Periodic orbits energy spectrum

This equation shows that only the closed classical trajectories [x(t) = x(0) and x(t) = x(0)] should be taken into account, and the energy spectrum is determined by these periodic orbits [Gutzwiller 1967 Balian and Bloch 1974 Miller 1975a Rajaraman 1975]. 

Strongly open potentials that have no energy minimum but instead a saddle point, a maximum, or possibly no equilibrium point. In such potentials, the dissociation process is direct and very fast so that the lifetimes are very short. If there is no equilibrium point, it is possible that no resonance exists. We may expect sparse spectra that are regular, or irregular depending on the spectrum of interfering periodic orbits. 

The relation (10.4.50) establishes a direct connection between a quantum spectrum characterized by discrete energy levels and the actions of classical periodic orbits. We now show how to actually compute the scaled energy levels Ej. 

However, in the sodium atom, An = 0 is also allowed. Thus the 3s —> 3p transition is allowed, although the 3s —> 4s is forbidden, since in this case A/ = 0 and is forbidden. Taken together, the Bohr model of quantized electron orbitals, the selection rules, and the relationship between wavelength and energy derived from particle-wave duality are sufficient to explain the major features of the emission spectra of all elements. For the heavier elements in the periodic table, the absorption and emission spectra can be extremely complicated - manganese and iron, for example, have about 4600 lines in the visible and UV region of the spectrum. 

In the three-branch horseshoe, the periodic oibit 0 is hyperbolic with reflection and has a Maslov index equal to no = 3 while the off-diagonal orbits 1 and 2 are hyperbolic without reflection with the Maslov index n = 2 [10]. Fitting of numerical actions, stability eigenvalues, and rotation numbers to polynomial functions in E can then be used to reproduce the analytical dependence on E. The resonance spectrum is obtained in terms of the zeros of (4.16) in the complex energy surface. 

Fig. 12. Photoelectron spectra of sodiumthiosulfate NajS203 under different instrumental conditions. The sulphur peak at higher binding energies corresponds to the more positive central sulphur atom. Fig. 12 A is a high sensitivity scan ( A = 100 eV) over a period of 100 s, while Fig. 12 B was run at higher resolution (/TA = 30 eV) and increased observation time (200 s). Note the spin orbit splitting of the sulphur 2p-doubIet appearing in the second spectrum. The noise of the digitally filtered line is demonstrated by the accompanying point plot...

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