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WOLFRAM NOTEBOOK
Quantum-Mechanical Particle in an Equilateral Triangle
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 (x,y)=sin-2sincos corresponds to an energy eigenvalue =. The general solutions have the form (x,y} with and , with energies =+pq+. 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.
-ψ{x,y}ψ(x,y)
2
ℏ
2m
2
▽
=E
ψ(x,y)=0
a
ψ
0
4πy
3
a2πy
3
a2πx
a
E
0
2
2
h
3m
2
a
ψ
p,q
q=0,,,1,,,2,⋯
1
3
2
3
4
3
5
3
p=q+1,q+2,⋯
E
p,q
2
p
2
q
E
0
C
3v
A
1
A
2
E
p,0
1,0
A
1
p,q
A
1
A
2
E
In this Demonstration, contour plots of the wavefunctions (x,y) 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.
ψ
p,q
p
q
q
p
.)
ψ=0
Vibration of an equilateral-triangular plate with fixed edges gives a classical analog of this problem with the same solutions.