Mesoscopic Physics and Quantum Chaos
Diffraction in quantum systems
Semiclassical approximations are methods for the investigation of quantum systems in the limit of short wavelengths. They allow a physical understanding of quantum phenomena by relating them to properties of the corresponding classical system. However, in cases where the potential changes abruptly on the scale of a wave length usual semiclassical approximations become inaccurate. Examples are impurities in semiconductors, the core of Rhyberg atoms, magnetic flux lines, or corners in billiard models. In order to describe the diffraction of wave functions on discontinuities or singularities of the potential semiclassical methods have to be extended.
Semiclassical treatment of diffraction
Diffraction and spectral statistics
One way of characterising a complex quantum system is
to consider statistical distributions of its high
lying eigen energies. It has been found that these
distributions reflect the nature of the underlying
classical system. In particular, if the classical
system is chaotic its spectral statistics coincide
with those of eigenvalues of random matrices.
This agreement has been observed in numerous
numerical investigations. Semiclassical methods can
be used to explain this agreement in certain regimes.
In one project we investigate whether singularities
of the potential lead to modifications of these
results. This is done by semiclassically determining
the influence of diffraction on spectral statistics.
Of particular interest is the question whether
diffraction can lead to deviations from random