Geometry and Dynamics of Gaussian Wave Packets and their Wigner Transforms
Geometry and Dynamics of Gaussian Wave Packets and their Wigner Transforms
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高斯波包的几何和动力学及其维格纳变换
DOI：
10.1063/1.4995233
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
C. Tronci
中科院分区：
文献类型：
--
作者：
T. Ohsawa;C. Tronci
我们发现的动力学的高斯波包和相应的高斯维格纳函数的动力学之间的关系，从哈密顿/辛的观点。主要结果表明，辛群在Siegel上半空间上的自然作用所对应的动量映射产生了相应的Gaussian Wigner函数的协方差矩阵。这一事实，结合Kostant的coadjoint轨道覆盖定理，建立了两个动力学之间的辛/泊松几何连接。哈密顿公式自然会引起对波包和维格纳函数动力学中势项的修正，从而导致与传统经典方程略有不同的方程组。我们数值研究的校正项的效果，并证明它提高了作为一个近似的动态观测值的期望值的动态的准确性。
We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant's coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables.