Effective Feynman propagators and Schrödinger equations for processes coupled to many degrees of freedom

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Abstract

AT&T Bell Laboratories. This paper presents a new approach to quantum evolution in the presence of a quantum bath. We develop an equation of motion for an observed system evolving under the influence of an unobserved quantum bath. The methodology we follow uses operator expansions of the Feynman propagator. Corrections to the zeroth order approximation are corrections to an adiabatic approximation. In this paper we explicitly develop an approximation which is infinite order in bath and system coupling, but first order in system degree of freedom. This infinite order approximation is found through a resummation of an infinite class of terms in the operator expansion. We first present a simplified single time (Markovian) version of the theory. We then present a derivation for including the effects of memory. The approach developed in this paper also has the potential for systematic improvement. In other words, while the bath and system coupling in this calculation is treated to infinite order, the system itself is only treated to first order. We will briefly discuss how these higher order corrections can be found. Finally, we present a test calculation of the our approach with comparison to exact results. For a two-dimensional test problem with potential much like that for collinear H + H2 the effective one-dimensional approximation we apply produces essentially exact results.

Original languageEnglish (US)
Pages (from-to)5952-5957
Number of pages6
JournalThe Journal of Chemical Physics
Volume96
Issue number8
StatePublished - 1992
Externally publishedYes

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degrees of freedom
baths
propagation
approximation
Equations of motion
Data storage equipment
operators
expansion
bells
equations of motion
derivation
methodology

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics

Cite this

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title = "Effective Feynman propagators and Schr{\"o}dinger equations for processes coupled to many degrees of freedom",
abstract = "AT&T Bell Laboratories. This paper presents a new approach to quantum evolution in the presence of a quantum bath. We develop an equation of motion for an observed system evolving under the influence of an unobserved quantum bath. The methodology we follow uses operator expansions of the Feynman propagator. Corrections to the zeroth order approximation are corrections to an adiabatic approximation. In this paper we explicitly develop an approximation which is infinite order in bath and system coupling, but first order in system degree of freedom. This infinite order approximation is found through a resummation of an infinite class of terms in the operator expansion. We first present a simplified single time (Markovian) version of the theory. We then present a derivation for including the effects of memory. The approach developed in this paper also has the potential for systematic improvement. In other words, while the bath and system coupling in this calculation is treated to infinite order, the system itself is only treated to first order. We will briefly discuss how these higher order corrections can be found. Finally, we present a test calculation of the our approach with comparison to exact results. For a two-dimensional test problem with potential much like that for collinear H + H2 the effective one-dimensional approximation we apply produces essentially exact results.",
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PY - 1992

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N2 - AT&T Bell Laboratories. This paper presents a new approach to quantum evolution in the presence of a quantum bath. We develop an equation of motion for an observed system evolving under the influence of an unobserved quantum bath. The methodology we follow uses operator expansions of the Feynman propagator. Corrections to the zeroth order approximation are corrections to an adiabatic approximation. In this paper we explicitly develop an approximation which is infinite order in bath and system coupling, but first order in system degree of freedom. This infinite order approximation is found through a resummation of an infinite class of terms in the operator expansion. We first present a simplified single time (Markovian) version of the theory. We then present a derivation for including the effects of memory. The approach developed in this paper also has the potential for systematic improvement. In other words, while the bath and system coupling in this calculation is treated to infinite order, the system itself is only treated to first order. We will briefly discuss how these higher order corrections can be found. Finally, we present a test calculation of the our approach with comparison to exact results. For a two-dimensional test problem with potential much like that for collinear H + H2 the effective one-dimensional approximation we apply produces essentially exact results.

AB - AT&T Bell Laboratories. This paper presents a new approach to quantum evolution in the presence of a quantum bath. We develop an equation of motion for an observed system evolving under the influence of an unobserved quantum bath. The methodology we follow uses operator expansions of the Feynman propagator. Corrections to the zeroth order approximation are corrections to an adiabatic approximation. In this paper we explicitly develop an approximation which is infinite order in bath and system coupling, but first order in system degree of freedom. This infinite order approximation is found through a resummation of an infinite class of terms in the operator expansion. We first present a simplified single time (Markovian) version of the theory. We then present a derivation for including the effects of memory. The approach developed in this paper also has the potential for systematic improvement. In other words, while the bath and system coupling in this calculation is treated to infinite order, the system itself is only treated to first order. We will briefly discuss how these higher order corrections can be found. Finally, we present a test calculation of the our approach with comparison to exact results. For a two-dimensional test problem with potential much like that for collinear H + H2 the effective one-dimensional approximation we apply produces essentially exact results.

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