高斯波包在半经典 regime 下的到达时间
Time of arrival in the semiclassical regime for Gaussian wave packets
- University of Massachusetts, Boston（马萨诸塞大学波士顿分校）
- University of the West of Scotland（西苏格兰大学）
- Somerville High School（萨默维尔高中）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究针对半经典 regime 下高斯波包的到达时间分布，推导出平均到达时间及标准差的闭式表达式，并应用于自由落体和陷阱模型，通过数值验证，扩展了早期预测。
AI中文摘要：
我们研究了半经典 regime 下高斯波包在一般一维二次哈密顿量（即具有时间相关频率的受迫谐振子）中的到达时间（TOA）分布。通过将高斯概率密度展开为狄拉克 delta 函数导数的级数，并利用分布与经典轨迹复合的标准规则，我们获得了领先阶半经典平均到达时间及其标准差紧凑的闭式表达式。平均值获得一个由经典速度和加速度以及 Ermakov 宽度的时间导数决定的 $\sigma^2$ 量级的量子偏移，而展宽简化为简单形式 $\Delta\mathcal{T}_x\simeq\sigma(t_x)/v_x$。后者蕴含一个仅依赖于经典运动方程基本解的时间-位置不确定关系。我们将该框架应用于自由落体和两种重力下随时间变化的开口陷阱模型，并通过 TOA 分布的精确数值积分验证了每个表达式。这些结果将早期的自由落体预测扩展到无法忽略陷阱势的现实场景。
英文摘要：
We study the time-of-arrival (TOA) distribution of a Gaussian wave packet in the semiclassical regime for a general one-dimensional quadratic Hamiltonian, namely a forced harmonic oscillator with time-dependent frequency. Expanding the Gaussian probability density as a series in derivatives of the Dirac delta and using the standard rules for the composition of distributions with the classical trajectory, we obtain compact closed-form expressions for the leading semiclassical mean arrival time and its standard deviation. The mean acquires a quantum shift of order $σ^2$ set by the classical velocity and acceleration and by the time derivative of the Ermakov width, while the spread reduces to the simple form $Δ\mathcal{T}_x\simeqσ(t_x)/v_x$. The latter implies a time--position uncertainty relation that depends only on the fundamental solutions of the classical equation of motion. We apply the framework to free fall and to two models of a time-dependent opening trap under gravity, and we validate every expression against exact numerical integration of the TOA distribution. These results extend earlier free-fall predictions to realistic settings in which the trapping potential cannot be neglected.
