Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Another definition for time delay

Heide Narnhofer

Published Nov 15, 1980• 31 citations

Abstract

Time delay is defined by geometrical considerations which work in classical as well as in quantum mechanics, and its connection with the $S$ matrix and the virial is proven for potentials with $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}V(x)$ vanishing as ${r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\epsilon}}$ for $r\ensuremath{\rightarrow}\ensuremath{\infty}$.

DOI: 10.1103/physrevd.22.2387

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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Another definition for time delay

Heide Narnhofer

Published Nov 15, 1980• 31 citations

Publisher

Abstract

Time delay is defined by geometrical considerations which work in classical as well as in quantum mechanics, and its connection with the

S

matrix and the virial is proven for potentials with

V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})

and

\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}V(x)

vanishing as

{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\epsilon}}

for

r\ensuremath{\rightarrow}\ensuremath{\infty}

DOI: 10.1103/physrevd.22.2387

Related papers

On the Hilbert-space approach to classical time delay

D. Bollé, J. D’Hondt

Journal of Physics A Mathematical and GeneralJul 1, 19817 citations

Concerning the Notion of "Time Interval" in S -Matrix Theory

M. L. Goldberger, Kenneth Watson

The notion of time duration is considered within the framework of $S$ -matrix theory. It is shown that Wigner's definition of the time delay, $\ensuremath{-}i\ensuremath{\hbar}[(\frac{d\mathrm{ln}S}{\mathrm{dE}})]$, for a scattering process permits one to define time duration in a "coarse-grained" sense.The notion of time duration is considered within the framework of $S$-matrix theory. It is shown that Wigner's definition of the time delay, $\ensuremath{-}i\ensuremath{\hbar}[(\frac{d\mathrm{ln}S}{\mathrm{dE}})]$, for a scattering process permits one to define time duration in a "coarse-grained" sense.

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Generalized definition of time delay in scattering theory

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Time delay for one-dimensional quantum systems with steplike potentials

Werner O. Amrein, Ph. Jacquet

This paper concerns time-dependent scattering theory and in particular the concept of time delay for a class of one-dimensional anisotropic quantum systems. These systems are described by a Schr\"odinger Hamiltonian $H=\ensuremath{-}\ensuremath{\Delta}+V$ with a potential $V(x)$ converging to different limits ${V}_{\ensuremath{\ell}}$ and ${V}_{r}$ as $x\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ and $+\ensuremath{\infty}$, respectively. Due to the anisotropy, they exhibit a two-channel structure. We first establish the existence and properties of the channel wave and scattering operators by using the modern Mourre approach. We then use scattering theory to show the identity of two apparently different representations of time delay. The first one is defined in terms of sojourn times while the second one is given by the Eisenbud-Wigner operator. The identity of these representations is well known for systems where $V(x)$ vanishes as $\ensuremath{\mid}x\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ $({V}_{\ensuremath{\ell}}={V}_{r})$. We show that it remains true in the anisotropic case ${V}_{\ensuremath{\ell}}\ensuremath{\ne}{V}_{r}$, i.e., we prove the existence of the time-dependent representation of time delay and its equality with the time-independent Eisenbud-Wigner representation. Finally, we use this identity to give a time-dependent interpretation of the Eisenbud-Wigner expression, which is commonly used for time delay in the literature.This paper concerns time-dependent scattering theory and in particular the concept of time delay for a class of one-dimensional anisotropic quantum systems. These systems are described by a Schr\"odinger Hamiltonian $H=\ensuremath{-}\ensuremath{\Delta}+V$ with a potential $V(x)$ converging to different limits ${V}_{\ensuremath{\ell}}$ and ${V}_{r}$ as $x\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ and $+\ensuremath{\infty}$, respectively. Due to the anisotropy, they exhibit a two-channel structure. We first establish the existence and properties of the channel wave and scattering operators by using the modern Mourre approach. We then use scattering theory to show the identity of two apparently different representations of time delay. The first one is defined in terms of sojourn times while the second one is given by the Eisenbud-Wigner operator. The identity of these representations is well known for systems where $V(x)$ vanishes as $\ensuremath{\mid}x\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ $({V}_{\ensuremath{\ell}}={V}_{r})$. We show that it remains true in the anisotropic case ${V}_{\ensuremath{\ell}}\ensuremath{\ne}{V}_{r}$, i.e., we prove the existence of the time-dependent representation of time delay and its equality with the time-independent Eisenbud-Wigner representation. Finally, we use this identity to give a time-dependent interpretation of the Eisenbud-Wigner expression, which is commonly used for time delay in the literature.

arXiv:quant-ph/0610198Physical Review AFeb 21, 200727 citations

Coulomb phase shift

Charles W. Clark

The relationship between the energy derivative of the Coulomb phase shift and the classical expression for the time delay is discussed.

American Journal of PhysicsAug 1, 19799 citations

Time Delay in Quantum Scattering

H. M. Nussenzveig

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsSep 15, 197242 citations

Time delay is defined by geometrical considerations which work in classical as well as in quantum mechanics, and its connection with the

S

matrix and the virial is proven for potentials with

V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}})

and

\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nabla}}V(x)

vanishing as

{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\epsilon}}

for

r\ensuremath{\rightarrow}\ensuremath{\infty}

The notion of time duration is considered within the framework of

S

-matrix theory. It is shown that Wigner's definition of the time delay, $\ensuremath{-}i\ensuremath{\hbar}[(\frac{d\mathrm{ln}S}{\mathrm{dE}})]$, for a scattering process permits one to define time duration in a "coarse-grained" sense.

V(x)

converging to different limits ${V}_{\ensuremath{\ell}}$ and

{V}_{r}

as $x\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ and $+\ensuremath{\infty}$, respectively. Due to the anisotropy, they exhibit a two-channel structure. We first establish the existence and properties of the channel wave and scattering operators by using the modern Mourre approach. We then use scattering theory to show the identity of two apparently different representations of time delay. The first one is defined in terms of sojourn times while the second one is given by the Eisenbud-Wigner operator. The identity of these representations is well known for systems where

V(x)

vanishes as $\ensuremath{\mid}x\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ $({V}_{\ensuremath{\ell}}={V}_{r})$. We show that it remains true in the anisotropic case ${V}_{\ensuremath{\ell}}\ensuremath{\ne}{V}_{r}$, i.e., we prove the existence of the time-dependent representation of time delay and its equality with the time-independent Eisenbud-Wigner representation. Finally, we use this identity to give a time-dependent interpretation of the Eisenbud-Wigner expression, which is commonly used for time delay in the literature.