From localized to extended states in a time-dependent quantum model

  • Jorge V. José
III. Time Dependent Systems
Part of the Lecture Notes in Physics book series (LNP, volume 263)


The problem of a particle inside a rigid box with one of the walls oscillating periodically in time is studied quantum mechanically. In the classical limit, this model was introduced by Fermi in the context of cosmic ray physics. The classical solutions can go from being quasiperiodic to chaotic, as a function of the amplitude of the wall oscillation. In the quantum case, we calculate the spectral properties of the corresponding evolution operator, i.e.: the quasi-energy eigenvalues and eigenvectors. The specific form of the wall oscillation, e.g. \(\ell (t) = \sqrt {1 + 2\delta \left| t \right|}\), with |t| ≤ 1/2, and ℓ(t + 1) = ℓ(t) , is essential to the solutions presented here. It is found that as ℏ increases with δfixed, the nearest neighbor separation between quasi-energy eigenvalues changes from showing no energy level repulsion to energy level repulsion. This transition, from Poisson-like statistics to Gaussian-Orthogonal-Ensemble-like statistics is tested by looking at the distribution of quasi-energy level nearest neighboor separations and the Δ3(L) statistics. These results are also correlated to a transition between localized to extended states in energy space. The possible relevance of the results presented here to experiments in quasi-one-dimensional atoms is also discussed.


Extended State Dimensional Hilbert Space Weak Coupling Regime Infinite Dimensional Hilbert Space Wall Oscillation 
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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Jorge V. José
    • 1
  1. 1.Department of PhysicsNortheastern UniversityBoston

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