URN to cite this document: urn:nbn:de:bvb:355-epub-266972
Semiclassical approach to systems of identical particles.
Diplomarbeit, Universität Regensburg
For most quantum mechanical systems of physical interest, central properties like the energy spectrum are not exactly accessable by analytical methods. In single-particle systems, however, many of the quantum features can be analytically adressed in semiclassical approximations that have been developed over the last decades and are now well established and sophisticated.
This work presents the transfer of semiclassical methods to systems of many identical and indistinguishable particles, showing the conceptual differences to single-particle systems. The concepts of the counterpart of periodic orbit theory are adressed and a classical sum rule is developed as a many-body-extension of a similar sum rule for single-particle systems, which is an useful tool for estimating statistical properties of many-body energy spectra.
The central result is a novel analytical approach to the calculation of the mean density of states in many-body billiard systems. The presented method makes explicit the intrinsic geometry inherent in the symmetrization postulate and, in the spirit of the usual Weyl expansion for the smooth part of the density of states in single-particle confined systems, the result takes the form of a sum over clusters of particles moving freely around manifolds in configuration space invariant under elements of the group of permutations.
Being asymptotic, the approximation gives increasingly better results for large excitation energies and comparison shows that it coincides with the celebrated Bethe estimate in the appropriate region. Moreover, the construction gives the correct high energy asymptotics expected from general considerations, and shows that the emergence of the fermionic ground state is actually a consequence of an extremely delicate large cancellation effect.
Remarkably, the expansion in cluster zones is naturally incorporated for systems of interacting particles, opening the road to address the fundamental problem about the interplay between exchange symmetry and interactions in many-body systems of identical particles.
|Item Type:||Thesis of the University of Regensburg (Diplomarbeit)|
|Date:||30 November 2012|
|Referee:||Prof. Dr. Klaus Richter|
|Date of exam:||31 May 2011|
|Additional information (public):||See also the subsequent preprint publication on the Weyl law for identical particles including physical boundary effects and a detailed formal connection to the Bethe estimate:
|Institutions:|| Physics > Institute of Theroretical Physics > Chair Professor Richter > Group Klaus Richter|
|Research groups and research centres:||Not selected|
|74.20.Fg, 75.10.Jm, 71.10.Li, 73.21.La||PACS|
|Keywords:||semiclassics, many-body physics, weyl expansion, HOdA sum rule, spectral statistics, identical particles|
|Subjects:||500 Science > 530 Physics|
|Refereed:||Yes, this version has been refereed|
|Created at the University of Regensburg:||Yes|
|Deposited On:||30 Nov 2012 09:02|
|Last Modified:||30 Nov 2012 09:02|