Symmetry and localization

Recently, interesting symmetry effects have been discovered in connection with the quantum kicked rotor (see, e.g., Dittrich and Smilansky (1991a,b), Bliimel and Smilansky (1992), Thaha et al. (1993), Thaha and Bliimel (1994)). The issue concerns the influence of symmetries and their destruction on the localization length of the quantum kicked rotor. Can these symmetry effects be observed in atomic and molecular physics We think that this issue is important and propose the search for the influence of symmetry on the localization length of dynamically localizing systems as an interesting and important topic for future research. Therefore, the purpose of this section is to sketch briefly the essence of these recent discoveries as an incentive for their application in atomic and molecular physics. 

The Hamiltonian (5.3.2) of the quantum kicked rotor is time reversal invariant. The question is, what happens to the localization length of the kicked rotor if a time reversal violating interaction is switched on The difficulty here is to add time reversal violating terms to (5.3.2) in such a way that the classical properties of the Hamiltonian are not affected. This is important, since otherwise a change in the localization length is not surprising since it can always, at least partially, be blamed on the 

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Pichard, J.-L., Sanquer, M., Slevin, K. and Debray, P. (1990). Broken symmetries and localization lengths in Anderson insulators theory and experiment, Phys. Rev. Lett. 65, 1812-1815. 

Nevertheless, by choosing an alternative approach to the proof of a more general problem, based on the natural convergence properties of electron densities, this difficulty can be circumvented [50], leading to the holographic electron density theorem on quantum mechanically correct, boundaryless molecular electron densities [50d. This also implies some fundamental relations between local and global symmetries and local and global chirality properties of electron densities [50b of molecules. 

The XAS technique measures transitions between core levels of the atom of interest and its excited or continuum states. These levels and transitions reflect such fundamental properties of atoms that they are always present. Hence, it is always possible to observe the X-ray absorption spectrum of an atom in a system irrespective of the state (i.e., gaseous, liquid, ordered or disordered solid) for all atoms across the periodic table. From the element-specific X-ray absorption spectrum it is possible to determine information on oxidation state, local symmetry, and local metrical structure at resolution greater than that attainable from X-ray crystal... 

Fig. 9.19. Long thin molecules with strongly polar heads can form a phase which simultaneously has columnar symmetry and local smectic order. Molecules arrange themselves in smectic bilayers and polar heads jump periodically from one layer to the next. A 2-dimensional network is thereby formed, whilst the order in the third dimension remains liquid...

For the representation of quadratic force fields, spectro-scopists have long been using symmetry and local internal coordinates. In quantum chemistry the use of these coordinates, called natural internal coordinates in this context, received widespread acceptance after automatic generation of these coordinates for most molecular systems had... 

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