The first considers asymmetry-driven Hamiltonian ratchet effects in context of quantum chaos. It is shown, in the Liouville framework, that a biased net effective force can emerge in initially symmetric systems subject to a force of zero mean. This arises via an asymmetric system distortion in both classical and quantum mechanics. However, differences in time-periodicity and chaos between the two mechanics lead to differences in the evolution of the effective force in time. The second part obtains the canonical commutation relation of quantum mechanics assuming, in addition to the standard assumptions of the quantum Hilbert space, only that position and momentum are functionally independent. This exposes the fundamental interpretation of the commutation relation as an expression of independence, and thereby establishes its analogy with the classical Poisson bracket. Operator techniques used in these arguments are discussed at length.
The first considers asymmetry-driven Hamiltonian ratchet effects in context of quantum chaos. It is shown, in the Liouville framework, that a biased net effective force can emerge in initially symmetric systems subject to a force of zero mean. This arises via an asymmetric system distortion in both classical and quantum mechanics. However, differences in time-periodicity and chaos between the two mechanics lead to differences in the evolution of the effective force in time. The second part obtains the canonical commutation relation of quantum mechanics assuming, in addition to the standard assumptions of the quantum Hilbert space, only that position and momentum are functionally independent. This exposes the fundamental interpretation of the commutation relation as an expression of independence, and thereby establishes its analogy with the classical Poisson bracket. Operator techniques used in these arguments are discussed at length.
Previous work [Gong and Brumer, Phys. Rev. Lett. 97, 240602 (2006)] motivates this study as to how asymmetry-driven quantum ratchet effects can persist despite a corresponding fully chaotic classical phase space. A simple perspective of ratchet dynamics, based on the Heisenberg picture, is introduced. We show that ratchet effects are in principle of common origin in classical and quantum mechanics, although full chaos suppresses these effects in the former but not necessarily the latter. The relationship between ratchet effects and coherent dynamical control is noted.
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.
The possibility of quantum chaos for vectorial interactions is here described in some detail, along with its apparent suppression when the radiative corrections of quantum field theory (quantum fluctuations of the classical, “external” electromagnetic field) are introduced. Based on these quantum-mechanical ideas, an application is made to classical-chaotic systems, of perhaps the simplest form – a forced Duffing model, without damping – where it is found that the chaos is first suppressed and then (apparently) removed by introducing couplings to random and/or chaotic sources. This may be characterized as “quantum mechanics with ħ ∼ 1”, and suggests a brute-force method by which the chaos of a classical system may be at least diminished. A similar effect is noted for a different classical system that displays chaos – a pair of coupled oscillators – independently of any external forcing. First-quantization chaos The reader is now asked to return to the G c [ A ] representations for vectorial interactions of the previous chapter; for simplicity, the arguments of this chapter are presented only for QED in a relativistic context, but have obvious generalizations to non-relativistic QED, and to relativistic QCD. The map (5.31) defines the quantity Ω μ ( s′ ), which is needed for the explicit solution of (5.33). It is the existence of such a map which carries with it the inescapable possibility of chaotic behavior, at least in the present context of vectorial interactions in potential theory. The analysis used here is given directly in terms of proper time τ, of which x μ (τ) is a function.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dtdP, P=mdtdV, and [X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
The conventional phase space of classical physics treats space and time differently, and this difference carries over to field theories and quantum mechanics (QM). In this paper, the phase space is enhanced through two main extensions. First, we promote the time choice of the Legendre transform to a dynamical variable. Second, we extend the Poisson brackets of matter fields to a spacetime symmetric form. The ensuing ``spacetime phase space'' is employed to obtain an explicitly covariant version of Hamilton equations for relativistic field theories. A canonical-like quantization of the formalism is then presented in which the fields satisfy spacetime commutation relations and the foliation is quantum. In this approach, the classical action is also promoted to an operator and retains explicit covariance through its nonseparability in the matter-foliation partition. The problem of establishing a correspondence between the new noncausal framework (where fields at different times are independent) and conventional QM is solved through a generalization of spacelike correlators to spacetime. In this generalization, the Hamiltonian is replaced by the action, and conventional particles by off-shell particles. When the foliation is quantized, the previous map is recovered by conditioning on foliation eigenstates, in analogy with the Page and Wootters mechanism. We also provide an interpretation of the correspondence in which the causal structure of a given theory emerges from the quantum correlations between the system and an environment. This idea holds for general quantum systems and allows one to generalize the density matrix to an operator containing the information of correlators both in space and time.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.