Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Physical Society of Japan

On the Irreversible Behavior of Coupled Oscillators With Application to Time of Flight

Youzou Kogure

Published Jan 1, 1961• 9 citations
Publisher

Abstract

The irreversible behavior of simple systems is studied. An example is a system composed of coupled many harmonic oscillators. It is shown that the amplitude of a specified oscillator under appropriate initial condition, and the response function vanish after a long time, as the number of oscillators tends to infinity. Another example is a system of gaseous particle which is subject to collisions due to fixed scattering centers. It is shown that if there is only one scatterer the occupation number for the initial state changes periodically, and if there are many scatterers the occupation number exhibits exponential decay.

DOI
10.1143/jpsj.16.14

Related papers

Quantum Dissipation in Open Harmonic Systems: --Operator Solution and Application to Decay Process--

I. Joichi, S. Matsumoto, M. Yoshimura

A finite number of harmonic oscillators coupled to infinitely many environment oscillators is fundamental to the problem of understanding quantum dissipation of a small system immersed in a large environment. Exact operator solution as a function of time is given to this problem, by using diagonalized dynamical variable of the entire system, the small system plus the environment. The decay law of prepared initial configuration is worked out in greatest detail. A clear separation of the exponential- and the power-law decay period is made possible by our method. Behavior of physical quantities at asymptotically late times can be understood in terms of the overlap probability of the system variable with the diagonal variable of the entire system.

arXiv:cond-mat/9612235•Progress of Theoretical Physics•Jul 1, 1997•19 citations

Chaotic and irreversible properties of quantum scattering systems

Clasine van Winter

Recent results for a free particle are generalized to N-particle quantum systems. Chaotic and irreversible behavior occurs in scattering states that belong to a certain Hilbert space K2 with a preferred time direction pointing to the future. At positive times the time evolution of positive observables exhibits quantum analogs of sensitive dependence on initial conditions, topological transitivity, and existence of a dense set of periodic points. A mixture of states in K2 can be described in terms of a density operator with thermodynamical entropy that increases to its least upper bound when the time tends to infinity.

Journal of Mathematical Physics•May 1, 2001•1 citation

Non-Markovian Irreversible Behavior in a Simple Model

Arturo López

The soluble model of an oscillator coupled to a scalar field is used as an example of irreversible behavior. By studying the reduced density operator for the oscillator, a generalized Fokker-Planck-Kramers-Chandrasekhar equation for the Wigner distribution function is derived. The diagonal matrix elements of the reduced density operator satisfy a generalized non-Markovian master equation. By application of a time-averaged method, the ordinary Pauli master equation is derived. The time evolution of the occupation probabilities of the oscillator levels has been numerically computed, and compared with the solutions of the Pauli equation.

Physical Review A•Aug 1, 1970•4 citations

An investigation of a class of soluble many body problems

Gregg Stanley Edwards

This thesis is a report of results so far obtained in an investigation of the properties and characteristics of simple interaction systems whose Hamiltonians are quadratic in their generalized coordinates. The exact solutions of two system types are presented. The first is composed of a simple harmonic oscillator coupled to a uniform lattice of coupled simple harmonic oscillators, or, in the small amplitude approximation, to a beaded string. The second is composed of an arbitrary distribution of oscillators coupled to string. The quantization of linear systems such as those described above is discussed with the result that the solution to the equivalent quantum system in terms of operators is just that of the corresponding classical solution as a function of its initial conditions when these are interpreted as quantum operators. The energy flow along the linear chain is examined in the discussion of the non-exponential decay in dispersive systems. This is compared with the original discussion by Trammell (1956) who pointed out the non-exponential decay of quantum systems and the corresponding behavior of dispersive classical systems.

•Jan 1, 1963•0 citations

Minimal irreversible quantum mechanics. The decay of unstable states

Arbó, Diego G., Castagnino, Mario A., Gaioli, Fabián H. +1 more

Brownian motion is modelled by a harmonic oscillator (Brownian particle) interacting with a continuous set of uncoupled harmonic oscillators. The interaction is linear in the coordinates and the momenta. The model has an analytical solution that is used to study the time evolution of the reduced density operator. It is derived in a closed form, in the one-particle sector of the model. The irreversible behavior of the Brownian particle is described by a reduced density matrix.

•Jan 1, 2000•15 citations

TEMPORAL BEHAVIOR OF QUANTUM MECHANICAL SYSTEMS

Hiromichi Nakazato, Mikio Namiki, Saverio Pascazio

The temporal behavior of quantum mechanical systems is reviewed. We mainly focus our attention on the time development of the so-called “survival” probability of those systems that are initially prepared in eigenstates of the unperturbed Hamiltonian, by assuming that the latter has a continuous spectrum. The exponential decay of the survival probability, familiar, for example, in radioactive decay phenomena, is representative of a purely probabilistic character of the system under consideration and is naturally expected to lead to a master equation. This behavior, however, can be found only at intermediate times, for deviations from it exist both at short and long times and can have significant consequences. After a short introduction to the long history of the research on the temporal behavior of such quantum mechanical systems, the short-time behavior and its controversial consequences when it is combined with von Neumann’s projection postulate in quantum measurement theory are critically overviewed from a dynamical point of view. We also discuss the so-called quantum Zeno effect from this standpoint. The behavior of the survival amplitude is then scrutinized by investigating the analytic properties of its Fourier and Laplace transforms. The analytic property that there is no singularity except a branch cut running along the real energy axis in the first Riemannian sheet is an important reflection of the time-reversal invariance of the dynamics governing the whole process. It is shown that the exponential behavior is due to the presence of a simple pole in the second Riemannian sheet, while the contribution of the branch point yields a power behavior for the amplitude. The exponential decay form is cancelled at short times and dominated at very long times by the branch-point contributions, which give a Gaussian behavior for the former and a power behavior for the latter. In order to realize the exponential law in quantum theory, it is essential to take into account a certain kind of macroscopic nature of the total system, since the exponential behavior is regarded as a manifestation of a complete loss of coherence of the quantum subsystem under consideration. In this respect, a few attempts at extracting the exponential decay form on the basis of quantum theory, aiming at the master equation, are briefly reviewed, including van Hove’s pioneering work and his well-known “λ2T” limit. In the attempt to further clarify the mechanism of the appearance of a purely probabilistic behavior without resort to any approximation, a solvable dynamical model is presented and extensively studied. The model describes an ultrarelativistic particle interacting with N two-level systems (called “spins”) and is shown to exhibit an exponential behavior at all times in the weak-coupling, macroscopic limit. Furthermore, it is shown that the model can even reproduce the short-time Gaussian behavior followed by the exponential law when an appropriate initial state is chosen. The analysis is exact and no approximation is involved. An interpretation for the change of the temporal behavior in quantum systems is drawn from the results obtained. Some implications for the quantum measurement problem are also discussed, in particular in connection with dissipation.

arXiv:quant-ph/9509016•International Journal of Modern Physics B•Jan 30, 1996•208 citations
The irreversible behavior of simple systems is studied. An example is a system composed of coupled many harmonic oscillators. It is shown that the amplitude of a specified oscillator under appropriate initial condition, and the response function vanish after a long time, as the number of oscillators tends to infinity. Another example is a system of gaseous particle which is subject to collisions due to fixed scattering centers. It is shown that if there is only one scatterer the occupation number for the initial state changes periodically, and if there are many scatterers the occupation number exhibits exponential decay.
A finite number of harmonic oscillators coupled to infinitely many environment oscillators is fundamental to the problem of understanding quantum dissipation of a small system immersed in a large environment. Exact operator solution as a function of time is given to this problem, by using diagonalized dynamical variable of the entire system, the small system plus the environment. The decay law of prepared initial configuration is worked out in greatest detail. A clear separation of the exponential- and the power-law decay period is made possible by our method. Behavior of physical quantities at asymptotically late times can be understood in terms of the overlap probability of the system variable with the diagonal variable of the entire system.
Recent results for a free particle are generalized to N-particle quantum systems. Chaotic and irreversible behavior occurs in scattering states that belong to a certain Hilbert space K2 with a preferred time direction pointing to the future. At positive times the time evolution of positive observables exhibits quantum analogs of sensitive dependence on initial conditions, topological transitivity, and existence of a dense set of periodic points. A mixture of states in K2 can be described in terms of a density operator with thermodynamical entropy that increases to its least upper bound when the time tends to infinity.
The soluble model of an oscillator coupled to a scalar field is used as an example of irreversible behavior. By studying the reduced density operator for the oscillator, a generalized Fokker-Planck-Kramers-Chandrasekhar equation for the Wigner distribution function is derived. The diagonal matrix elements of the reduced density operator satisfy a generalized non-Markovian master equation. By application of a time-averaged method, the ordinary Pauli master equation is derived. The time evolution of the occupation probabilities of the oscillator levels has been numerically computed, and compared with the solutions of the Pauli equation.
This thesis is a report of results so far obtained in an investigation of the properties and characteristics of simple interaction systems whose Hamiltonians are quadratic in their generalized coordinates. The exact solutions of two system types are presented. The first is composed of a simple harmonic oscillator coupled to a uniform lattice of coupled simple harmonic oscillators, or, in the small amplitude approximation, to a beaded string. The second is composed of an arbitrary distribution of oscillators coupled to string. The quantization of linear systems such as those described above is discussed with the result that the solution to the equivalent quantum system in terms of operators is just that of the corresponding classical solution as a function of its initial conditions when these are interpreted as quantum operators. The energy flow along the linear chain is examined in the discussion of the non-exponential decay in dispersive systems. This is compared with the original discussion by Trammell (1956) who pointed out the non-exponential decay of quantum systems and the corresponding behavior of dispersive classical systems.
The temporal behavior of quantum mechanical systems is reviewed. We mainly focus our attention on the time development of the so-called “survival” probability of those systems that are initially prepared in eigenstates of the unperturbed Hamiltonian, by assuming that the latter has a continuous spectrum. The exponential decay of the survival probability, familiar, for example, in radioactive decay phenomena, is representative of a purely probabilistic character of the system under consideration and is naturally expected to lead to a master equation. This behavior, however, can be found only at intermediate times, for deviations from it exist both at short and long times and can have significant consequences. After a short introduction to the long history of the research on the temporal behavior of such quantum mechanical systems, the short-time behavior and its controversial consequences when it is combined with von Neumann’s projection postulate in quantum measurement theory are critically overviewed from a dynamical point of view. We also discuss the so-called quantum Zeno effect from this standpoint. The behavior of the survival amplitude is then scrutinized by investigating the analytic properties of its Fourier and Laplace transforms. The analytic property that there is no singularity except a branch cut running along the real energy axis in the first Riemannian sheet is an important reflection of the time-reversal invariance of the dynamics governing the whole process. It is shown that the exponential behavior is due to the presence of a simple pole in the second Riemannian sheet, while the contribution of the branch point yields a power behavior for the amplitude. The exponential decay form is cancelled at short times and dominated at very long times by the branch-point contributions, which give a Gaussian behavior for the former and a power behavior for the latter. In order to realize the exponential law in quantum theory, it is essential to take into account a certain kind of macroscopic nature of the total system, since the exponential behavior is regarded as a manifestation of a complete loss of coherence of the quantum subsystem under consideration. In this respect, a few attempts at extracting the exponential decay form on the basis of quantum theory, aiming at the master equation, are briefly reviewed, including van Hove’s pioneering work and his well-known “λ2T” limit. In the attempt to further clarify the mechanism of the appearance of a purely probabilistic behavior without resort to any approximation, a solvable dynamical model is presented and extensively studied. The model describes an ultrarelativistic particle interacting with N two-level systems (called “spins”) and is shown to exhibit an exponential behavior at all times in the weak-coupling, macroscopic limit. Furthermore, it is shown that the model can even reproduce the short-time Gaussian behavior followed by the exponential law when an appropriate initial state is chosen. The analysis is exact and no approximation is involved. An interpretation for the change of the temporal behavior in quantum systems is drawn from the results obtained. Some implications for the quantum measurement problem are also discussed, in particular in connection with dissipation.