Showing papers on "Canonical coordinates published in 1983"

The quantum dynamics of three particles in hyperspherical coordinates

Abstract: A derivation of the quantum mechanical wave equation for the three body problem expressed in hyperspherical coordinates is presented. The coordinates, due to Smith and Whitten and later modified by Johnson [B. R. Johnson, J. Chem. Phys. 73, 5051 (1980)], are used in this study. The analysis is presented from a point of view that emphasizes the role of the three‐dimensional configuration space that is associated with these coordinates. Three types of motion are analyzed: full three dimensional; motion restricted to a plane surface; and zero angular momentum motion.

The actual content of quantum theoretical kinematics and mechanics

Abstract: First, exact definitions are supplied for the terms: position, velocity, energy, etc. (of the electron, for instance), such that they are valid also in quantum mechanics. Canonically conjugated variables are determined simultaneously only with a characteristic uncertainty. This uncertainty is the intrinsic reason for the occurrence of statistical relations in quantum mechanics. Mathematical formulation is made possible by the Dirac-Jordan theory. Beginning from the basic principles thus obtained, macroscopic processes are understood from the viewpoint of quantum mechanics. Several imaginary experiments are discussed to elucidate the theory.

Generalized canonical transformations for time‐dependent systems

Abstract: We introduce the concept of generalized canonical transformations as symplectomorphisms of the extended phase space. We prove that any such transformation factorizes in a standard canonical transformation times another one that changes only the time variable. The theory of generating functions as well as that of Hamilton–Jacobi is developed. Some further applications are developed.

Molecular translation—rovibronic hamiltonian. II. Linear molecules

Abstract: The transformation of the translational—rovibronic (non-relativistic) hamiltonian from cartesian to generalized coordinates by the quantum-mechanical path outlined in the previous work has been carried out for linear molecules.

The description of molecular large-amplitude motions in terms of curvilinear coordinates associated with the reaction path

Abstract: As a generalization of an original idea by Miller et al., a set of coordinates suitable for the description of any N -atom reactive molecular system is given: (i) n + 1 coordinates allowing the description of any large-amplitude motion: (ii) 3 N -7- n local normal coordinates, and (iii) six rotation-translation coordinates. An ( n + 1)-dimensional manifold (referred to the n + 1 coordinates (i) and imbedded in the 3 N -dimensional configuration space) containing and generalizing the reaction path is introduced. The classical and quantum-mechanical hamiltonians in terms of the above set of coordinates are obtained by particularizing the method by Chapuisat et al. for deriving the classical hamiltonian in terms of any set of curvilinear coordinates. Finally, the geometrical construction of the ( n + 1)-dimensional manifold is sketched.

On relativistic quantum theory

Abstract: The construction of a formalism for the description of the dynamical properties of many-body systems which is consistent with special relativity, in the framework of classical or quantum theory, is still far from complete. Dime’s proposal,1 in 1949, of Hamiltonian-type theories, developed2,3 into an action-at-a-distance theory with the Lorentz group canonically represented. Currie, Jordan, and Sudarshan,4 however, showed that when the canonical coordinates are taken as the physical coordinates of point particles and their world lines are Lorentz invariant, the only possible dynamics is that of free particles. The history of these developments, and an invaluable collection of reprinted papers, has been provided by Kerner.4 Rohrlich and King5,6 have developed a Hamiltonian theory in which constraint equations control the masses of individual particles and also have the role of generating the evolution of the system.7 For N-body systems, the requirements of compatibility imply the existence of N-body forces (corresponding to the potential functions appearing in the constraint equations).

Transition to special canonical coordinates for systems with constraints

Abstract: When using the Dirac hamiltonization of Lagrange systems with constraints, it is convenient to perform a canonical transformation such that the constraints become linear combinations of only a subset of the new variables, while the primary constraints can be identified with some of the variables belonging to this subset. We prove the existence of such canonical transformation, as well as the possibility of separation of first-class constraints.

Analysis Of Three-Dimensional Time Varying Scene

Abstract: We present some of the highlights of the recent advancements in the uniqueness and estimation of 3-D motion parameters and object surface structure from perspective views, a key issue in the analysis of 3-D time varying scene. For estimating planar patch motion from two views, there are two solutions in general, unless the 3 x 3 matrix containing the canonical coordinates of a Lie group has multiple singular values. Closed form solutions are derived analytically. The solutions would be unique if three views are given. For curved surface motion, two theorems, one lemma and a collection of corollaries on the conditions of uniqueness of solution are given. Closed form solutions for the nonlinear motion equations are derived. Results of simulation and real experiments are discussed.

Canonical transformation method in classical electrodynamics

Abstract: The solutions of Maxwell's equations in the parabolic equation approximation is obtained on the basis of the canonical transformation method. The Hamiltonian form of the equations for the field in an anisotropic stratified medium is also examined. The perturbation theory for the calculation of the wave reflection and transmission coefficients is developed.

Canonical forms for nonlinear systems

Abstract: Necessary and sufficient conditions for transforming a nonlinear system to a controllable linear system have been established, and this theory has been applied to the automatic flight control of aircraft. These transformations show that the nonlinearities in a system are often not intrinsic, but are the result of unfortunate choices of coordinates in both state and control variables. Given a nonlinear system (that may not be transformable to a linear system), we construct a canonical form in which much of the nonlinearity is removed from the system. If a system is not transformable to a linear one, then the obstructions to the transformation are obvious in canonical form. If the system can be transformed (it is called a linear equivalent), then the canonical form is a usual one for a controllable linear system. Thus our theory of canonical forms generalizes the earlier transformation (to linear systems) results. Our canonical form is not unique, except up to solutions of certain partial differential equations we discuss. In fact, the important aspect of this paper is the constructive procedure we introduce to reach the canonical form. As is the case in many areas of mathematics, it is often easier to work with the canonical form than in arbitrary coordinate variables.

The problem of physical coordinates in predictive Hamiltonian systems

Abstract: In the Hamiltonian formulation of predictive relativistic systems, the canonical coordinates cannot be the physical positions. The relation between them is given by the individuality differential equations. However, due to the arbitrariness in the choice of Cauchy data, there is a wide family of solutions for these equations. In general, those solutions do not satisfy the condition of constancy of velocities’ moduli, and therefore we have to reparametrize the world lines into the proper time. We derive here a condition on the Cauchy data for the individuality equations which ensures the constancy of the velocities’ moduli and makes the reparametrization unnecessary.