Biomedical subjects

David J Tannor

Publications and source records attributed to David J Tannor.

3 recordsLinked to original sources

PubMed2006

Bohmian mechanics with complex action: a new trajectory-based formulation of quantum mechanics.

In recent years there has been a resurgence of interest in Bohmian mechanics as a numerical tool because of its local dynamics, which suggest the possibility of significant computational advantages for the simulation of large quantum systems. However, closer inspection of the Bohmian formulation reveals that the nonlocality of quantum mechanics has not disappeared-it has simply been swept under the rug into the quantum force. In this paper we present a new formulation of Bohmian mechanics in which the quantum action, S, is taken to be complex. This leads to a single equation for complex S, and ultimately complex x and p but there is a reward for this complexification-a significantly higher degree of localization. The quantum force in the new approach vanishes for Gaussian wave packet dynamics, and its effect on barrier tunneling processes is orders of magnitude lower than that of the classical force. In fact, the current method is shown to be a rigorous extension of generalized Gaussian wave packet dynamics to give exact quantum mechanics. We demonstrate tunneling probabilities that are in virtually perfect agreement with the exact quantum mechanics down to 10(-7) calculated from strictly localized quantum trajectories that do not communicate with their neighbors. The new formulation may have significant implications for fundamental quantum mechanics, ranging from the interpretation of non-locality to measures of quantum complexity.

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PubMed2006

Calculating multidimensional discrete variable representations from cubature formulas.

Finding multidimensional nondirect product discrete variable representations (DVRs) of Hamiltonian operators is one of the long standing challenges in computational quantum mechanics. The concept of a "DVR set" was introduced as a general framework for treating this problem by R. G. Littlejohn, M. Cargo, T. Carrington, Jr., K. A. Mitchell, and B. Poirier (J. Chem. Phys. 2002, 116, 8691). We present a general solution of the problem of calculating multidimensional DVR sets whose points are those of a known cubature formula. As an illustration, we calculate several new nondirect product cubature DVRs on the plane and on the sphere with up to 110 points. We also discuss simple and potentially very useful finite basis representations (FBRs), based on general (nonproduct) cubatures. Connections are drawn to a novel view on cubature presented by I. Degani, J. Schiff, and D. J. Tannor (Num. Math. 2005, 101, 479), in which commuting extensions of coordinate matrices play a central role. Our construction of DVR sets answers a problem left unresolved in the latter paper, namely, the problem of interpreting as function spaces the vector spaces on which commuting extensions act.

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PubMed2005

The non-Markovian quantum master equation in the collective-mode representation: application to barrier crossing in the intermediate friction regime.

The calculation of chemical reaction rates in the condensed phase is a central preoccupation of theoretical chemistry. At low temperatures, quantum-mechanical effects can be significant and even dominant; yet quantum calculations of rate constants are extremely challenging, requiring theories and methods capable of describing quantum evolution in the presence of dissipation. In this paper we present a new approach based on the use of a non-Markovian quantum master equation (NM-QME). As opposed to other approximate quantum methods, the quantum dynamics of the system coordinate is treated exactly; hence there is no loss of accuracy at low temperatures. However, because of the perturbative nature of the NM-QME it breaks down for dimensionless frictions larger than about 0.1. We show that by augmenting the system coordinate with a collective mode of the bath, the regime of validity of the non-Markovian master equation can be extended significantly, up to dimensionless frictions of 0.5 over the entire temperature range. In the energy representation, the scaling goes as the number of levels in the relevant energy range to the third power. This scaling is not prohibitive even for chemical systems with many levels; hence we believe that the current method will find a useful place alongside the existing techniques for calculating quantum condensed-phase rate constants.

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