Tetrahedron's Home Page
The hyperbolic tetrahedra are finding applications in cosmology
as well as in quantum chaos.
In the former case perturbations in a compact hyperbolic universe can be
simulated in order to compute the properties of the
cosmic microwave background.
In quantum chaos a tetrahedron is considered as
a three dimensional billiard in a noneuclidean space with constant negative
curvature.
A free point particle is confined within the tetrahedron
and is elastically reflected at the boundary.
The classical billiard possesses the Anosov property and thus shows the
strongest form of deterministic chaos (hard chaos in Gutzwiller's definition).
The quantum mechanics is governed by the Schroedinger equation.
It determines the quantal levels and the probability distribution of the
particle within the billiard.
The tetrahedral eigenfunctions are visualized for some
quantal levels belonging to the symmetry class possessing
Dirichlet boundary conditions on the surface of the tetrahedron.
More details about the tetrahedral billiards can be found in:
R. Aurich, J. Marklof,
Trace Formulae for Three-Dimensional
Hyperbolic Lattices and Application to a Strongly Chaotic
Tetrahedral Billiard,
Physica D 92 (1996) 101,
DESY report
95-009 (1995),
chao-dyn 9502001
[abs,
ps.gz]
Eigenfunctions of Tetrahedral billiard T8
229th eigenfunction of the tetrahedral billiard T8
(E=1501.7601)
The eigenfunction in color
(mpeg) and its modulus square
(mpeg).
256th eigenfunction of the tetrahedral billiard T8
(E=1603.5592)
The eigenfunction in color
(mpeg) and its modulus square
(mpeg).
285th eigenfunction of the tetrahedral billiard T8
(E=1704.5414)
The eigenfunction in color
(mpeg) and its modulus square
(mpeg).
315th eigenfunction of the tetrahedral billiard T8
(E=1803.7282)
The eigenfunction in color
(mpeg) and its modulus square
(mpeg).
343th eigenfunction of the tetrahedral billiard T8
(E=1904.2713)
The eigenfunction in color
(mpeg) and its modulus square
(mpeg).
375th eigenfunction of the tetrahedral billiard T8
(E=2000.1506)
The eigenfunction in color
(mpeg) and its modulus square
(mpeg).
752th eigenfunction of the tetrahedral billiard T8
(E=3033.0593)
The eigenfunction in color
(mpeg) and its modulus square
(mpeg).
753th eigenfunction of the tetrahedral billiard T8
(E=3034.9153)
The eigenfunction in color
(mpeg) and its modulus square
(mpeg).
The
tetrahedral billiard T8
is shown from the same perspective as the tetrahedral eigenfunctions.
Classical Dynamics betrayed by Eigenfunctions
The structure of the classical dynamics, e.g
is hidden within the eigenfunctions and can be revealed by careful
chosen sums over the eigenfunctions as discussed in detail in
R. Aurich, F. Steiner,
Orbit sum rules for the quantum wave functions of the strongly
chaotic Hadamard billiard in arbitrary dimensions ,
Ulm report ULM-TP/00-6
(December 2000).
Some of the shortest
periodic orbits
are shown in the tetrahedral fundamental cell.
The shortest periodic orbit is shown in red, the next two having
the same length are shown in blue, whereas the third shortest is
shown in orange. Some of the next ones are shown in grey.
A certain sum over the eigenfunctions reveals
the action of the rotation elements as shown
here.
The intensity increases from blue to red.
The chosen parameters L=0.5 and t=0.01 corresponds to
figure 5 in ULM-TP/00-6.
See there for more details.
A sum
over the eigenfunctions which reveals the contribution of the rotation
elements as well as the boost of the shortest periodic orbit.
However the periodic orbit shown as a blue tube is not clearly revealed
due to the large contribution of the rotation elements.
Subtraction of their contributions shows clearly the
shortest periodic orbit.
These two animations correspond to figures 6a and 6b in ULM-TP/00-6.
This
animation
corresponds to figure 7 in ULM-TP/00-6.
It is also obtained from the eigenfunctions and shows high intensities
near the third shortest periodic orbit.
All colored animations are computed using
Amira
from the Konrad-Zuse-Zentrum für Informationstechnik Berlin.
Here
is the Home Page of the Quantum Chaos Group in Ulm.
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Last modified: Mon Jan 24 14:56:22 2005
URL of this page: http://www.physik.uni-ulm.de/theo/qc/