非绝热量子流体力学几何
Geometry of nonadiabatic quantum hydrodynamics
摘要：
李群在辛流形上的哈密顿作用导致了一个动量映射，推广了在对称群情况下发生的诺特守恒量。然后，当一个哈密顿函数可以用这个动量映射来表示时，这个哈密顿函数就叫做“集体的”。在这里，我们推导出集体哈密顿量子分子动力学的李群是光滑可逆映射和幺正变换的组成的一系列模型。在这个过程中，不同的流体描述出现从不同的因式分解方案的波函数或密度算子。在导出这一系列量子流体模型之后，我们通过引入局部空间平滑来正则化有限$\hbar$的哈密顿量。在标准量子流体力学的情况下，拉格朗日路径的$\hbar\ne0$动力学可以推导为一个有限维正则哈密顿系统，用于演化称为“Bohmion”的奇异解，该奇异解遵循配置空间中的Bohmian轨迹。对于分子动力学模型，应用平滑过程的密度算子的一个新的因式分解，导致一个有限维的哈密顿系统的相互作用的多个（核）波密子和一个序列的电子量子态。
英文摘要：
The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether's conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called `collective'. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite $\hbar$ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the $\hbar\ne0$ dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called `Bohmions', which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.
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