IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v141y1987i1p81-110.html
   My bibliography  Save this article

On the consistent description of quantum relaxation processes: The harmonic oscillator

Author

Listed:
  • Eckhardt, W.

Abstract

The quantum-mechanical description of a relaxation process is critically reviewed. The initially reversible “universe” consists of a harmonic oscillator (system) interacting with N further harmonic oscillators (heat bath). The coupling is linear in both, system- and bath-coordinates. The underlying dynamical equations are exactly solvable. A reasonable quantum-mechanical treatment of relaxation processes should guarantee that for all times the canonical commutation relations deviate only in a higher order of the coupling constant from their reversible values. A hierarchy of descriptions with decreasing quantum-mechanical consistency is formulated and discussed: 1.(i) the exact irreversible solution in connection with a decoupling of system and bath at t = 0;2.(ii) the exact irreversible solution in connection with a conditional thermal average at t = 0;3.(iii) a semi-Markovian stochastic differential equation (SDE), where the dissipation is described by a single parameter while the fluctuation (i.e. the correlation function of the stochastic force includes the information on the continuous bath-spectrum;4.(iv) a complete Markovian SDE; dissipation and fluctuation are characterized by one single parameter. For the Drude model the descriptions (i)–(iii) are equivalent in relation to consistent results of first order in the dissipative parameters. On the contrary, description (iv) is unsuitable in the quantum-mechanical regime.

Suggested Citation

  • Eckhardt, W., 1987. "On the consistent description of quantum relaxation processes: The harmonic oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 141(1), pages 81-110.
    • Handle: RePEc:eee:phsmap:v:141:y:1987:i:1:p:81-110
    DOI: 10.1016/0378-4371(87)90263-9
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0378437187902639
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/0378-4371(87)90263-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Micheline Thomine-Desmazures, 1965. "L'investigation psychosociologique en milieu rural," Économie rurale, Programme National Persée, vol. 64(1), pages 19-24.
    2. Braun, E. & Godoy, S.V., 1977. "Quantum statistical effects of the motion of an oscillator interacting with a radiation field," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 86(2), pages 337-354.
    3. Caldirola, P. & Lugiato, L.A., 1982. "Connection between the Schrödinger equation for dissipative systems and the master equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 116(1), pages 248-264.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yannouleas, C., 1985. "Quantal Brownian motion from RPA dynamics: The master and Fokker-Planck equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 132(1), pages 1-27.
    2. Remaud, B. & Hernández, E.S., 1981. "On the phase-space dynamics of a time-dependent harmonic oscillator at finite temperature," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 107(3), pages 553-566.
    3. Braun, E., 1980. "A model of Brownian dynamics for colloidal suspensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 103(1), pages 325-342.
    4. Dekker, H., 1987. "Quantum mechanical barrier problems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 146(3), pages 387-395.
    5. Dekker, H., 1988. "Incoherent tunneling at weak bias: Strong quantum damping," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 154(1), pages 61-88.
    6. Gorini, Vittorio & Verri, Maurizio & Frigerio, Alberto, 1989. "Non-markovian behavior in low-temperature damping," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 161(2), pages 357-384.
    7. Dekker, H., 1991. "Multisite spin hopping analysis of multilevel dissipative quantum tunneling and coherence at finite temperatures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 179(1), pages 81-102.
    8. Lindenberg, Katja & Mohanty, Udayan & Seshadri, V., 1983. "Hamiltonian model for the Brownian motion of a rigid rotor," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 119(1), pages 1-16.
    9. Kagermann, H., 1982. "Stochastic equations arising from test particle problems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 116(1), pages 199-206.
    10. Yannouleas, C., 1984. "The RPA description of the quantum damped harmonic oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 125(2), pages 549-576.
    11. Godoy, S.V. & Braun, E., 1978. "Long time limit of correlation functions between massless modes interacting with a quantum oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 92(1), pages 245-254.
    12. Dekker, H., 1987. "Dissipative quantum mechanics: A proof of dynamical consistency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 144(2), pages 445-452.
    13. Caldeira, A.O. & Leggett, A.J., 1983. "Path integral approach to quantum Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 121(3), pages 587-616.
    14. Cáceres, Manuel O. & Chattah, Ana K., 1996. "On the Schrödinger-Langevin picture and the master equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 234(1), pages 322-340.
    15. Tegmark, Max & Yeh, Leehwa, 1994. "Steady states of harmonic oscillator chains and shortcomings of harmonic heat baths," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 202(1), pages 342-362.
    16. Bafaluy, J. & Rubí, J.M., 1988. "The “harmonic liquid” away from equilibrium," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 153(1), pages 129-146.
    17. Kagermann, H., 1981. "Stochastic equations arising from test particle problems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 105(3), pages 365-379.
    18. Dekker, H., 1985. "A contribution to the dynamical theory of radiating electrons," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 133(1), pages 1-34.
    19. Hernández, E.S. & Mizrahi, S.S., 1983. "Parametrization of open systems with effective quadratic Hamiltonians plus stochastic force," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 119(1), pages 159-176.
    20. Brustein, Ram & Marianer, Shlomo & Schwartz, Moshe, 1991. "Langevin memory kernel and noise from Lagrangian dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 175(1), pages 47-58.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:141:y:1987:i:1:p:81-110. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.