Bohmian mechanics with complex action: A new trajectorybased formulation of quantum mechanics
Abstract
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 disappearedit 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{sup 7} calculated from strictly localized quantum trajectories that do not communicatemore »
 Authors:
 Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, 76100 (Israel)
 Publication Date:
 OSTI Identifier:
 20864359
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Chemical Physics; Journal Volume: 125; Journal Issue: 23; Other Information: DOI: 10.1063/1.2400851; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; EQUATIONS; GAUSS FUNCTION; QUANTUM MECHANICS; SIMULATION; TRAJECTORIES; TUNNEL EFFECT; WAVE PACKETS
Citation Formats
Goldfarb, Yair, Degani, Ilan, and Tannor, David J. Bohmian mechanics with complex action: A new trajectorybased formulation of quantum mechanics. United States: N. p., 2006.
Web. doi:10.1063/1.2400851.
Goldfarb, Yair, Degani, Ilan, & Tannor, David J. Bohmian mechanics with complex action: A new trajectorybased formulation of quantum mechanics. United States. doi:10.1063/1.2400851.
Goldfarb, Yair, Degani, Ilan, and Tannor, David J. Thu .
"Bohmian mechanics with complex action: A new trajectorybased formulation of quantum mechanics". United States.
doi:10.1063/1.2400851.
@article{osti_20864359,
title = {Bohmian mechanics with complex action: A new trajectorybased formulation of quantum mechanics},
author = {Goldfarb, Yair and Degani, Ilan and Tannor, David J.},
abstractNote = {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 disappearedit 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{sup 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 nonlocality to measures of quantum complexity.},
doi = {10.1063/1.2400851},
journal = {Journal of Chemical Physics},
number = 23,
volume = 125,
place = {United States},
year = {Thu Dec 21 00:00:00 EST 2006},
month = {Thu Dec 21 00:00:00 EST 2006}
}

In their comment, Sanz and MiretArtes (SMA) describe previous trajectorybased formalisms based on the quantum HamiltonJacobi (QHJ) formalism. In this reply, we highlight our unique contributions: the identification of the smallness of the quantum force in the complex QHJ and its solution using complex trajectories. SMA also raise the question of how the term locality should be used in quantum mechanics. We suggest that at least certain aspects of nonlocality can depend on the method used to solve the problem.

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