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Falling of a quantum particle in an inverse square attractive potential

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Abstract

Evolution of a quantum particle in an inverse square potential is studied by analysis of the equation of motion for 〈r2〉. In such a way we identify the conditions of falling of a particle into the center. We demonstrate the existence of a purely quantum limit of falling, namely, a particle does not fall, when the coupling constant is smaller than a certain critical value. Also the time of falling of a particle into the center is estimated. Although there are no stationary energy levels for this potential, we show that there are quasi-stationary states which evolve with 〈r2〉 being constant in time. Our results are compared with measurements of neutral atoms falling in the electric field of a charged wire. Modifications of the experiment, which may help in observing quantum limit of falling, are proposed.

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Authors and Affiliations

  1. Department for Theoretical Physics, Ivan Franko National University of Lviv, 12, Drahomanov St., 79005, Lviv, Ukraine

    Vasyl M. Vasyuta & Volodymyr M. Tkachuk

Authors
  1. Vasyl M. Vasyuta
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  2. Volodymyr M. Tkachuk
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Correspondence to Vasyl M. Vasyuta.

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Vasyuta, V.M., Tkachuk, V.M. Falling of a quantum particle in an inverse square attractive potential. Eur. Phys. J. D 70, 267 (2016). https://doi.org/10.1140/epjd/e2016-70463-3

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  • DOI: https://doi.org/10.1140/epjd/e2016-70463-3

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