electric fields, the system remains integrable with SO(2)xSO(2) symmetry.

In magnetic fields, the system is non-separable but admits exact solutions as power series.

Abstract

The Hamiltonian of a pure hydrogen atom possesses the SO(4) symmetry group generated by the integrals of motion: the angular momentum and the Runge-Lenz vector. The pure hydrogen atom is a supersymmetric and superintegrable system, since the Hamilton-Jacobi and the Schr\"odinger equations are separable in several different coordinate systems and has an exact analytical solution. The Schr\"odinger equation for a hydrogen atom in a uniform electric field (Stark effect) is separable in parabolic coordinates. The system has two conserved quantities: z-projections of the generalized Runge-Lenz vector and of the angular momentum. The problem is integrable and has the symmetry group SO(2)xSO(2). The ion of the hydrogen molecule (problem of two Coulomb centers) has similar symmetry group SO(2)xSO(2) generated by two conserved z-projections of the generalized Runge-Lenz and of the angular momentum on the internuclear axis. The corresponding Schr\"odinger equation is separable in the elliptical coordinates. For the hydrogen atom in a uniform magnetic field, the respective Schr\"odinger equation is not separable. The problem is non-separable and non-integrable and is considered as a representative example of quantum chaos that cannot be solved by any analytical method. Nevertheless, an exact analytical solution describing the quantum states of a hydrogen atom in a uniform magnetic field can be obtained as a convergent power series in two variables, the radius and the sine of the polar angle. The energy levels and wave functions for the ground and excited states can be calculated exactly, with any desired accuracy, for an arbitrary strength of the magnetic field. Therefore, the problem can be considered superintegrable, although it does not possess supersymmetry and additional integrals of motion.