Canonical transformation

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

Strong coupling expansion of the extended Hubbard model with spin-orbit coupling

Abstract: We study the strong coupling limit of the extended Hubbard model in two dimensions. The model consists of hopping, on-site interaction, nearest-neighbor interaction, spin-orbit coupling, and Zeeman spin splitting. While the study of this model is motivated by a search for topological phases, and in particular, a topological superconductor, the methodology we develop may be useful for a variety of systems. We begin our treatment with a canonical transformation of the Hamiltonian to an effective model, which is appropriate when the (quartic) interaction terms are larger than the (quadratic) kinetic and spin-orbit coupling terms. We proceed by analyzing the strong coupling model variationally. Since we are mostly interested in a superconducting phase, we use a Gutzwiller projected BCS wave function as our variational state. To continue analytically, we employ the Gutzwiller approximation and compare the calculated energy with Monte Carlo calculations. Finally, we determine the topology of the ground state and map out the topology phase diagram.

Functions Whose Poisson Brackets Are Constants

Abstract: A local normal form under canonical transformation is found for n independent functions of 2n variables, with the condition that the Poisson bracket of each pair of the functions be constant. The normal form, closely related to work of Lie, is used to prove a conjecture of Avez on 1‐forms in involution and to obtain a criterion for n independent functions of 2n variables to be extendable to a canonical coordinate system. The last result has been obtained in different ways by Lie and Kruskal.

Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant

Abstract: We consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be anisotropic and the magnetic field is applied along z-axis. A canonical transformation is invoked to remove the interaction term and the system is reduced to a model contains two uncoupled harmonic oscillators. Two classes of real and complex quadratic invariants (constants of motion) are obtained. We employ the Lie algebraic technique to find the most general solution for the wave-function for both real and complex invariants. The quadratic invariant is also used to derive two classes of creation and annihilation operators from which the wave-functions in the coherent states and number states are obtained. Some discussion related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself is also given.

Canonical analysis of Brans-Dicke theory addresses Hamiltonian inequivalence between the Jordan and Einstein frames

Abstract: The Jordan and Einstein frames are studied under the light of the Hamiltonian formalism. Dirac's constraint theory for Hamiltonian systems is applied to Brans-Dicke theory in the Jordan frame. In both the Jordan and Einstein frames, Brans-Dicke theory has four secondary first class constraints and their constraint algebra is closed. We show, contrary to what is generally believed, the Weyl (conformal) transformation, between the two frames, is not a canonical transformation, in the sense of the Hamiltonian formalism. This addresses quantum mechanical inequivalence as well. A canonical transformation is shown.

Morse, Lennard-Jones, and Kratzer Potentials: A Canonical Perspective with Applications

Abstract: Canonical approaches are applied to classic Morse, Lennard-Jones, and Kratzer potentials. Using the canonical transformation generated for the Morse potential as a reference, inverse transformations allow the accurate generation of the Born–Oppenheimer potential for the H2+ ion, neutral covalently bound H2, van der Waals bound Ar2, and the hydrogen bonded one-dimensional dissociative coordinate in a water dimer. Similar transformations are also generated using the Lennard-Jones and Kratzer potentials as references. Following application of inverse transformations, vibrational eigenvalues generated from the Born–Oppenheimer potentials give significantly improved quantitative comparison with values determined from the original accurately known potentials. In addition, an algorithmic strategy based upon a canonical transformation to the dimensionless form applied to the force distribution associated with a potential is presented. The resulting canonical force distribution is employed to construct an algorith...