Interference in Bohmian Mechanics with Complex Action

In recent years, intensive effort has gone into developing numerical tools for exact quantum mechanical calculations that are based on Bohmian mechanics. As part of this effort we have recently developed as alternative formulation of Bohmian mechanics in which the quantum action, S, is taken to be complex [JCP {125}, 231103 (2006)]. In the alternative formulation there is a significant reduction in the magnitude of the quantum force as compared with the conventional Bohmian formulation, at the price of propagating complex trajectories. In this paper we show that Bohmian mechanics with complex action is able to overcome the main computational limitation of conventional Bohmian methods -- the propagation of wavefunctions once nodes set in. In the vicinity of nodes, the quantum force in conventional Bohmian formulations exhibits rapid oscillations that pose severe difficulties for existing numerical schemes. We show that within complex Bohmian mechanics, multiple complex initial conditions can lead to the same real final position, allowing for the description of nodes as a sum of the contribution from two or more crossing trajectories. The idea is illustrated on the reflection amplitude from a one-dimensional Eckart barrier. We believe that trajectory crossing, although in contradiction to the conventional Bohmian trajectory interpretation, provides an important new tool for dealing with the nodal problem in Bohmian methods

Phase Space Approach to Solving the Time-independent Schr\"odinger Equation

We propose a method for solving the time independent Schr\"odinger equation based on the von Neumann (vN) lattice of phase space Gaussians. By incorporating periodic boundary conditions into the vN lattice [F. Dimler et al., New J. Phys. 11, 105052 (2009)] we solve a longstanding problem of convergence of the vN method. This opens the door to tailoring quantum calculations to the underlying classical phase space structure while retaining the accuracy of the Fourier grid basis. The method has the potential to provide enormous numerical savings as the dimensionality increases. In the classical limit the method reaches the remarkable efficiency of 1 basis function per 1 eigenstate. We illustrate the method for a challenging two-dimensional potential where the FGH method breaks down.Comment: 5 figures. Includes supplementary material. arXiv admin note: substantial text overlap with arXiv:1010.258

Mixed Quantum/Classical Theory of Rotationally and Vibrationally Inelastic Scattering in Space-fixed and Body-fixed Reference Frames

We formulated the mixed quantum/classical theory for rotationally and vibrationally inelastic scattering process in the diatomic molecule + atom system. Two versions of theory are presented, first in the space-fixed and second in the body-fixed reference frame. First version is easy to derive and the resultant equations of motion are transparent, but the state-to-state transition matrix is complex-valued and dense. Such calculations may be computationally demanding for heavier molecules and/or higher temperatures, when the number of accessible channels becomes large. In contrast, the second version of theory requires some tedious derivations and the final equations of motion are rather complicated (not particularly intuitive). However, the state-to-state transitions are driven by real-valued sparse matrixes of much smaller size. Thus, this formulation is the method of choice from the computational point of view, while the space-fixed formulation can serve as a test of the body-fixed equations of motion, and the code. Rigorous numerical tests were carried out for a model system to ensure that all equations, matrixes, and computer codes in both formulations are correct

Non-adiabatic dynamics of molecules in optical cavities

Strong coupling of molecules to the vacuum field of micro cavities can modify the potential energy surfaces opening new photophysical and photochemical reaction pathways. While the influence of laser fields is usually described in terms of classical field, coupling to the vacuum state of a cavity has to be described in terms of dressed photon-matter states (polaritons) which require quantized fields. We present a derivation of the non-adiabatic couplings for single molecules in the strong coupling regime suitable for the calculation of the dressed state dynamics. The formalism allows to use quantities readily accessible from quantum chemistry codes like the adiabatic potential energy surfaces and dipole moments to carry out wave packet simulations in the dressed basis. The implications for photochemistry are demonstrated for a set of model systems representing typical situations found in molecules

A mapping approach to synchronization in the "Zajfman trap". II: the observed bunch

We extend a recently introduced mapping model, which explains the bunching phenomenon in an ion beam resonator for two ions [Geyer, Tannor, J. Phys. B 37 (2004) 73], to describe the dynamics of the whole ion bunch. We calculate the time delay of the ions from a model of the bunch geometry and find that the bunch takes on a spherical form at the turning points in the electrostatic mirrors. From this condition we derive how the observed bunch length depends on the experimental parameters. We give an interpretation of the criteria for the existence of the bunch, which were derived from the experimental observations by Pedersen et al [Pedersen etal, Phys. Rev. A 65 042704].Comment: 25 pages, 6 figures; added new section 5 and clarified text; submitted to J. Phys.

Cumulative Reaction Probability in Terms of Reactant-Product Wave Packet Correlation Functions

We present new expressions for the cumulative reaction probability (N(E)), cast in terms of time-correlation functions of reactant and product wave packets. The derivation begins with a standard trace expression for the cumulative reaction probability, expressed in terms of the reactive scattering matrix elements in an asymptotic internal basis. By combining the property of invariance of the trace with a wave packet correlation function formulation of reactive scattering, we obtain an expression for N(E) in terms of the correlation matrices of incoming and outgoing wave packets which are arbitrary in the internal coordinates. This formulation, like other recent formulations of N(E), allows calculation of the quantum dynamics just in the interaction region of the potential, and removes the need for knowledge of the asymptotic eigenstates. However, unlike earlier formulations, the present formulation is fully compatible with both exact and approximate methods of wave packet propagation. We illustrate this by calculating N(E) for the collinear hydrogen exchange reaction, both quantally and semiclassically. These results indicate that the use of wave packet cross-correlation functions, as opposed to a coordinate basis and flux operators, regularizes the semiclassical calculation, suggesting that the semiclassical implementation described here may be applied fruitfully to systems with more degrees of freedom