Showing papers on "Canonical transformation published in 1978"
TL;DR: In this article, the authors study the classical canonical transformations to action and angle variables for the repulsive and attractive oscillator and the free particle and find the ambiguity group for the three problems mentioned, which connects points in one phase space that are mapped on a single point in the other.
67 citations
TL;DR: In this paper, the canonical formalism of a singular-Lagrangian model describing the interaction between two relativistic particles is studied and the covariance and quantization of the model are discussed.
Abstract: The canonical formalism of a singular-Lagrangian model describing the interaction between two relativistic particles is studied. Instead of following the Dirac method, we make use of a canonical transformation that enables us to work in the complete phase space. The covariance and the quantization of the model are discussed.
40 citations
TL;DR: In this article, the authors examined the relationship between the two representations and showed that the correct transformation cannot be strictly canonical because of the presence of projection operators, which has some relevance to the Arima-Iachello model.
31 citations
TL;DR: In this article, the authors analyze the representations in quantum mechanics of classical canonical transformations that are non-bijective, i.e., not one-to-one onto, and show that the canonical transformation can be related to the conformal transformation w=zκ, which again is κ to 1 and where a group Cκ also appears.
Abstract: In the present paper we analyze the representations in quantum mechanics of classical canonical transformations that are nonbijective, i.e., not one‐to‐one onto. We take as the central example the canonical transformation that changes the Hamiltonian of a one‐dimensional oscillator of frequency κ−1 into one of frequency k−1 where κ, k are relatively prime integers. For the particular case k=1, the mapping of the original phase space (x,p) onto the new one (x, p) is κ to 1 and the equivalent points in (x,p) are related by a cyclic group Cκ of linear canonical transformations. When formulating this problem in Bargmann Hilbert space, the canonical transformation can be related to the conformal transformation w=zκ, which again is κ to 1 and where a group Cκ also appears. This cyclic group proves fundamental for the determination of representations of the conformal transformation in Bargmann Hilbert space. To begin with, it suggests that while we can take in the original Bargmann Hilbert space a single compo...
28 citations
TL;DR: In this article, an analytical approach for the linear Jahn-Teller effect E −e is presented, which yields analytical expressions for both the ground state and the lowest five excited states over the weak and intermediate coupling range (0?k2?3).
Abstract: An analytical approach is presented for the linear Jahn–Teller effect E–e. This approach yields analytical expressions for both the ground state and the lowest five excited states over the weak and intermediate coupling range (0?k2?3). The method is based on canonical transformations combined with perturbation theory. A comparison of our results with existing numerical solutions to this problem shows good agreement within the weak and intermediate coupling range. Ham’s reduction factors p and q as well as the oscillating behavior of the energy levels for large quantum number n are also discussed.
24 citations
TL;DR: In this article, a canonical transformation of the single-particle Hamiltonian is proposed to eliminate those interaction terms which are linear in the fields, and the new entity (the "oscillation centre") then has no first-order jittering motion, and generalized ponderomotive forces appear as nonlinear terms in the transformed Hamiltonian.
Abstract: A novel approach to the theory of nonlinear mode coupling in hot magnetized plasma is presented. The formulation retains the conceptual simplicity of the familiar ponderomotive-scalar-potential method, but removes the approximations. The essence of the approach is a canonical transformation of the single-particle Hamiltonian, designed to eliminate those interaction terms which are linear in the fields. The new entity (the 'oscillation centre') then has no first-order jittering motion, and generalized ponderomotive forces appear as nonlinear terms in the transformed Hamiltonian. This viewpoint is applied to derive a compact symmetric formula for the general three-wave coupling coefficient in hot uniform magnetized plasma, and to extend the conventional ponderomotive-scalar-potential method to the domain of strongly magnetized plasma.
21 citations
20 citations
TL;DR: In this paper, a method of quantization for models described by singular Lagrangians is proposed, in which a set of variables, suitable for the quantization, is obtained by means of a canonical transformation.
Abstract: In this work we propose a method of quantization for models described by singular Lagrangians. A set of variables, suitable for the quantization, is obtained by means of a canonical transformation. The quantum theory is carried out by working in the complete phase space; in other words, we do not eliminate the variables corresponding to the second-class constraints, as suggested by Dirac. The expectation values of the constraints are required to be zero and the problem of the consistency of these conditions is studied.
16 citations
01 Jan 1978
16 citations
01 Dec 1978
TL;DR: In this paper, a method for determining semiclassical eigenvalues and the canonical transformation relating good and harmonic action-angle variables for nonseparable molecular vibrations is developed, which makes use of Fourier expansions of the harmonic action and good angle variables in terms of harmonic angle variables (for fixed good actions).
Abstract: A method for determining semiclassical eigenvalues and the canonical transformation relating “good” and harmonic action-angle variables for nonseparable molecular vibrations is developed The method makes use of Fourier expansions of the harmonic action and good angle variables in terms of the harmonic angle variables (for fixed good actions) The coefficients in these expansions are determined by requiring that each expansion, when truncated at N terms, be exactly satisfied at N appropriately chosen times during the integration of a trajectory for the system of interest Applications of this method are made to the determination of semi-classical eigenvalues for several model two mode systems which have been extensively studied using other semiclassical methods Essentially exact agreement with these earlier calculations is obtained We then study the dependence of cartesian coordinates and harmonic actions on good angle variables for two of the models Weak correlation between motions in different modes is found, and this leads to a simple but reasonably accurate method for decoupling the two modes based on motional time scale separations
13 citations
TL;DR: In this article, three simple models are discussed in which the linear or parabolic super-Hamiltonian can be cast, essentially by a canonical transformation, into an equivalent hyperbolic form.
Abstract: Three types of super‐Hamiltonians occur in generally covariant field theories: linear in the momenta (hypersurface kinematics), parabolic in the momenta (parametrized field theories on a given Riemannian background), and hyperbolic in the momenta (geometrodynamics). Three simple models are discussed in which the linear or parabolic super‐Hamiltonian can be cast, essentially by a canonical transformation, into an equivalent hyperbolic form: (1) The scalar field propagating on a (1+1) ‐dimensional flat Minkowskian background, (2) hypersurface kinematics on a (1+n) ‐dimensional flat Minkowskian background, and (3) geometrodynamics of a (1+2) ‐dimensional vacuum spacetime. The implications for constraint quantization are mentioned.
TL;DR: In this paper, it was shown that a transformation to a new Hamiltonian independent of position is in general impossible using a trans-formation satisfying this requirement; the best that can be done being a Hamiltonian which is almost everywhere in normal form, but which retains potential barriers of infinitesimal width to reflect trapped particles.
Abstract: The topological constraints imposed on a canonical transformation by the requirement of continuous connection to the identity are considered. In the case of a particle interacting with a periodic potential, it is shown that a transformation to normal form (new Hamiltonian independent of position) is in general impossible using a trans- formation satisfying this requirement; the best that can be done being a transformation to a Hamiltonian which is almost everywhere in normal form, but which retains potential barriers of infinitesimal width to reflect trapped particles. A new Hamiltonian-Jacobi theory based on Lie operator techniques is presented and its relation to the usual theory established. It is suggested that the new method enables the motion of a particle in a random potential to be transformed into an almost Markovian random walk.
01 Jan 1978
TL;DR: In this paper, the feasibility of applying the Lie transform method to the problem of the physical libration of the Moon is investigated, and the Hamiltonian of the problem is brought under a form suitable for perturbation technique.
Abstract: The feasibility of applying the Lie transform method to the problem of the physical libration of the Moon is investigated. By a succession of canonical transformations, the Hamiltonian of the problem is brought under a form suitable for perturbation technique. The mean value of the inclination of the angular momentum upon the ecliptic and the frequencies of the free libration are computed.
TL;DR: In this article, Pfaff's equations of motion were considered as the Euler-Lagrange equations derived from the linear Lagrangian rather than the usual Lagrangians which is quadratic in the velocity components.
Abstract: In this article we study a form of equations of motion which is different from Lagrange's and Hamilton's equations: Pfaff's equations of motion. Pfaff's equations of motion were published in 1815 and are remarkably elegant as well as general, but still they are much less well known. Pfaff's equations can also be considered as the Euler-Lagrange equations derived from the linear Lagrangian rather than the usual Lagrangian which is quadratic in the velocity components. The article first treats the theory of changes of variables in Pfaff's equations and the connections with canonical equations as well as canonical transformations. Then the applications to the perturbed two-body problem are treated in detail. Finally, the Pfaffians are given in Hill variables and Scheifele variables. With these two sets of variables, the use of the true anomaly as independent variable is also considered.
TL;DR: In this article, a new infinite-order perturbation approach for treating strongly anisotropic magnets was developed based on a canonical transformation of a given system into a new system with an effective two-ion anisotropy which can be treated by conventional spinwave techniques.
Abstract: We have developed a new infinite‐order perturbation approach for treating strongly anisotropic magnets. This formalism is based on a canonical transformation of a given system into a new system with an effective two‐ion anisotropy which, for example, can be treated by conventional spin‐wave techniques. An expression for the spin‐wave energy for the general conical magnetic moment configuration with arbitrary single‐ion anisotropy has been derived using this formalism. It was found that an analysis of the transverse spin‐wave spectra can at most result in a determination of a renormalized exchange integral and two renormalized anisotropy constants. A comparison of the theoretical predictions of this formalism with experimentally determined spin‐wave spectra indicate that a substantial part of the large two‐ion anisotropy, which had been previously introduced to describe the spin‐wave spectra in the heavy rare earth series, actually resulted from the fitting procedure which was used and not from any physica...
TL;DR: In this paper, a new canonical transformation of a Schrieffer-Wolff (S-W) type is applied to the Wolff model at the strong correlation limit to derive an antiferromagnetic (kinetic) exchange interaction between the electrons on the impurity site and the host sites.
Abstract: A new canonical transformation of a Schrieffer-Wolff (S-W) type is applied to the Wolff model at the strong correlation limit to derive an antiferromagnetic (kinetic) exchange interaction between the electrons on the impurity site and the host sites. The problem of the singularity in the original S-W transformation is removed. The condition for the formation of localized moments is obtained by solving the Green's function with a moment conserving decoupling scheme. If localized moments exist, the high temperature susceptibility, derived in the mean field theory, has a Curie-Weiss form χ = C/(T + θ) with θ > 0, in agreement with experiments.
TL;DR: In this paper, an exact propagator for a damped oscillator with a time-dependent frequency and a perturbative force was obtained, based on a simple relation between S and the generating function of a canonical transformation involving an explicitly timedependent invariant.
TL;DR: In this article, a canonical transformation of the usual conjugate coordinates and momenta into a set of four variables of the action-angle type was proposed, and the new canonical equations were solved by a method of iteration, and then the solution was used to calculate the change of the adiabatic invariant.
Abstract: The change of the action integral of a charged particle spiraling in a slowly varying longitudinal magnetic field is investigated. The method used to solve this two‐dimensional problem is a generalization of Vandervoort’s analysis of a one‐dimensional system. It is based on a canonical transformation of the usual conjugate coordinates and momenta into a set of four variables of the action‐angle type. The new canonical equations are solved by a method of iteration, and the solution is used to calculate the change of the adiabatic invariant. Our results are analogous to those obtained with the approximations derived by Hertweck and Schluter and by Chandrasekhar. We have compared our analytic results to those obtained by numerical integration of the equation of motion in the case of a magnetic solenoid with a slowly decreasing field.
TL;DR: The canonical transformation of Lee, Low and Pines (1953) is suitably applied to the bound polaron Hamiltonian to obtain its energy levels in a series form with interesting bounds as discussed by the authors.
Abstract: The canonical transformation of Lee, Low and Pines (1953) is suitably applied to the bound polaron Hamiltonian to obtain its energy levels in a series form with interesting bounds and to discuss the hitherto ill-understood role of the intermediate continuum states in an earlier calculation of the energy levels.
01 Jan 1978
TL;DR: In this paper, a generalized Hamiltonian formalism is established which is invariant not only under canonical transformations but also under arbitrary transformations, and the dependent variables, coordinates and momenta, as well as the independent variable are allowed to be transformed.
Abstract: A generalized Hamiltonian formalism is established which is invariant not only under canonical transformations but under arbitrary transformations. Moreover the dependent variables, coordinates and momenta, as well as the independent variable are allowed to be transformed. This is to say that instead of the physical time t another independent variable s is used, such that t becomes a dependent variable or, more precisely, an additional coordinate. The formalism under consideration permits also to include nonconservative forces.
TL;DR: In this paper, a second-order perturbation theory is used to calculate the retardation of intermolecular forces of the inductive type, which is analogous to those of Craig and Power (1969) for the dispersive type of forces but requires both a further canonical transformation of Henneberger form and an addition to the attractive two-photon exchange term of a mixed photon-Coulombic contribution.
Abstract: The calculation of retarded intermolecular forces of the inductive type, normally based on fourth-order perturbation theory or its equivalent, is shown to be possible using only second-order perturbation theory after a series of canonical transformations are used to simplify the initial Hamiltonian. The method is analogous to those of Craig and Power (1969) for the dispersive type of intermolecular forces but requires both a further canonical transformation of Henneberger form and an addition to the attractive two-photon exchange term of a mixed photon-Coulombic contribution which is repulsive. The overall retardation correction is a repulsive potential which is additional to the well known R-4 attraction.
01 Jan 1978
TL;DR: In this paper, a canonical transformation is employed to implement a conformal transformation of the configuration variables of general relativity, so that the spatial constraints become algebraic in the trace of the momentum density.
Abstract: A canonical transformation is employed to implement a conformal transformation of the configuration variables of general relativity. The transformation is so chosen that the spatial constraints become algebraic in the trace of the momentum density. The temporal constraint is then found to have the form of York and O'Murchadha. The role played by the York coordinate condition in decoupling the constraint equations is examined, and a procedure to solve the constraint equations without employing such a coordinate condition is sketched.
TL;DR: In this paper, a canonical transformation is used to investigate a generalized Wolff-Clogston model of dilute alloys including small but finite off-diagonal matrix elements of the impurity potential.
TL;DR: In this paper, the authors investigated time-dependent canonical transformations as applied to polynomial hamiltonians and constructed an invariant tensor for the time-independent three-dimensional isotropic oscillator.
Abstract: This thesis is an investigation of time-dependent canonical transformations as applied to polynomial hamiltonians. The motivation came from two sources. The first was the work of Lewis [74, 7 5, 16, 7 7] on an exact invariant for the time-dependent harmonic oscillator. The second was the construction by Fradkin ([?]; see also [4]) of an invariant tensor for the time-independent three dimensional isotropic oscillator. This tensor, together with the angular momentum, was used as a basis for the generators of the dynamical symmetry group of that system, namely SU(3) .