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The exact factorization of an electron-nuclear wavefunction [A. Abedi, N. T.Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)] allows us todefine the rigorous nuclear time-dependent Schr\"{o}dinger equation (TDSE) witha time-dependent potential-energy surface (TDPES) that fully accounts for thecoupling to the electronic motion and drives the nuclear wavepacket dynamics.Here, we study whether the propagation of multiple classical trajectories canreproduce the quantum nuclear motion in strong-field processes when theirmotions are governed by the quantum Hamilton-Jacobi equation derived byapplying Bohmian mechanics to this exact nuclear TDSE. We demonstrate thatmultiple classical trajectories propagated by the force from the gradient ofthe exact TDPES plus the Bohmian quantum potential can reproduce thestrong-field dissociation dynamics of a one-dimensional model of the H$_2^+$molecule. Our results show that the force from the Bohmian quantum potentialplays a non-negligible role in yielding quantum nuclear dynamics in thestrong-field process studied here, where ionization and/or splitting of nuclearprobability density occurs.

Bohmian mechanics in the exact factorization of electron-nuclear wavefunction