Skip to main content
SpringerLink
Log in

Semiclassical quantum mechanics

I. The ℏ→0 limit for coherent states

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the ℏ→0 limit of the quantum dynamics generated by the HamiltonianH(ℏ)=−(ℏ2/2m)Δ+V. We prove that the evolution of certain Gaussian states is determined asymptotically as ℏ→0 by classical mechanics. For suitable potentialsV inn≧3 dimensions, our estimates are uniform in time and our results hold for scattering theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arnold, V.I.:Mathematical methods of classical mechanics. Berlin, Heidelberg, New York: Springer 1978

    Google Scholar 

  2. Birkhoff, G., Rota, G.-C.: Ordinary differential equations. London, Toronto, Waltham, MA: Blaisdell Publishing Company 1962

    Google Scholar 

  3. Eckmann, J.-P., Sénéor, R.: The Maslov-WKB method for the (an-) harmonic oscillator. Arch. Rational Mech. Anal.61, 151–173 (1976)

    Google Scholar 

  4. Enss, V.: Asymptotic completeness for quantum mechanical potential scattering. I. Short range potentials. Commun. Math. Phys.61, 285–291 (1978)

    Google Scholar 

  5. Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems. Commun. Math. Phys.66, 37–76 (1979)

    Google Scholar 

  6. Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems. II. Preprint. Université de Paris-Sud, Orsay (1979)

    Google Scholar 

  7. Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys.35, 265–277 (1974)

    Google Scholar 

  8. Hunziker, W.: TheS-matrix in classical mechanics. Commun. Math. Phys.8, 282–299 (1968)

    Google Scholar 

  9. Loomis, L., Sternberg, S.: Advanced calculus. Reading, Mass.: Adison-Wesley 1968

    Google Scholar 

  10. Maslov, V.P.: Théorie des perturbations et méthodes asymptotiques. Paris: Dunod 1972

    Google Scholar 

  11. Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. I. Functional analysis. New York, London: Academic Press 1972

    Google Scholar 

  12. Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. III. Scattering theory. New York, London: Academic Press 1979

    Google Scholar 

  13. Siegel, C.L.: Vorlesungen über Himmelsmechanik. Berlin, Göttingen, Heidelberg: Springer 1956

    Google Scholar 

  14. Simon, B.: Wave operators for classical particle scattering. Commun. Math. Phys.23, 37–48 (1971)

    Google Scholar 

  15. Yajima, K.: The quasi-classical limit of quantum scattering theory. Commun. Math. Phys.69, 101–130 (1979)

    Google Scholar 

  16. Yajima, K.: The quasi-classical limit of quantum scattering theory. II. Long range scattering. Preprint, University of Virginia (1979)

Download references

Author information

Authors and Affiliations

  1. The Rockefeller University, 10021, New York, NY, USA

    G. A. Hagedorn

Authors
  1. G. A. Hagedorn
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by B. Simon

Supported in part by the National Science Foundation under Grant PHY 78-08066

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hagedorn, G.A. Semiclassical quantum mechanics. Commun.Math. Phys. 71, 77–93 (1980). https://doi.org/10.1007/BF01230088

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01230088

Keywords

Search

Navigation