Semiclasscal Quantization and Quantum-classical Correspondence of Two-dimensional Oscillators

A. O. Bolivar

Xi Li, SongJJ, XingYZ +1 more

Using the classical quantization way in a two dimensional integrable system and quantized energies periodic orbits correspondence relationship, the long range correlation among the quantum lavels has been discussed in detail. Making use of Berry Tabor trace formula, the quantization conditions of the action of the periodic orbits in two dimensional integrable systems have been obtained. Furthermore, considering the periodicity conditions for the periodic orbits on reasonable torus, the correspondence relations between quantum levels and classical periodic orbits in the two dimensional uncoupled oscillators have been established. It is also shown that there exist the long range correlations among these quantum levels which correspond to the group of the classical periodic orbits with same topology M (M 1,M 2) . To concert state the new points, an example has been given.

Hiroshi Kuratsuji, Shinji Iida

Xin Jun-Li, Shen Jun-Xia

On the basis of quantum-classical correspondence for two-dimensional anisotropic oscillator, we study quantum-classical correspondence for two-dimensional rotation and translation harmonic oscillator system from both quantum-classical orbits and geometric phases. Here, the two one-dimensional oscillators refer to a common harmonic oscillator and a rotation and translation harmonic oscillator. In terms of the generalized gauge transformation, we obtain the stationary Lissajous orbits and Hannay's angle. On the other hand, the eigenfunctions and Berry phases are derived analytically with the help of time-dependent gauge transformation. We may draw the conclusion that the nonadiabatic Berry phase in the original gauge is-n times the classical Hannay's angle, here n is the eigenfunction index. As a matter of fact, the quantum geometric phase and the classical Hannay's angle have the same nature according to Berry. Finally, by using the SU(2) coherent superposition of degenerate two-dimensional eigenfunctions for a fixed energy value, we construct the stationary wave functions and show that the spatial distribution of wave-function probability clouds is in excellent accordance with the classical orbits, indicating the exact quantum-classical correspondence. We also demonstrate the quantum-classical correspondences for the geometric phase-angle and the quantum-classical orbits in a unified form.

Tasko P. Grozdanov, Subhash Saini, Howard S. Taylor

The semiclassical method based on the Einstein-Brillouin-Keller quantization of invariant phase-space tori and the hypothesis of adiabatic invariance of classical action variables is applied in the calculation of highly excited energy levels of two-dimensional coupled oscillator systems. The results obtained are in good agreement with quantum-mechanical calculations even in the regions where the corresponding classical dynamics becomes apparently irregular. General limitations of the method are discussed.

Mustapha, Maamache, H.T. Jeong +2 more