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Time-delay of classical and quantum scattering processes: a conceptual overview and a general definition

Massimiliano Sassoli de Bianchi

Published Jan 1, 2012• 12 citations
OpenAlex

Abstract

We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of "fuzzy" and "free-flight" sojourn times, that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they shouldn't be called "times".

Related papers

Time-delay of classical and quantum scattering processes: a conceptual overview and a general definition

Massimiliano Sassoli de Bianchi

Abstract We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of “fuzzy” and “free-flight” sojourn times that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they should not be called “times.”

arXiv:1010.5329•Open Physics•Dec 12, 2011•12 citations

Time-delay of classical and quantum scattering processes: a conceptual overview and a general

Massimiliano Sassoli de Bianchi

We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of “fuzzy” and “free-flight” sojourn times that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they should not be called “times.”

•Jan 1, 2012•0 citations

Time delay for an abstract quantum scattering process

Serge Richard

In this short review paper, we discuss the concept of time delay for an abstract quantum scattering system. Its definition in terms of sojourn times is explained as well as its identity with the so-called Eisenbud-Wigner time delay. Necessary and natural conditions for such a construction are introduced and thoroughly discussed. Assumptions and statements are precisely formulated but proofs are contained in two companion papers written in collaboration with R. Tiedra de Aldecoa.

arXiv:1103.3901•Mar 20, 2011•0 citations

Time delay in tunneling: Transmission and reflection time delays

Wojciech Jaworski, David M. Wardlaw

The problem of time delay in one-dimensional quantum-mechanical scattering by a potential barrier is studied in the framework of time-dependent scattering theory. It is shown how the concept of sojourn time can be used to define three separate time delays: the time delay for transmission through the barrier, the time delay for reflection, and the total time delay (i.e., averaged over transmission and reflection). In addition to the anticipated dependence on the S matrix and its energy derivative, the time delays depend on certain parameters which can be interpreted as describing positions of detectors in a gedanken-experiment measurement. The problem of defining separate time delays for transmission and reflection appears to be rooted in the foundations of quantum theory and its interpretation.

Physical Review A•Apr 1, 1988•69 citations

Time delay in tunneling: Sojourn-time approach versus mean-position approach

Wojciech Jaworski, David M. Wardlaw

Our previous results [Phys. Rev. A 37, 2843 (1988)], based on the concept of sojourn time, for the transmission and reflection time delays in one-dimensional scattering by a potential barrier are compared to the corresponding results of E. H. Hauge, J. P. Falck, and T. A. Fjedly [Phys. Rev. B 36, 4203 (1987)], based on the idea of following the time evolution of the mean position of wave packets. It is shown that the mean-position approach agrees with the more general sojourn-time approach in the limit of well-defined momentum, and that, in general, the two approaches appear to be related in a classical-like manner.

Physical Review A•Nov 1, 1988•18 citations

The concept of a time-of-sojourn operator and spreading of wave packets

Wojciech Jaworski

The concept of sojourn time and a sojourn time operator, aimed at describing the length of time spent by a quantum mechanical system in a given subspace of states, is investigated. A general rigorous definition to the sojourn time operator is given and some of its properties are studied. In particular, it is shown that the usual Born’s probability interpretation of the associated spectral measure yields strange, if not paradoxical, results, resembling the well-known quantum mechanical Zeno paradox. Also a specific example of a free nonrelativistic particle is considered. Here it is proven that the probability Pt(Ω) of the particle being present in a volume Ω at time t cannot vanish on a set of t having nonzero measure. This implies that the sojourn time in Ω never vanishes, and that zero is never an eigenvalue of the sojourn time operator. It is also shown that for a very general class of sets Ω, including all bounded sets, the sojourn time turns out to be bounded with a bound independent of the initial state of the particle. Correspondingly, the sojourn time operator turns out to be a bounded one.

Journal of Mathematical Physics•Jul 1, 1989•12 citations
We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of "fuzzy" and "free-flight" sojourn times, that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they shouldn't be called "times".
Abstract We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of “fuzzy” and “free-flight” sojourn times that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they should not be called “times.”
We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of “fuzzy” and “free-flight” sojourn times that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they should not be called “times.”
The problem of time delay in one-dimensional quantum-mechanical scattering by a potential barrier is studied in the framework of time-dependent scattering theory. It is shown how the concept of sojourn time can be used to define three separate time delays: the time delay for transmission through the barrier, the time delay for reflection, and the total time delay (i.e., averaged over transmission and reflection). In addition to the anticipated dependence on the S matrix and its energy derivative, the time delays depend on certain parameters which can be interpreted as describing positions of detectors in a gedanken-experiment measurement. The problem of defining separate time delays for transmission and reflection appears to be rooted in the foundations of quantum theory and its interpretation.
Our previous results [Phys. Rev. A 37, 2843 (1988)], based on the concept of sojourn time, for the transmission and reflection time delays in one-dimensional scattering by a potential barrier are compared to the corresponding results of E. H. Hauge, J. P. Falck, and T. A. Fjedly [Phys. Rev. B 36, 4203 (1987)], based on the idea of following the time evolution of the mean position of wave packets. It is shown that the mean-position approach agrees with the more general sojourn-time approach in the limit of well-defined momentum, and that, in general, the two approaches appear to be related in a classical-like manner.
The concept of sojourn time and a sojourn time operator, aimed at describing the length of time spent by a quantum mechanical system in a given subspace of states, is investigated. A general rigorous definition to the sojourn time operator is given and some of its properties are studied. In particular, it is shown that the usual Born’s probability interpretation of the associated spectral measure yields strange, if not paradoxical, results, resembling the well-known quantum mechanical Zeno paradox. Also a specific example of a free nonrelativistic particle is considered. Here it is proven that the probability Pt(Ω) of the particle being present in a volume Ω at time t cannot vanish on a set of t having nonzero measure. This implies that the sojourn time in Ω never vanishes, and that zero is never an eigenvalue of the sojourn time operator. It is also shown that for a very general class of sets Ω, including all bounded sets, the sojourn time turns out to be bounded with a bound independent of the initial state of the particle. Correspondingly, the sojourn time operator turns out to be a bounded one.