Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time

1965/05/24 by Jon H. Shirley, J.H. Shirley · 2,880 citations
Computer Science · Mathematics · Physics and Astronomy · #Classical mechanics #Covariant Hamiltonian field theory #Degenerate energy levels #Excited state #Formalism (music) #Hamiltonian (control theory) #Hamiltonian system #Mathematical physics #Mathematics #Physics #Quantum #Quantum Information and Cryptography #Quantum electrodynamics #Quantum field theory #Quantum mechanics #Quantum optics and atomic interactions #Schrödinger equation #Semiclassical physics #Spectroscopy and Quantum Chemical Studies

paper · doi:10.1103/physrev.138.b979

published in Physical Review 138(4B), B979-B987 (American Institute of Physics)

openalex publication_date 1965/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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.

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