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American Journal of Physics -- December 2011 -- Volume 79, Issue 12, pp. 1218

Paradoxical reflection in quantum mechanics

Pedro L. Garrido1, Sheldon Goldstein2, Jani Lukkarinen3, and Roderich Tumulka4

1Departamento de Electromagnetismo y Física de la Materia, Institute Carlos I for Theoretical and Computational Physics, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
2Departments of Mathematics, Physics, and Philosophy, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
3Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, FI-00014 Helsingin yliopisto, Finland
4Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019

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We discuss a phenomenon of elementary quantum mechanics that is counterintuitive, non-classical, and apparently not widely known: the reflection of a particle at a downward potential step. In contrast, classically, particles are reflected only at upward steps. The conditions for this effect are that the wavelength is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. The phenomenon is suggested by non-normalizable solutions to the time-independent Schrödinger equation. We present numerical and mathematical evidence that it is also predicted by the time-dependent Schrödinger equation. The paradoxical reflection effect suggests and we confirm mathematically that a particle can be trapped for a long time (though not indefinitely) in a region surrounded by downward potential steps, that is, on a plateau.

© 2011 American Association of Physics Teachers

ACKNOWLEDGMENTS

The authors thank the Institut des Hautes Études Scientifiques at Bures-sur-Yvette, France, where the idea for this article was conceived, for hospitality. For discussions on the topic the authors thank, in particular, Federico Bonetto (Georgia Tech), Ovidiu Costin (Ohio State University), and Herbert Spohn (TU München). The work of S. Goldstein was supported in part by NSF Grant (No. DMS-0504504). The work of J. Lukkarinen was supported by the Academy of Finland and by the Deutsche Forschungsgemeinschaft project Sp 181/19-2. The work of R. Tumulka was supported by the European Commission through its 6th Framework Programme “Structuring the European Research Area” and contract Nr. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action.

Article Outline

  1. INTRODUCTION
  2. STATIONARY ANALYSIS OF THE RECTANGULAR STEP
  3. SOFT STEP
  4. WAVE PACKETS
    1. Numerical simulation
    2. Is it for real?
  5. PARAMETER DEPENDENCE
  6. THE CLASSICAL LIMIT
  7. A PLATEAU AS A TRAP
  8. EIGENFUNCTIONS WITH COMPLEX ENERGY
  9. WAVE PACKETS ON THE PLATEAU
  10. CONCLUSIONS

KEYWORDS and PACS

Keywords

particle traps, physics education, quantum theory, Schrodinger equation

PACS

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History
Received Aug 2008
Accepted Aug 2011

PUBLICATION DATA

ISSN

0002-9505 (print)  
1943-2909 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Association of Physics Teachers

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1119/1.3636408

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