Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time

Phys. Rev. 138, B979 – Published 24 May 1965
Jon H. Shirley


The interaction of a quantum system with an oscillating field is studied in a formalism which replaces the semiclassical time-dependent Hamiltonian with a time-independent Hamiltonian represented by an infinite matrix. The formalism is developed as a mathematical equivalent to the semiclassical treatment, and interpreted as a classical approximation to the quantum treatment of the field. Combined with a perturbation theory for two nearly degenerate states, the formalism provides a convenient method for determining resonance transition probabilities including frequency shifts and multiple quantum transitions. The theory is illustrated by a detailed study of the simple case of a two-state system excited by a strong oscillating field.


  • Received 8 November 1963
  • Revised 18 January 1965
  • Published in the issue dated May 1965

© 1965 The American Physical Society

Authors & Affiliations

Jon H. Shirley*

  • California Institute of Technology, Pasadena, California

  • *Present address: National Bureau of Standards, Radio Standards Laboratory, Boulder, Colorado.

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