The correspondence limit is illustrated for the periodically kicked pendulum. The classical dynamics of this system can be represented by a discrete map. A corresponding quantum map is derived and then rendered in the Ehrenfest formulation for expectation values. The Ehrenfest representation is studied for a minimum uncertainty Gaussian wave packet. It is shown that as \ensuremath{\Elzxh} gets very small, followed by a decrease in the variance of the wave packet, the quantum map shadows the classical map with an error approaching zero for a length of time approaching infinity. The Gaussian form of the wave packet is preserved by the time evolution provided ${\mathrm{\ensuremath{\omega}}}_{0}$T\ensuremath{\ll}1. This constraint implies that the classical map is predominantly not chaotic with very small regions of very weak chaos. As ${\mathrm{\ensuremath{\omega}}}_{0}$T approaches 1, where the classical map becomes strongly chaotic, the propagation in time of the Gaussian wave packet completely breaks down. The possible significance of this breakdown is discussed.
The correspondence limit is illustrated for the periodically kicked pendulum. The classical dynamics of this system can be represented by a discrete map. A corresponding quantum map is derived and then rendered in the Ehrenfest formulation for expectation values. The Ehrenfest representation is studied for a minimum uncertainty Gaussian wave packet. It is shown that as \ensuremath{\Elzxh} gets very small, followed by a decrease in the variance of the wave packet, the quantum map shadows the classical map with an error approaching zero for a length of time approaching infinity. The Gaussian form of the wave packet is preserved by the time evolution provided \ensuremathω0T\ensuremath{\ll}1. This constraint implies that the classical map is predominantly not chaotic with very small regions of very weak chaos. As \ensuremathω0T approaches 1, where the classical map becomes strongly chaotic, the propagation in time of the Gaussian wave packet completely breaks down. The possible significance of this breakdown is discussed.
The initial quantum dynamics of an initially well-localized, minimum-uncertainty Gaussian wave packet is studied analytically for the periodically kicked pendulum. The general results presented here include our earlier analytic result [Phys. Rev. A 41, 2952 (1990)] as a special case. We find that as long as the packet remains localized, it remains approximately Gaussian, while its center and widths evolve in time approximately according to, respectively, the classical map and a variance map which utilize the instantaneous values of the classical Jacobi matrix. When the classical map is very strongly chaotic, we predict that the average kinetic energy of the well-localized wave packet does not initially grow diffusively (linearly with time). We show that, instead, in the presence of chaos, the deviations initially grow, on the average, exponentially at a rate equal to the corresponding classical Liapunov exponent under the appropriate conditions. The time scale (Ehrenfest time) on which the Ehrenfest-Hamilton correspondence holds in the presence of chaos is estimated and is slightly different from a previous estimate made by Chirikov, Israilev, and Shepelyansky [Sov. Sci. Rev. C 2, 209 (1981)]. Numerical evidence in support of these analytical results is presented.
The behavior of the periodically kicked pendulum is determined classically and quantum mechanically. Our earlier analytic study [Phys. Rev. A 41, 2952 (1990)] is confirmed and extended by the numerical results presented here. We are able to construct minimum uncertainty, Gaussian wave packets that maintain their form, and shadow the corresponding classical motion for long times when the classical motion is nonchaotic or only weakly chaotic. However, when the classical motion is robustly chaotic, the quantum-mechanical probability distribution grows macroscopically broad exponentially fast and the quantum-classical correspondence breaks down.
The trajectory predicted by Bohm's quantum equation of motion, using an initially angle-localized minimum-uncertainty Gaussian wave packet, is compared with the trajectory predicted by Newton's classical equation of motion for a macroscopic periodically kicked pendulum system. We find that, in violation of the strong form of the correspondence principle, the Bohmian trajectory does not, in general, agree or correspond with the Newtonian trajectory for all times. The Bohm-Newton trajectory correspondence holds as long as the wave packet remains localized in angle. Delocalization of the wave packet in angle triggers the breakdown of the correspondence. The breakdown occurs exponentially quickly if either the Newtonian trajectory that initially coincides with the Bohmian trajectory or one of its classical neighbors is chaotic. In contrast, the breakdown occurs much slower if neither the Newtonian trajectory nor its neighbors are chaotic. These behaviors of the Bohm-Newton trajectory correspondence are generic, or universal, for the class of macroscopic Hamiltonian dynamical systems with bound coordinate space since the kick pendulum is a prototypical member. We predict that neither Bohm's equation of motion nor Newton's correctly predicts the experimentally observed and measured trajectory of a macroscopic system after the breakdown of the correspondence, implying that the all-times agreement required by the strong form of the correspondence principle is unnecessary.
The problem ``what is the quantum signature of a classically chaotic system'' is studied for the periodically kicked top. We find that the quantum variances initially grow exponentially if the corresponding classical description is chaotic. The rate of growth is connected to the corresponding classical Jacobi matrix and, thereby, to the local, classical, transient expansion rate. These connections were recently established in an analysis of the kicked pendulum for the correspondence between quantum Husimi-O'Connell-Wigner distributions and classical Gaussian ensembles. Here, we present closely related results for the kicked top by using generalized coherent states. An explanation is given for why this quantum signature of classical chaos was missed in earlier studies of the kicked top.
Two approaches to quantum-classical correspondence are distinguished according to the classical dynamical theory with which quantum theory is compared. The first of these, Ehrenfest correspondence, defines a dynamical regime in which the quantum expectation values follow approximately a classical trajectory. The second of these, Liouville correspondence, applies when the quantum probability distributions remain well approximated by a density in the classical phase space. The former applies only for narrow states, whereas the latter may remain valid even for quantum states that have spread to the system size. A spin model is adopted for this correspondence study because the quantum state is discrete and finite- dimensional, and thus no artificial truncation of the Hilbert space is required. The quantum time-evolution is given by a discrete unitary mapping. The corresponding classical model is volume-preserving (non-dissipative) and the time-evolution is given by a symplectic map. In classically chaotic regimes, the widths of initially narrow quantum states grow, on average, exponentially with time, until saturation at the system size. This initial spreading rate is well approximated by the classical Lyapunov exponent when the accessible classical phase space is predominantly chaotic. Because of the exponential growth rate of the quantum variance, the Ehrenfest regime is delimited by a break-time that grows logarithmically with increasing quantum numbers. The small differences between quantum expectation values and corresponding Liouville averages of dynamical variables also grow exponentially, initially, if the classical behaviour is chaotic. This exponential rate is independent of h and consistently larger than the classical Lyapunov exponent by at least a factor of two. Interestingly, this exponential growth rate does not continue until these differences approach magnitudes of order the system size, but crosses over to power-law growth (or simply saturates), when the differences have reached magnitudes no larger than order ħ. Because of the exponential growth of these quantum-classical differences, the Liouville regime is also delimited by a break-time that grows logarithmically with increasing quantum numbers. However, due to the early saturation of the exponential growth of the differences, the Liouville log break-time applies only in a restricted domain. After spreading to the system size, the quantum states relax to an approximately time-independent state. This is characterised by equilibrium distributions that are subject to rapidly oscillating fluctuations about the coarse-grained classical steady-state. If the accessible classical phase space is predominantly chaotic, then the equilibrium quantum probability distributions are nearly microcanonical. Otherwise the steady-state quantum probability distributions accurately reflect the details of the KAM surfaces in the classical phase space. The equilibrium regime quantum fluctuations, which are quantum-classical differences, are shown to approach zero as a negative power of the quantum numbers. Since the differences between quantum expectation values and individual classical trajectories grow exponentially in time to a magnitude that scales with the system size, the Ehrenfest regime is delimited by an unrestricted log break-time. The log break-time may be short compared to experimental time-scales for some chaotic macroscopic bodies, and, as a result, an observable breakdown of Ehrenfest correspondence may arise for some macroscopic systems. Although the Liouville regime is also delimited by a log break-time, an observable breakdown of correspondence between quantum mechanics and Liouville mechanics for macroscopic bodies is not likely since their differences, expressed relative to the dimensions of the observable, approach zero in the limit of large quantum numbers.
In this thesis, a comparison between quantum and classical mechanics was made for two systems whose classical behaviour can be chaotic.The general correspondence between classical mechanics and quantum mechanics in the classical regime has always been assumed but never proved.Recent research in the field of quantum chaos has brought this assumption into question.For some systems, such as the kicked rotor in the classically chaotic regime, quantum mechanics does not display the classical behaviour of unbounded momentum.It has been suggested that quantum mechanics will fail to produce classical chaos for a broad range of simple systems.A comparison of quantum and classical mechanics was made using numerical simulations, without recourse to taking limits such as 7i going to zero.The two systems studied were a driven pendulum and a pair of pendula joined by a spring.The quantum and classical expectation values for a number of observables were computed, as were the probability distributions in phase space.For periodic motion of the driven pendulum it was found that the theories agreed well for times up to 150 driving periods.Similar agreement was found for initial conditions whose classical behaviour was chaotic, but only for times up to a few tens of driving periods.For longer times, it was found that the quantum system displayed a boundedness in position and/or momentum that was absent in the classical system.A possible explanation was given in terms of the quasienergy expansion.In the coupled-pendula system no such disagreements between the theories were found.I would like to thank Dr. Ballentine for his continued support during this project and his help in editing and revising this thesis.I would like to thank the members of my committee, Drs.Bechhoefer, Boa1 and Loss for their many suggestions.For many helpful discussions, I would like to thank Dominic Mimnagh and Jim Zibin.