Past exam of the options I course of the University of Oxford / 2012 / J (Quantum Theory)

The interior time-independent Schrödinger equation for a free particle isConfinement by the rectangular infinite well imposes Dirichlet boundary conditions: $\psi =0$ at $x=0,b$ and $y=0,\sqrt{5}b$. The wave function is square-integrable and may be normalized to unit probability on the rectangle.

Separation of variables gives products of the one-dimensional well states,The Hamiltonian is the sum of the independent $x$ and $y$ kinetic-energy operators, so the one-dimensional energies add. Put $C={\pi }^2{\hslash }^2/(2m{b}^2)$. ThenFor $n=1$, increasing $\ell$ gives $6C/5,9C/5,14C/5,21C/5$. The smallest energy with $n\ge 2$ is also $21C/5$, attained at $(2,1)$; all $n\ge 3$ give larger energies. Thus the ground and first three distinct excited energy levels areThe largest listed energy therefore has energy-level degeneracy $\boxed{2}$, since the two product eigenfunctions are linearly independent and no other pair solves $5n^2+{\ell }^2=21$.