Reflection time as an eigenvalue problem

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 T R is linear and diagonal in energy space. The eigenvalues are given by τ 2 (E) = ℏ \documentclass{article}\pagestyle{empty}\begin{document}$ \left\{ {\frac{{\partial \rho (E)}}{{\partial E}} + \frac{{\sin \rho (E)}}{{2E}}} \right\} $\end{document} , where ρ( E ) is the phase shift for stationary reflection at fixed scattering energy E . The quantum mechanical reflection time τ R ( E ) has a well defined WKB‐limit. For low scattering energies τ R ( E ) is proportional to \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt E $\end{document} if weakly bound states are absent in the reflection region. However, if there is an empty weakly bound state it will transiently trap a slowly moving particle, causing a \documentclass{article}\pagestyle{empty}\begin{document}$ 1/\sqrt E $\end{document} divergence of the reflection time for E →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.