Canonical transformation

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

Time dependence of operators in minimal and multipolar nonrelativistic quantum electrodynamics. II. Analysis of the functional forms of operators in the two frameworks

Abstract: The relationship between the multipolar and minimal Hamiltonians through a canonical transformation is used to analyze the time dependence of quantum operators in the two formalisms. It is shown that operators dependent on particle position, the vector potential, and their time derivatives have the same time dependence. However, operators such as the photon number and the atomic population number evolve differently in the two cases. Expressions correct to second order in the electric-dipole moments for these are given. The expectation values of the photon number operator are calculated in the two formalisms and are used to predict natural line shapes. These theoretical shapes differ. The observed shape will depend on the particular setup of the experiment involving the radiative decay of an excited atom. A comparison with the theoretical predictions will determine which of the two frameworks is most appropriate to describe the decay. Finally, the energy density of the electromagnetic field in the neighborhood of an atom is calculated within the two formalisms.

Canonical maps near separatrix in Hamiltonian systems.

Abstract: A systematic and rigorous method to construct symplectic maps near separatrix of generic Hamiltonian systems subjected to time-periodic perturbations is developed. It is based on the method of canonical transformation of variables to construct Hamiltonian maps [S. S. Abdullaev, J. Phys. A 35, 2811 (2002)]. Using canonical transformation of variables and the first-order approximation for the generating function, the general form of mapping in terms of time and energy variables is obtained. Different limiting cases of the mapping are considered. The method is illustrated for simple Hamiltonian systems with one and a large number of saddle points. It is also applied to derive mappings for the periodic-driven Morse oscillator describing the process of stochastic excitation and dissociation of diatomic molecules. The so-called canonical Kepler map is derived for the one-dimensional hydrogen atom in a microwave field.

Global dynamics of a flexible asymmetrical rotor

Abstract: Global dynamics of a flexible asymmetrical rotor resting on vibrating supports is investigated. Hamilton’s principle is used to derive the partial differential governing equations of the rotor system. The equations are then transformed into a discretized nonlinear gyroscopic system via Galerkin’s method. The canonical transformation and normal form theory are applied to reduce the system to the near-integrable Hamiltonian standard forms considering zero-to-one internal resonance. The energy-phase method is employed to study the chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. In both the Hamiltonian and the dissipative perturbation, the homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated. In the case of damping dissipative perturbation, the existence of generalized Silnikov-type multi-pulse orbits which are homoclinic to fixed points on the slow manifold is examined and the parameter region for which the dynamical system may exhibit chaotic motions in the sense of Smale horseshoes is obtained analytically. The global results are finally interpreted in terms of the physical motion of axis orbit. The present study indicates that the existence of multi-pulse homoclinic orbits provides a mechanism for how energy may flow from the high-frequency mode to the low-frequency mode when the rotor system operates near the first-order critical speed.

Geometry and Duality in Supersymmetric sigma-Models

Abstract: The Supersymmetric Dual Sigma Model (SDSM) is a local field theory introduced to be nonlocally equivalent to the Supersymmetric Chiral nonlinear sigma-Model (SCM), this dual equivalence being proven by explicit canonical transformation in tangent space. This model is here reconstructed in superspace and identified as a chiral-entwined supersymmetrization of the Dual Sigma Model (DSM). This analysis sheds light on the Boson-Fermion Symphysis of the dual transition, and on the new geometry of the DSM.