 While \(t_{E}\) represents an important limit for how long quantum state vectors can be expected to shadow classical trajectories, there are interesting behaviors in the semi-classical quantum models corresponding to classical chaotic systems on longer time scales. By performing some more detailed analysis, Tomsovic and Heller (1993) showed that comparing the full quantum solutions with suitably chosen semi-classical solutions for some billiards problems provided excellent agreement well after \(t_{E}\) including fine details of the energy spectra. For their techniques semi-classical mechanics remains accurate for modeling quantum systems up to a time that scales with \((h/2\pi)^{-1/2}\). 