The particle in an equilateral triangle is the simplest quantum-mechanical problem that has a nonseparable but exact analytic solution. The Schrödinger equation can be written with on and outside an equilateral triangle of side . The ground-state solution corresponds to an energy eigenvalue . The general solutions have the form with and , with energies . The Hamiltonian transforms under the symmetry group so eigenfunctions belong to one of the irreducible representations , or . The states labeled by quantum numbers , including the ground state , are nondegenerate with symmetry . All other integer combinations give degenerate pairs of and states. Noninteger quantum numbers belong to twofold degenerate levels.