In this work we present a simple and elegant approach to study the adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator which is described by the generalized Caldirola–Kanai Hamiltonian, in both classical and quantum contexts. Based on time-dependent dynamical invariants, we find that the geometric phase acquired when the damped oscillator evolves adiabatically in time provides a direct connection between the classical Hannay’s angle and the quantum Berry’s phase. In addition, we solve the time-dependent Schrödinger equation for this system and calculate various quantum properties of the damped generalized harmonic one, such as coherent states, expectation values of the position and momentum operators, their quantum fluctuations and the associated uncertainty product.

中文翻译：

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广义阻尼谐振子的绝热和非绝热演化

在这项工作中，我们提出了一种简单而优雅的方法来研究广义阻尼谐振子的绝热和非绝热演化，该谐振子由广义 Caldirola-Kanai Hamiltonian 在经典和量子环境中描述。基于时间相关的动力学不变量，我们发现当阻尼振荡器随时间绝热演化时获得的几何相位提供了经典汉奈角和量子贝里相位之间的直接联系。此外，我们求解了该系统的时间相关薛定谔方程，并计算了阻尼广义谐波一的各种量子特性，例如相干态、位置和动量算子的期望值、它们的量子涨落和相关的不确定性积。