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.

In this Demonstration, contour plots of the wavefunctions

are displayed when you select the quantum numbers

and

. (If you change

, you must also change

Except for the ground state, only the contours

, representing the nodes of the wavefunction, are drawn. The contour plots might take a few seconds to generate.