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.
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.
  • Contributed by: Wai-Kee Li (Chinese University of Hong Kong) and S. M. Blinder


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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.
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