Relating the Lorentzian and exponential: Fermi’s approximation, the Fourier transform, and causality

A. Bohm, N. L. Harshman, and H. Walther
Phys. Rev. A 66, 012107 – Published 25 July 2002
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Abstract

The Fourier transform is often used to connect the Lorentzian energy distribution for resonance scattering to the exponential time dependence for decaying states. However, to apply the Fourier transform, one has to bend the rules of standard quantum mechanics; the Lorentzian energy distribution must be extended to the full real axis <E< instead of being bounded from below 0<~E< (Fermi’s approximation). Then the Fourier transform of the extended Lorentzian becomes the exponential, but only for times t>~0, a time asymmetry which is in conflict with the unitary group time evolution of standard quantum mechanics. Extending the Fourier transform from distributions to generalized vectors, we are led to Gamow kets, which possess a Lorentzian energy distribution with <E< and have exponential time evolution for t>~t0=0 only. This leads to probability predictions that do not violate causality.

  • Received 6 March 2002

DOI:https://doi.org/10.1103/PhysRevA.66.012107

©2002 American Physical Society

Authors & Affiliations

A. Bohm1, N. L. Harshman2, and H. Walther3

  • 1Physics Department, University of Texas at Austin, Austin, Texas 78712
  • 2Department of Physics and Astronomy, Rice University, Houston, Texas 77005
  • 3Max-Planck Institut für Quantenoptik und Sektion Physik, Universität München, 85748 Garching, Germany

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Vol. 66, Iss. 1 — July 2002

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