Quantum Molecular Trajectory and Its Statistical Properties

Abstract

Despite the quantum nature of molecules, classical mechanics is often employed to describe molecular motions that play a fundamental role in a wide range of phenomena including chemical reactions. This is due to the need of assigning well-defined positions to the atomic nuclei during the time evolution of the system in order to describe unambiguously the molecular motions, whereas quantum mechanics provides information on probabilistic nature only. One would like to employ a quantum molecular trajectory that defines rigorously the instantaneous nuclear positions and, simultaneously, guarantees the conservation of all quantum mechanics predictions unlike the classical trajectory. We argue that such a quantum molecular trajectory can be formally defined and we prove that it corresponds to a single Bohm trajectory. Our analysis establishes a clear correspondence between the statistical properties of the trajectory and the quantum expectation values. The obvious and undeniable benefit is that of dealing with a quantum methodology fully characterizing the molecular motions without any reference to classical mechanics.

Quantum Stochastic Trajectories: The Fokker–Planck–Bohm Equation Driven by the Reduced Density Matrix

“…We introduce also the invariant manifold Ω P for the ensemble of possible dynamical states X(t), with reference to a specific set P of populations. 7 A generic point of Ω P will be denoted as x=(q,α). The invariant manifold Ω P represents the natural support for the distribution on the dynamical states, more specifically the time dependent probability density ρ Ω P (x,t) to be normalized by integration over Ω P .…”

“…The invariant manifold Ω P represents the natural support for the distribution on the dynamical states, more specifically the time dependent probability density ρ Ω P (x,t) to be normalized by integration over Ω P . As a consequence of the deterministic dynamics of X(t), such a distribution follows the Liouville equation 7,42…”

“…x q P j N j j ( , ) 1 j (7) so that the pure state wave function at a given time t is recovered by inserting the corresponding values of the phases Ψ(q,t) = ψ P (q,α)| α=A(t) . In eq 4, as well as in the next equations, the dot symbol is employed to denote the scalar product between the components of the generic gradient operator ∇ and of the generic velocity field Λ(•).…”

“…Recently we have proposed what has been called the quantum molecular trajectory to determine the time dependent particle positions without contradicting at the same time the predictions of quantum mechanics. 7 The later are recovered in terms of statistical properties of the quantum molecular trajectory. For example, the expectation values are formally related to the time averages of the corresponding physical observables along the trajectory.…”

“…By employing projection operator techniques, like Mori−Zwanzig procedures in classical mechanics, 34−37 we derive the stochastic equations for the set of relevant degrees of freedom from the Liouville representation of the Schrodinger−Bohm dynamics. 7 By considering the open system coordinates as the only relevant variables, in ref 38, we have derived a Smoluchowski type equation, the socalled Smoluchowski−Bohm equation, for the distribution of the Bohm coordinates of the open system. In this framework the dynamics of the wave function does not appear explicitly besides its role as a source of the noise.…”