We consider the problem of a free particle in a rigid box with one wall fixed and the other oscillating periodically in time. The spectral properties of the evolution operator are calculated. We find that by a tuning of Planck's constant, the statistics of the quasienergy spectrum go continuously from showing Poisson to Gaussian-orthogonal-ensemble-like properties. This change is related to a transition from localized to extended states in energy space. We make the conjecture that recent results of microwave excitations on quasi-one-dimensional higly excited hydrogen atoms may be related to this work. PACS numbers; 05.45. + b, 02.90.+p, 03.20.+i, 03.65-w An important fundamental question of current interest is to find the characteristics of a quantum problem defined by a Hamiltonian //which, at the classical level, shows chaotic solutions. Of interest in this paper is the conjecture that different statistical properties of the spectrum of a quantum problem may be correlated with the transition between periodic and aperiodic classical behavior. 1 The basic idea is that Wigner's level repulsion corresponds to nonintegrability, whereas integrability implies Poisson statistics. 2 Convincing evidence supporting the validity of this conjecture was presented by Bohigas, Giannoni, and Schmit, 3 following earlier tests given by other authors. 4 Bohigas, Giannoni, and Schmit studied the quantum Sinai's billiard and stadium models, which are known to be classically K systems. These models are described by time-independent Hamiltonians. Their main finding is that the spectrum of //has an energy-level distribution that corresponds to the Gaussian orthogonal ensemble (GOE), studied extensively in random matrix theory. This type of level statistics has been used extensively to describe successfully the spectra of complicated nuclei, atomic, and molecular systems. 2 Studies of polynomial Hamiltonians with chaotic and periodic regions in phase space have led to numerical evidence showing that the spectrum of //has GOE statistics in the chaotic regions and Poisson statistics in the periodic ones. 5 Furthermore, there is evidence that ZTs without time-reversal invariance have spectra with Gaussianunitary-ensemble statistics. 6 Recently, Berry has provided a semiclassical analytic proof that associates a GOE to nonintegrable regions of phase space, and Poisson properties to integrable regions. 7 We can conclude that, for the time-independent quantum problems studied thus far, the conjecture mentioned above is supported.The next question to ask is if the conjecture has any meaning for time-dependent quantum problems. A large body of work has been produced from the study of the periodically kicked quantum rigid-rotor model (PKQRRM). 8 " 10 This model has been studied extensively in the classical case, and its behavior is considered the prototype for a two-degree-of-freedom nonlinear problem 11 : It has both periodic and chaotic regions in phase space that can be changed by variation of the coupling-constant parameter in the model. ...
A lattice model for equilibrium polymerization which allows for loop polymers is presented with critical behavior described by n = 1, rather thanra-*0 predicted by the theory of Wheeler, Kennedy, and Pfeuty. The specific heat of sulfur above and below T c is consistent with Ising critical behavior.
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