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Keywords:

  • Tunneling time;
  • reflection time;
  • Goos-Hänchen effect;
  • classical-limit quantum mechanics

Abstract

Starting with wave packets, an eigenvalue equation is derived for the amount of time a particle needs for reflection from an impermeable potential barrier. The corresponding Hermitian operator TR is linear and diagonal in energy space. The eigenvalues are given by τ2(E) = ℏ equation image, where ρ(E) is the phase shift for stationary reflection at fixed scattering energy E. The quantum mechanical reflection time τR(E) has a well defined WKB-limit. For low scattering energies τR(E) is proportional to equation image if weakly bound states are absent in the reflection region. However, if there is an empty weakly bound state it will transiently trap a slowly moving particle, causing a equation image divergence of the reflection time for E[RIGHTWARDS ARROW]0. We work out and illustrate the theory for various reflecting potentials. The method allows for a consistent treatment of the quantum mechanical Goos-Hänchen time delay.