Canonical transformation

About: Canonical transformation is a research topic. Over the lifetime, 1854 publications have been published within this topic receiving 38019 citations.

Canonical transforms. III. Configuration and phase descriptions of quantum systems possessing an sl (2,R) dynamical algebra

Abstract: The purpose of this article is to present a detailed analysis on the quantum mechnical level of the canonical transformation between coordinate‐momentum and number‐phase descriptions for systems possessing an sl (2,R) dynamical algebra, specifically, the radial harmonic oscillator and pseudo‐Coulomb systems. The former one includes the attractive and repulsive oscillators and the free particle, each with an additional ’’centrifugal’’ force, while the latter includes the bound, free and threshold states with an added ’’centrifugal’’ force. This is implemented as a unitary mapping—canonical transform—between the usual Hilbert space L2 of quantum mechanics and a new set of Hilbert spaces on the circle whose coordinate has the meaning of a phase variable. Moreover, the UIR’s D+k of the universal covering group of SL (2,R) realized on the former space are mapped unitarily onto the latter.

The Hamiltonian Formulation

Abstract: In this chapter we return to the Hamiltonian formulation of the NS model in order to discuss the basic transformation of the inverse scattering method from the Hamiltonian standpoint. We shall describe the Poisson structure on the scattering data of the auxiliary linear problem induced through f from the initial Poisson structure defined in Chapter I. Under the rapidly decreasing or finite density boundary conditions, the NS model proves to be a completely integrable system, with f defining a transformation to action-angle variables. In particular, we will show that the integrals of the motion introduced in Chapter I are in involution. In these terms scattering of solitons amounts to a simple canonical transformation.

Resonances of the SD oscillator due to the discontinuous phase

Abstract: Resonance phenomena of a harmonically excited system with mul-tiple potential well play an important role in nonlinear dynamics research. In this paper, we investigate the resonant behaviours of a discontinuous dynamical system with double well potential derived from the SD oscillator to gain better understanding of the transition of resonance mechanism. Firstly, the time dependent Hamiltonian is obtained for a Duffing type discontinuous system modelling snap-through buckling. This system comprises two subsystems connected at x = 0, for which the system is discontinuous. We construct a series of generating functions and canonical transformations to obtain the canonical form of the system to investigate the complex resonant behaviours of the system. Furthermore, we introduce a composed winding number to explore complex resonant phenomena. The formulation for resonant phenomena given in this paper generalizes the formulation of n Omega0 = m Omega used in the regular perturbation theory, where n and m are relative prime integers, Omega 0 and Omega are the natural frequency and external frequencies respectively. Understanding the resonant behaviour of the SD oscillator at the discontinuous phase enables us to further reveal the vibrational energy transfer mechanism between smooth and discontinuous nonlinear dynamical systems

Influence of the External Classical Field on the Entanglement of a Two-Level Atom

Abstract: The atomic and the field entropies of a two-level atom, which is additionally driven by an external classical field are investigated. Under a certain canonical transformation for the excited and ground states the system is transformed into the usual JCM. Using the equations of motion in the Heisenberg picture exact solutions for the time-dependent dynamical operators are obtained. The entanglement between atom-field system is studied by using the atomic and the field entropies. Also we use the concurrence to detect the sudden death phenomenon and the relationship between entropies and the concurrence of the entanglement are discussed. It is shown that the amount of entanglement, the atomic and the field entropies of the subsystem can be improved by controlling the external classical field.