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.
