Skip to main content
SpringerLink
Log in

A Microscopic Derivation of the Quantum Mechanical Formal Scattering Cross Section

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

Abstract

We prove that the empirical distribution of crossings of a “detector” surface by scattered particles converges in appropriate limits to the scattering cross section computed by stationary scattering theory. Our result, which is based on Bohmian mechanics and the flux-across-surfaces theorem, is the first derivation of the cross section starting from first microscopic principles.

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. Albeverio S., Gesztesy F., Høegh-Krohn R, Holden H., (1988) Solvable Models in Quantum Mechanics. Berlin Heidelberg New York, Springer

    MATH  Google Scholar 

  2. Amrein W.O., Jauch J.M., Sinha K.B., (1977) Scattering Theory in Quantum Mechanics. London, W. A. Benjamin, Inc.

    MATH  Google Scholar 

  3. Berndl K. Zur Existenz der Dynamik in Bohmschen Systemen, Ph.D. thesis, Ludwig-Maximilians-Universität München, 1994

  4. Berndl K., Dürr D., Goldstein S., Peruzzi G., Zanghì N. (1995) On the global existence of Bohmian mechanics. Commun. Math. Phys. 173(3): 647–673

    Article  ADS  MATH  Google Scholar 

  5. Bohm D. A suggested interpretation of the quantum theory in terms of “hidden” variables I, II. Phys. Rev. 85, 166–179, 180–193 (1952)

    Google Scholar 

  6. Combes J.-M., Newton R.G., Shtokhamer R. (1975) Scattering into cones and flux across surfaces. Phys. Rev. D 11(2): 366–372

    Article  ADS  Google Scholar 

  7. Dürr D., (2001) Bohmsche Mechanik als Grundlage der Quantenmechanik. Berlin Heidelberg New York, Springer

    MATH  Google Scholar 

  8. Dürr D., Goldstein S., Moser T., Zanghì, N.: What does quantum scattering theory physically describe? In preparation

  9. Dürr D., Goldstein S., Teufel S., Zanghì N. (2000) Scattering theory from microscopic first principles. Physica A 279: 416–431

    Article  Google Scholar 

  10. Dürr D., Goldstein S., Zanghì N. (1992) Quantum Equilibrium and the Origin of Absolute Uncertainty. J. Stat. Phys. 67, 843–907

    Article  ADS  MATH  Google Scholar 

  11. Dürr D., Moser T., Pickl P. (2006) The Flux-Across-Surfaces Theorem under conditions on the scattering state. J. Phys. A: Math. Gen. 39, 163–183

    Article  ADS  MATH  Google Scholar 

  12. Dürr D., Pickl P. (2003) Flux-across-surfaces theorem for a Dirac particle. J. Math. Phys. 44(2): 423–465

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Dürr D., Teufel S. On the exit statistics theorem of many particle quantum scattering. In: Blanchard P., Dell’Antonio, G.F. eds. Multiscale Methods in Quantum Mechanics: Theory and experiment, Boston: Birkhäuser, 2003

  14. Ikebe T. (1960) Eigenfunction expansion associated with the Schrödinger operators and their applications to scattering theory. Arch. Rat. Mech. Anal. 5, 1–34

    Article  MathSciNet  MATH  Google Scholar 

  15. Jensen A., Kato T. (1979) Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46(3): 583–611

    Article  MathSciNet  MATH  Google Scholar 

  16. Kato T. (1951) Fundamental Properties Of Hamiltonian Operators Of Schrödinger Type. Trans. Amer. Math. Soc. 70(1): 195–211

    Article  MathSciNet  MATH  Google Scholar 

  17. Newton R.G., (1982) Scattering Theory of Waves and Particles, Second Edition. Berlin Heidelberg New York, Springer

    MATH  Google Scholar 

  18. Pearson D.B., (1988) Quantum Scattering and Spectral Theory. San Diego, Academic Press

    MATH  Google Scholar 

  19. Reed M., Simon B., (1979) Methods Of Modern Mathematical Physics III: Scattering Theory. San Diego, Academic Press

    MATH  Google Scholar 

  20. Reed M., Simon B. Methods Of Modern Mathematical Physics I: Functional Analysis. Revised and enlarged ed., San Diego: Academic Press, 1980

  21. Teufel S. The flux-across-surfaces theorem and its implications for scattering theory. Ph.D. thesis, Ludwig-Maximilians-Universität München, 1999

  22. Teufel S., Dürr D., Münch-Berndl K. (1999) The flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs. J. Math. Phys. 40(4): 1901–1922

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Teufel S., Tumulka R. (2005) A Simple Proof for Global Existence of Bohmian Trajectories. Commun. Math. Phys. 258(2): 349–365

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Tumulka R. Closed 3-Forms and Random Worldlines. Ph.D. thesis, Ludwig-Maximilians-Universität München, 2001

  25. Weinberg S., (1996) Quantum Theory of Fields Volume I: Foundations. Cambridge, Cambridge University Press

    Google Scholar 

  26. Yajima K. (1995) The W k,p-continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47(3): 551–581

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut der Universität München, Theresienstr. 39, 80333, München, Germany

    D. Dürr & T. Moser

  2. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    S. Goldstein

  3. Dipartimento di Fisica, Università di Genova, Sezione INFN Genova, Via Dodescanesco 33, 16146, Genova, Italy

    N. Zanghì

Authors
  1. D. Dürr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Goldstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Moser
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. N. Zanghì
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Dürr.

Additional information

Communicated by A. Kupiainen

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dürr, D., Goldstein, S., Moser, T. et al. A Microscopic Derivation of the Quantum Mechanical Formal Scattering Cross Section. Commun. Math. Phys. 266, 665–697 (2006). https://doi.org/10.1007/s00220-006-0053-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-006-0053-x

Keywords

Search

Navigation