# Schrödinger equation and degeneracy of atomic orbitals

How does the application of the Schrödinger equation to model a system, such as a particle in a box, help us understand the origin of the degeneracy of atomic orbitals?

• It doesn't. Unless we apply it to a model system which is a hydrogen atom, that is. Commented Apr 10, 2021 at 10:20
• I would argue it does if you consdier a 2D (or 3D etc.) box and consider under what conditions you can get degeneracies. But I also would argue this isn't the world's greatest ever exam question. Commented Apr 10, 2021 at 10:37
• I'd say the simplest model possessing degeneracy is the particle on a ring, which is quasi 1D for fixed radius.
– Paul
Commented Apr 10, 2021 at 12:19

• This is not right. Consider the Hamiltoninan $H=-\partial_x^2-\partial_y^2+(x-y)^2$ with Dirichlet boundaries at $x,y=\pm1$. It's invariant with respect to reflection $x\leftrightarrow y$, but the symmetry group is abelian, so you don't get degeneracy (except maybe accidental). Commented Apr 10, 2021 at 18:55