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International Journal of Theoretical Physics

, Volume 54, Issue 6, pp 1882–1907 | Cite as

Three-dimensional Quantum Slit Diffraction and Diffraction in Time

  • M. Beau
  • T. C. Dorlas
Article

Abstract

We derive a self-consistent formula for the quantum propagator of the quantum diffraction by a reflecting (or non-absorbing) screen with an arbitrary aperture, in the transmission region. To achieve this, we use the Green function approach (Brukner and Zeilinger, Phys. Rev. A. 56(5), 3804–3824 1997) that we extend to mixed boundary conditions (including Dirichlet and Neumann) on the shutter’s screen and to any initial wave functions, such as Gaussian wave packets. To illustrate our results, we apply this method to the famous rectangular slit diffraction problem. It allows us to take into account the effect of the quantum nature of the motion perpendicular to the screen where the diffraction-in-time phenomenon appears in this direction. Then we derive corrections to the standard semi-classical formula and the diffraction pattern. We also point out situations in which this might be observable. In particular, we discuss the diffraction in space and time in the presence of gravity.

Keywords

Quantum diffraction Diffraction in space Diffraction in time Slit experiment Semiclassical approximation Constant gravity field 

References

  1. 1.
    Jönsson, C.: Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten. Z. Phys. 161 (4), 454–474 (1961)CrossRefADSGoogle Scholar
  2. 2.
    Jönsson, C.: Electron Diffraction at Multiple Slits. Am. J. Phys. 42 (1), 4–11 (1974)CrossRefADSGoogle Scholar
  3. 3.
    Donati, O., Missiroli, G. F., Pozzi, G.: An Experiment on Electron Interference. Am. J. Phys. 41, 639–644 (1973)CrossRefADSGoogle Scholar
  4. 4.
    Merli, P. G., Missiroli, G. F., Pozzi, G.: On the statistical aspect of electron interference phenomena. Am. J. Phys. 44 (3), 306–307 (1976)CrossRefADSGoogle Scholar
  5. 5.
    Tonomura, A., Endo, J., Ezawa, H., Matsuda, T., Kawasaki, T.: Demonstration of single-electron buildup of an interference pattern. Am. J. Phys. 57 (2), 117–120 (1989)CrossRefADSGoogle Scholar
  6. 6.
    Feynman, R. P., Leighton, R. B., Sands, M. L.: The Feynman Lectures on Physics. Addison-Wesley, Reading, MA (1965)MATHGoogle Scholar
  7. 7.
    Bach, R., Pope, D., Liou, S.-H., Batelaan, H.: Controlled double-slit electron diffraction. New J. Phys. 15, 033018 (2013)CrossRefADSGoogle Scholar
  8. 8.
    Frabboni, S., Frigeri, C., Gazzadi, G. C., Pozzi, G.: Two and three slit electron interference and diffraction experiments. Am. J. Phys. 79 (6), 615–618 (2011)CrossRefADSGoogle Scholar
  9. 9.
    Zeilinger, A., Gähler, R., Shull, C. G., Treimer, W., Mampe, W.: Single- and double-slit diffraction of neutrons. Rev. Mod. Phys. 60 (4), 1067–73 (1988)CrossRefADSGoogle Scholar
  10. 10.
    Shimizu, F., Shimizu, K., Takuma, H.: Double-slit interference with ultracold metastable neon atoms. Phys. Rev. A. 46 (1), R17–R20 (1992)CrossRefADSGoogle Scholar
  11. 11.
    Nairz, O., Arndt, M., Zeilinger, A.: Quantum interference experiments with large molecules. Am. J. Phys. 71 (4), 319–325 (2003)CrossRefADSGoogle Scholar
  12. 12.
    Feynman, R. P., Hibbs, A. R., 3rd edn: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)MATHGoogle Scholar
  13. 13.
    Barut, A. O., Basri, S.: Path integrals and quantum interference. Am. J. Phys. 60 (10), 896–899 (1992)CrossRefADSGoogle Scholar
  14. 14.
    Beau, M.: Feynman path integral and electron diffraction slit experiments. Eur. Journ. Phys. 33 (15), 1023–1039 (2012)CrossRefADSMATHGoogle Scholar
  15. 15.
    Zecca, A.: Two-Slit Diffraction Pattern for Gaussian Wave Packets. Int. J. Theo. Phys. 38 (3), 911–918 (1999)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Moshinsky, M.: Diffraction in Time. Phys. Rev. 88 (3), 625–631 (1952)CrossRefADSMATHMathSciNetGoogle Scholar
  17. 17.
    Szriftgiser, P., Guéry-Odelin, D., Arndt, M., Dalibard, J.: Atomic Wave Diffraction and Interference Using Temporal Slits. Phys. Rev. Lett. 77 (1), 4–7 (1996)CrossRefADSGoogle Scholar
  18. 18.
    Hils, Th., Hils, Al: Matter-wave optics in the time domain: Results of a cold-neutron experiment. Phys. Rev. A 58 (6), 4784 (1998)CrossRefADSGoogle Scholar
  19. 19.
    Brukner, C., Zeilinger, A.: Diffraction of matter waves in space and in time. Phys. Rev. A. 56 (5), 3804–3824 (1997)CrossRefADSGoogle Scholar
  20. 20.
    Morse, P. M., Feshbach, H.: Methods of Theoretical Physics. McGraw-Hill, New York (1953)MATHGoogle Scholar
  21. 21.
    Goussev, A.: Huygens-Fresnel-Kirchhoff construction for the quantum propagator with application to diffraction in space and time. Phys. Rev. A. 85 (1), 013626–013636 (2012)CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Goussev, A.: Time diffraction: an exact model. Phys. Rev. A. 87 (5), 053621 (2013)CrossRefADSGoogle Scholar
  23. 23.
    del Campo, A., Muga, J. G.: Single-particle matter wave pulses. J. Phys. A: Math. Gen. 38 (45), 9803–9819 (2005)CrossRefADSMATHMathSciNetGoogle Scholar
  24. 24.
    del Campo, A., Muga, J. G., Moshinsky, M.: Time modulation of atom sources. J. Phys. B: At. Mol. Opt. Phys. 40, 975 (2007)CrossRefADSGoogle Scholar
  25. 25.
    Kälbermann, G.: Single- and double-slit scattering of wavepackets. J. Phys. A: Math. Gen. 35 (21), 4599–4616 (2002)CrossRefADSMATHGoogle Scholar
  26. 26.
    Kälbermann, G.: Diffraction of wave packets in space and time. J. Phys. A: Math. Gen. 34 (33), 6465–6480 (2001)CrossRefADSMATHGoogle Scholar
  27. 27.
    Kälbermann, G.: Wave packet diffraction in the KronigPenney model. J. Phys. A: Math. Gen. 35 (4), 1045–1053 (2002)CrossRefADSMATHGoogle Scholar
  28. 28.
    del Campo, A., García-Calderón, G., Mugad, J.G.: Quantum transients. Phys. Rep. 476 (1–3), 1–50 (2009)CrossRefADSGoogle Scholar
  29. 29.
    Torrontegui, E., Muoz, J., Ban, Y., Muga, J. G.: Explanation and observability of diffraction in time, Vol. 83, p 043608 (2011)Google Scholar
  30. 30.
    Baker, B. B., Copson, E. T., 2nd edn: The mathematical theory of Huygens’ principle. Clarendon Press, Oxford (1950)MATHGoogle Scholar
  31. 31.
    Gondran, M., Gondran, A.: Numerical simulation of the double slit interference with ultracold atoms. Am. J. Phys. 73 (6), 507–515 (2005)CrossRefADSGoogle Scholar
  32. 32.
    Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions. Dover, New York (1965)Google Scholar
  33. 33.
    Born, M., Wolf, E., 4th edn: Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Pergamon, Oxford (1969)Google Scholar
  34. 34.
    Erdelyi, A.: Tables of Integral Transforms, vol. 1. McGraw-Hill Inc, US (1954)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Dublin Institute for Advanced StudiesSchool of Theoretical PhysicsDublinIreland

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