Wavefunctions in chaotic systems
Topics:
Exceptional states,
Rate of quantum ergodicity,
Momentum space representation
In ergodic systems it is proven that almost all
eigenfunctions become equidistributed in the semiclassical
limit (quantum ergodicity theorem, see the paper
Rate of quantum ergodicity in Euclidean billiards
for an introduction).
As consequence also at finite energies wavefunctions tend to be
uniform (well, of course only as uniform as is permitted by the quantum
oscillations).
For example the look like (grey-scale plot with black corresponding
to high probability):
Exceptional states
Still there is the possibility of exceptional wavefunctions
which are not equidistributed.
Prominent examples are so-called
scars which localize
around unstable periodic orbits and
bouncing ball modes
which arise in billiards with parallel walls:
Rate of quantum ergodicity
For ergodic systems it is of interest to know the speed by which
eigenfunctions become equidistributed; this
is described by the rate of quantum ergodicity.
It turns out that exceptional eigenfunctions may have an important
influence on this rate.
See the paper
Rate of quantum ergodicity in Euclidean
billiards for more details.
Momentum space representation
Most commonly one uses the position space representation of
eigenstates. However, also the momentum representation
provides useful insight.
The following figures show eigenstates
in position and momentum representation (3D plots)
and their grey-scale projections below.
Beneath the radially integrated angular momentum
distribution
I(φ) and for comparison
the Husimi Poincaré representation
Hn(s,p)
is shown
116th eigenstate in the cardioid billiard
567th eigenstate in the cardioid billiard
References:
Last modified: 27 October 2004, 16:21:54
Impressum, © Arnd Bäcker
