AI 中文总结
研究\(N\)个经典轨迹有限束在固定辛面积条件下的情况，推导保持条件且功率为零的约束力，设定\(\kappa = \hbar^2/4\)确定面积尺度，高斯大\(N\)极限下质心和宽度方程有特定结果，明确\(N\)的意义。
AI 中文摘要
我们研究了\(N\)个经典轨迹的有限束，其相空间协方差受固定辛面积条件约束。我们推导了能保持此条件且束平均功率为零的约束力。设\(\kappa = \hbar^2/4\)仅确定面积尺度：轨迹仍为经典的，有限束不必是高斯的。在高斯大\(N\)极限下，质心和宽度方程与变分高斯波包动力学的方程一致。这里\(N\)指定有限束表示，而非莫伊尔展开或\(\hbar\)展开中的阶数。
英文摘要
Variational Gaussian wave-packet dynamics is usually obtained by restricting quantum evolution to Gaussian wave packets. We instead construct finite-$N$ dynamics for a labeled bundle of classical trajectories under a constraint fixing the symplectic-area scale of its covariance. At finite $N$, the bundle itself is the dynamical state: it may be non-Gaussian, and the potential force on each trajectory is evaluated from the full potential without a local harmonic approximation. For sequences whose initial empirical distributions approach a smooth Gaussian density as $N \to \infty$, the limiting relative motion becomes linear and preserves Gaussianity. With the area scale set to $\hbar/2$, the limiting centroid and width equations coincide with those of the Gaussian time-dependent variational principle, and the limiting phase-space density coincides with the corresponding Gaussian Wigner density for matched initial data. Here $N$ counts trajectories, not orders in a Moyal or $\hbar$ expansion. This phase-space correspondence reveals two complementary constructions of the same Gaussian dynamics: a variational reduction of quantum evolution and the Gaussian large-$N$ limit of a constrained classical trajectory bundle.
