Diffraction in Quantum Systems


Martin Sieber

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 Rhydberg 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

Semiclassical approximations in quantum mechanics are the analog of geometrical optics approximations for electromagnetic waves. Consider, for example, a light source and a metallic plate (black) as shown in Fig. 1. In geometrical optics the intensity and phase of light at an observation point are given in terms of a ray from the source to the observation point (red). The observation region is divided into an illuminated region and a shadow region which is inaccessible to direct rays. In order to describe the diffraction of light on the edge of the plate geometrical optics has to be extended. This is done by introducing an additional kind of rays (blue) that go to the observation point via the edge of the plate. The underlying theory is Keller's geometrical theory of diffraction.

Fig. 1
Similar methods are applied for describing diffraction in quantum systems in the limit of small wave lengths. Fig. 2 shows a system that is confined to a finite region which contains a magnetic flux line. In the general semiclassical theory the energy spectrum of the quantum system is approximated in terms of periodic orbits (red). The diffraction of wave functions on the flux line requires the introduction of an additional set of trajectories that start from the flux line and return to it (blue).
  • M. Sieber, Semiclassical Treatment of Diffraction in Billiard Systems with a Flux Line, Phys. Rev. E 60 (1999) 3982-3991.
  • M. Sieber, N. Pavloff and C. Schmit Uniform Approximation for Diffractive Contributions to the Trace Formula in Billiard Systems, Phys. Rev. E 55 (1997) 2279-2299.

Fig. 2

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 investigated whether diffraction on singularities of the potential can lead to modifications of these results. In particular, we examined whether the addition of a point scatterer to a system can change its spectral statistics. It was shown for the first terms in a semiclassical expansion that there is no change in the spectral statistics if the classical system is chaotic.

  • M. Sieber, Spectral Statistics in Chaotic Systems with a Point Interaction, J. Phys. A 33 (2000) 6263-6278.
  • M. Sieber, Geometrical Theory of Diffraction and Spectral Statistics, J. Phys. A 32 (1999) 7679-7689.