Siegel der Universität

Universität zu Köln
Mathematisch-Naturwissenschaftliche Fakultät
Fachgruppe Physik

I. Physikalisches Institut


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Semi Classical Aproach


Useful descriptions of the intermediate regime between the well-understood quantum mechanical "world'' and the classical limit include the use of semi-classical calculations, which we use to determine the rotational energy spectra of different molecules at high J-quantum numbers. Basis of the semiclassical approach is the rotational energy surface (RES, cf. Fig. 1) which is found by writing the quantum mechanical Hamiltonian in terms of a "classical vector" (Jx, Jy,Jz)T using two angles and the fixed length |J|.

To get to quantization conditions, one can use the so-called WKB- or Sommerfeld quantization rules, which were first applied to standard problems like the quantum harmonic oscillator. Here we can use them analogously to find conditions for the quantization of the energy levels.

An alternative approach makes use of results of quantum chaos theory. This apporach is completely coordinate free and involves only geometrical and topological features of the classical dynamics, which makes it more useful than the heuristic picture of the earlier works. In that approach one uses the Gutzwiller trace formula for the description of the density of states, where the poles signals the energy levels. Going one step further, we can use the symmetry projected Green function in the derivation of the density of states and hence find different quantization conditions for the various representations of the molecular symmetry group.



Figure 1: The rotational energy surface (RES) for rigid H2O for rotational quantum number J=20. This is a typical example of a RES for a rigid asymetric top molecule. Notice that this is a surface of constant J. To find the energy, one has to find the intersection line with a sphere of constant energy.


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