### 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 + H_{2} the effective one-dimensional approximation we apply produces essentially exact results.

Original language | English (US) |
---|---|

Pages (from-to) | 5952-5957 |

Number of pages | 6 |

Journal | The Journal of Chemical Physics |

Volume | 96 |

Issue number | 8 |

State | Published - 1992 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

**Effective Feynman propagators and Schrödinger equations for processes coupled to many degrees of freedom.** / Schwartz, Steven D.

Research output: Contribution to journal › Article

*The Journal of Chemical Physics*, vol. 96, no. 8, pp. 5952-5957.

}

TY - JOUR

T1 - Effective Feynman propagators and Schrödinger equations for processes coupled to many degrees of freedom

AU - Schwartz, Steven D

PY - 1992

Y1 - 1992

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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UR - http://www.scopus.com/inward/citedby.url?scp=22244487707&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:22244487707

VL - 96

SP - 5952

EP - 5957

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 8

ER -