Quantum maps

M. V. Berry, N. L. Balazs, M. Tabor, A. Voros

Research output: Contribution to journalArticlepeer-review

372 Scopus citations


We quantize area-preserving maps M of the phase plane q, p by devising a unitary operator U transforming states | φn〉 into | φn+1〉. The result is a system with one degree of freedom q on which to study the quantum implications of generic classical motion, including stochasticity. We derive exact expressions for the equation iterating wavefunctions ψn(q), the propagator for Wigner functions Wn(q,p), the eigenstates of the discrete analog of the quantum harmonic oscillator, and general complex Gaussian wave packets iterated by a U derived from a linear M. For | ψn〉 associated with curves Ln in q, p, we derive a semiclassical theory for evolving states and stationary states, analogous to the familiar WKB method. This theory breaks down when Ln gets so complicated as to develop convolutions of area h{stroke} or smaller. Such complication is generic; its principal morphotologies are"whorls" and "tendrils," associated respectively with elliptic and hyperbolic fixed points of M. Under U, ψn(q) eventually transforms into a new sort of wave that no longer perceives the details of Ln. For all regimes, however, the smoothed | ψn(q)|2 appears semiclassically appears to be given accurately by the smoothed projection of Ln onto the q axis, both smoothings being over a de Broglie wavelength. The classical, quantum, and semiclassical theory is illustrated by computations on the discrete quartic oscillator map. We display for the first time stochastic wavefunctions, dominated by dense clusters of caustics and characterized by multiple scales of oscillation.

Original languageEnglish (US)
Pages (from-to)26-63
Number of pages38
JournalAnnals of Physics
Issue number1
StatePublished - Sep 15 1979

ASJC Scopus subject areas

  • Physics and Astronomy(all)


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