WOLFRAM|DEMONSTRATIONS PROJECT

Quantum-Mechanical Particle in an Equilateral Triangle

​
q =
1
p =
q+1
q+2
q+3
q+4
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
-
2
ℏ
2m
2
▽
ψ{x,y}
=E
ψ(x,y)
with
ψ(x,y)=0
on and outside an equilateral triangle of side
a
. The ground-state solution
ψ
0
(x,y)=sin
4πy
3
a
-2sin
2πy
3
a
cos
2πx
a
corresponds to an energy eigenvalue
E
0
=
2
2
h
3m
2
a
. The general solutions have the form
ψ
p,q
(x,y}
with
q=0,
1
3
,
2
3
,1,
4
3
,
5
3
,2,⋯
and
p=q+1,q+2,⋯
, with energies
E
p,q
=
2
p
+pq+
2
q

E
0
. The Hamiltonian transforms under the symmetry group
C
3v
so eigenfunctions belong to one of the irreducible representations
A
1
,
A
2
,
or
E
. The states labeled by quantum numbers
p,0
, including the ground state
1,0
, are nondegenerate with symmetry
A
1
. All other integer combinations
p,q
give degenerate pairs of
A
1
and
A
2
states. Noninteger quantum numbers belong to twofold degenerate
E
levels.
In this Demonstration, contour plots of the wavefunctions
ψ
p,q
(x,y)
are displayed when you select the quantum numbers
p
and
q
. (If you change
q
, you must also change
p
.)
Except for the ground state, only the contours
ψ=0
, 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.