Quantum-Mechanical Particle in an Equilateral Triangle
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.[more]
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.
Vibration of an equilateral-triangular plate with fixed edges gives a classical analog of this problem with the same solutions.[less]
Snapshot 1: contour plot of ground state
Snapshot 2: degenerate pair of , states
Snapshot 3: lowest-energy states
Reference: W.-K. Li and S. M. Blinder, "Solution of the Schrödinger Equation for a Particle in an Equilateral Triangle," Journal of Mathematical Physics, 26(11), 1985 pp. 2784–2786.