# 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.