Semiclassical quantization of truncated potentials

M. V. Berry, Kieron Burke

Research output: Contribution to journalArticle (Academic Journal)peer-review

6 Citations (Scopus)
146 Downloads (Pure)


An infinite potential well, truncated at finite height, provides a simple model for studying the effect of nonanalyticity on semiclassical approximations. An exact quantization condition for the bound states separates the effects associated with the untruncated well from those of the truncation. Because the truncation occurs beyond the classical turning points, it has no effect to any finite order in powers of Planck's constant. The truncation contribution is exponentially small and depends on the potential in the classically forbidden region. The contribution associated with the well, when consistently approximated beyond all semiclassical orders, also leads to a small exponential, depending on the potential in the classically allowed region. Both exponentially small contributions can be extracted by asymptotic analysis, with explicit results in the simple case of a linear well. This combination of several different semiclassical techniques could be pedagogically useful as an exercise in teaching physical asymptotics at the postgraduate level.

Original languageEnglish
Article number065403
Number of pages13
JournalEuropean Journal of Physics
Issue number6
Early online date30 Aug 2019
Publication statusPublished - 11 Oct 2019


  • asymptotics
  • divergent series
  • singularities
  • small exponentials


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