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Lecture notes in physics

Adiabatic invariants, resonances, and multidimensional semiclassical quantization

Frank R. Johnston, Philip Pechukas

Published Jan 1, 1987• 3 citations
Publisher
DOI
10.1007/3-540-17894-5_338

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Journal of Physics A Mathematical and General•Sep 21, 1993•1 citation
The formulation of a general strategy for implementing semiclassical quantization using the adiabatic-switching method, presented in the preceding paper for two-dimensional systems, is generalized to multidimensional systems. The method is shown to produce accurate semiclassical eigenvalues for the realistic polynomial potential describing SO2{\mathrm{SO}}_{2}SO2​ as well as for model potentials for other multidimensional systems. The results are compared with semiclassical results obtained with use of other methods.
Invariance of the actions which are the classical counterparts of the quantum numbers is very often postulated in adiabatic switching calculations but never checked carefully. In the particular case of a free particle moving in a prolate cavity deformed adiabatically at constant volume, time evolution of the classical actions is presented. Starting from semiclassical energies in a spherical cavity, the classical actions are calculated for each trajectory at each step of the switching procedure. This can be performed for any deformation since the system is separable in suitable coordinates. Invariance of the actions is not well verified for few values of the deformation. If it is easily understood why the actions invariance fails when the semiclassical solution cross the separatrix for the L z =0 levels, such failures are in principle not expected for the L z not=0 levels for which the trajectories belong to only one topology. The intrinsic frequencies of the motion are thus calculated for each trajectory at each step of the switching. Then, it is shown that failures in the action invariance occur exactly for values of the deformation for which two intrinsic frequencies become commensurate. For these particular values, the semiclassical trajectories are resonant for the L z not=0 levels and periodic for the L z =0 levels. By an appropriate canonical transformation on the action variables, a better invariant than the actions themselves has been built in the neighbourhood of each resonance. Apart from the particular values of the deformation where the radial and the angular frequencies become commensurate, action invariance is verified with a very good accuracy during the adiabatic switching.