Wavefunctions in mixed systems
Topics:
Amplitude distribution,
Autocorrelation function,
Bifurcations
In mixed systems the classical phase space consists
of both regular and irregular regions.
Poincare section for the limaçon billiard at ε=0.3:
The structure of phase space is reflected in the behaviour of
the wavefunctions of the quantum systems.
Amplitude distribution
To understand the behaviour of wavefunctions
in mixed systems one considers their
statistical behaviour.
One of the simplest statistics is the
amplitude distribution.
It turns out that irregular wavefunctions (i.e. those
which are supported on an irregular component) are
not Gaussian distributed in general.
Using a
restricted random wave model
the resulting distribution was derived in the paper
Amplitude distribution of eigenfunctions
in mixed systems.
Autocorrelation function
For irregular states in mixed systems one obtains a prediction
for the autocorrelation function
(see the paper
Autocorrelation function of eigenstates
in chaotic and mixed systems)
which is substantially
different from the
J0(r) behaviour
in ergodic systems.
The figure below shows an eigenfunction in the
limaçon billiard (ε=0.3, approx 130516
th state),
the corresponding Poincaré Husimi representation
and the classical angular momentum distribution and
finally the resulting autocorrelation function.
Thus, even for irregular states the autocorrelation function
clearly differs from the J0(r) behaviour!
Bifurcations
Bifurcations of periodic orbits may have a substantial
influence on the autocorrelation function.
This is illustrated in the following figure
(taken from the paper
Orbit bifurcations
and wavefunction autocorrelations)
where for the perturbed cat maps
the (absolute value of the) autocorrelation function
c(l)
is shown as a function of the perturbation parameter κ.
At bifurcations of periodic orbits and
l-caustics
strongly enhanced peaks are observed.
References:
Last modified: 27 October 2004, 16:21:54
Impressum, © Arnd Bäcker
