2012/03/12 by Robert Salazar, R. Salazar, Gabriel Téllez +9
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Mathematical Dynamics and Fractals #Quantum Physics (quant-ph) #Quantum chaos and dynamical systems #nlin.CD #quant-ph
paper · pdf · doi:10.48550/arxiv.1205.4990
arxiv created 2012/03/12 · openalex publication_date 2012/03/12 · arxiv updated 2012/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We analyse the classical and quantum behaviour of a particle trapped in a diamond shaped billiard. We defined this billiard as a half stadium connected with a triangular billiard. A parameter ξ which gradually change the shape of the billiard from a regular equilateral triangle (ξ=1) to a diamond (ξ=0) was used to control the transition between the regular and chaotic regimes. The classical behaviour is regular when the control parameter ξ is one; in contrast, the system is chaotic when ξ≠ 1 even for values of ξ close to one. The entropy grows fast as ξ is decreased from 1 and the Lyapunov exponent remains positive for ξ<1. The Finite Difference Method was implemented in order to solve the quantum problem. The energy spectrum and eigenstates were numerically computed for different values of the control parameter. The nearest-neighbour spacing distribution is analysed as a function of ξ, finding a Poisson and a Gaussian Orthogonal Ensemble(GOE) distribution for regular and chaotic regimes respectively. Several scars and bouncing ball states are shown with their corresponding classical periodic orbits. Along the document the classical chaos identifiers are computed to show that system is chaotic. On the other hand, the quantum counterpart is in agreement with the Bohigas-Giannoni-Schmit conjecture and exhibits the standard features for chaotic billiard such as the scarring of the wavefunction.