谐振子系统的量子-经典轨道、Berry相及Hannay角

Correspondences between quantum and classical orbits Berry phases and Hannay angles for harmonic oscillator system

从量子-经典轨道和几何相两方面,研究了二维旋转平移谐振子系统的量子-经典对应.通过广义规范变换得到了Lissajous经典周期轨道和Hannay角.另外,使用含时规范变换解析推导了旋转平移谐振子系统Schr?dinger方程的本征波函数和Berry相,得出结论:原规范中的非绝热Berry相是经典Hannay角的-n倍.最后,使用SU(2)自旋相干态叠加,构造一稳态波函数,其波函数的概率云很好地局域于经典轨道上,满足几何相位和经典轨道同时对应.

On the basis of quantum-classical correspondence for two-dimensional anisotropic oscillator, we study quantumclassical correspondence for two-dimensional rotation and translation harmonic oscillator system from both quantumclassical 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.