Logo Ulm

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.
--> Valid HTML 4.01! Last modified: Mon Jan 24 14:56:22 2005
URL of this page: http://www.physik.uni-ulm.de/theo/qc/