Classical and quantum mechanics with chaos

Abstract

This thesis is concerned with the study, classically and quantum mechanically, of the square billiard with particular attention to chaos in both cases. Classically, we show that the rotating square billiard has two regular limits with a mixture of order and chaos between, depending on an energy parameter, E. This parameter ranges from -2w(^2) to oo, where w is the angular rotation, corresponding to the two integrable limits. The rotating square billiard has simple enough geometry to permit us to elucidate that the mechanism for chaos with rotation or curved trajectories is not flyaway, as previously suggested, but rather the accumulation of angular dispersion from a rotating line. Furthermore, we find periodic cycles which have asymmetric trajectories, below the value of E at which phase space becomes disjointed. These trajectories exhibit both left and right hand curvatures due to the fine balance between Centrifugal and Coriolis forces. Quantum mechanically, we compare the spectral analysis results for the square billiard with three different theoretical distribution functions. A new feature in the study is the correspondence we find, by utilising the Berry-Robnik parameter q, between classical E and a quantum rotation parameter w. The parameter q gives the ratio of chaotic quantum phase volume which we can link to the ratio of chaotic phase volume found classically for varying values of E. We find good correspondence, in particular, the different values of q as w is varied reflect the births and subsequent destructions of the different periodic cycles. We also study wave packet dynamics, necessitating the adaptation of a one dimensional unitary integration method to the two dimensional square billiard. In concluding we suggest how this work may be used, with the aid of the chaotic phase volumes calculated, in future directions for research work.