Supersymmetric Quantum Mechanics For Atomic Electronic Systems

TL;DR

This paper extends supersymmetric quantum mechanics to three-dimensional atomic systems, providing a new analytical framework for hydrogen and helium atoms that captures their quantum states more comprehensively.

Contribution

It introduces a vector superpotential approach in SUSY-QM for three-dimensional atomic systems, enabling detailed analysis of hydrogen and helium atoms beyond radial approximations.

Findings

Vector superpotential formalism for 3D atomic systems

Analytical expressions for excited states of hydrogen

Application of SUSY-QM to helium atom

Abstract

We employ our new approach to non-relativistic supersymmetric quantum mechanics (SUSY-QM), (J. Phys. Chem. A 114, 8202(2010)) for any number of dimensions and distinguishable particles, to treat the hydrogen atom in full three-dimensional detail. In contrast to the standard one-dimensional radial equation SUSY-QM treatment of the hydrogen atom, where the superpotential is a scalar, in a full three-dimensional treatment, it is a vector which applies regardless of the electron angular momentum. The original scalar Schrodinger Hamiltonian operator is factored into vector "charge" operators: Q and Q^{dagger}. Using these operators, the first sector Hamiltonian is written as H1 = Q^{\dagger} * Q. The second sector Hamiltonian is a tensor given by H2 = Q Q^{\dagger} and is isospectral with H1. The second sector ground state, psi^{(2)}_0, can be used to obtain the excited state wave functions of the first sector by application of the adjoint charge operator. Alternatively, Q applied to analytical, sector one excited states yield analytical results for the sector two vector eigenstates. Several of these are plotted for illustration. We then adapt the aufbau principle to show this approach can be applied to treat the helium atom.