Bohmian dynamics

An analysis related to the classical limit of the Bohmian dynamics [56] has been proposed in Ref. [38]. It considers the time variation of the local probability current in each channel defined as... 

This suggests that to determine the path solutions in Eq. (4.88), j4(R, t) should be known beforehand. This is a nonlinear process. However, the Bohmian dynamics is known to be a very important reformulation of the Schrodinger equation with respect to the interpretation of quantum mechanics. Much study has been devoted to practical methods to numerically realize the Bohmian dynamics in the context of chemical dynamics (see [48f] and references therein). 

The Bohmian formalism has been recently generalized to cases of multistate systems that couple nonadiabatically [100]. Readers are suggested to consult the original papers for the explicit form of the coupled equations of motion. Since the Bohmian dynamics is very difficult to solve even for a single potential energy surface, its extension to nonadiabatic dynamics requires a number of specific approximations and assumptions. 

Sophya Garaschuk, Vitaly Rassolov, Oleg Prezhdo, Semiclassical Bohmian Dynamics. 

In the case of a many-body problem, the Bohmian mechanics for an A-body dynamics follows from the one for a single system, but replacing Equation 8.25 by... 

These new trajectories are the so-called reduced quantum trajectories [30], which are only explicitly related to the system reduced density matrix. The dynamics described by Equation 8.42 leads to the correct intensity (time evolution of which is described by Equation 8.40) when the statistics of a large number of particles are considered. Moreover, Equation 8.42 reduces to the well-known expression for the velocity held in Bohmian mechanics, when there is no interaction with the environment. 

The question then arises if a convenient mixed quantum-classical description exists, which allows to treat as quantum objects only the (small number of) degrees of freedom whose dynamics cannot be described by classical equations of motion. Apart in the limit of adiabatic dynamics, the question is open and a coherent derivation of a consistent mixed quantum-classical dynamics is still lacking. All the methods proposed so far to derive a quantum-classical dynamics, such as the linearized path integral approach [2,6,7], the coupled Bohmian phase space variables dynamics [3,4,9] or the quantum-classical Li-ouville representation [11,17—19], are based on approximations and typically fail to satisfy some properties that are expected to hold for a consistent mechanics [5,19]. 

Burghardt, I. Dynamics of coupled Bohmian and phase-space variables a moment approach to mixed quantum-classical dynamics. J. Chem. Phys. 122 94103 (2005). 

Hybrid Quantum/Classical Dynamics Using Bohmian Trajectories... 

Hybrid Quantum/Classical Dynamics Using Bohmian Trajectories Equation (22) is then equivalent to... 

Figure 1. The time dependent scattering probability P, Eq. (40) for the model problem detailed in the text obtained for the incident energy of 10 kJ/mol using exact quantum dynamics (Quantum), mean-field (MF), quantized mean-field (QMF) the Bohmian quantum-classical technique (Bohmian), and the generalized surface hopping technique of Ref. [13] (SH).

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