Showing papers on "Canonical transformation published in 1983"
TL;DR: In this article, the relationship of the multipolar to the minimal-coupling formalism is examined when both the electron and the radiation are second-quantized fields, and both the Lagrangian and Hamiltonian formulations are considered: in the former the transformation between the two is a point transformation on the electron field coordinates, while in the latter it is a canonical transformation.
Abstract: The multipolar formalism is commonly used as the starting point in chemical physics and quantum optics for discussion of the interaction of radiation with atoms and molecules. The relationship of the multipolar to the minimal-coupling formalism is examined when both the electron and the radiation are second-quantized fields. Both the Lagrangian and Hamiltonian formulations are considered: in the former the transformation between the two is a point transformation on the electron field coordinates, while in the latter it is a canonical transformation. The resulting equations of motion are Maxwell's equations, in terms of the basic and auxiliary fields, for the electromagnetic field and Schr\"odinger equations for charges in an electromagnetic field with the coupling given through the multipole moments. That the Schr\"odinger equation is different from that which arises in the minimal-coupling formalism is a natural consequence of the use of new field coordinates. The theory is extended to a system of molecules anticipating the discussion of intermolecular energies in paper III (the second succeeding paper).
71 citations
TL;DR: A Mathieu canonical transformation is proposed whereby the new variables separate naturally into (i) a coordinate system on any reduced manifold of constant angular momentum, and (ii) a quadruple made of a pair of ignorable longitudes together with their conjugate momenta.
Abstract: In application of the Reduction Theorem to the general problem ofn (>-3) bodies, a Mathieu canonical transformation is proposed whereby the new variables separate naturally into (i) a coordinate system on any reduced manifold of constant angular momentum, and (ii) a quadruple made of a pair of ignorable longitudes together with their conjugate momenta. The reduction is built from a binary tree of kinetic frames Explicit transformation formulas are obtained by induction from the top of the tree down to its root at the invariable frame; they are based on the unit quaternions which represent the finite rotations mapping one vector base onto another in the chain of kinetic frames. The development scheme lends itself to automatic processing by computer in a functional language.
60 citations
TL;DR: In this paper, noncanonical, Lie algebraic, hamiltonian structures in the eulerian description of ideal continuum mechanics are shown to be compatible with nearly canonical structure in the lagrangrian description.
52 citations
TL;DR: In this paper, the authors present sufficient conditions for strong local optimality in the generalized problem of Bolza, which can be applied to both the calculus of variations and to optimal control problems, as well as problems with nonsmooth data.
Abstract: This paper presents sufficient conditions for strong local optimality in the generalized problem of Bolza. These conditions represent a unification, in the sense that they can be applied to both the calculus of variations and to optimal control problems, as well as problems with nonsmooth data. Also, this paper brings to light a new point of view concerning the Jacobi condition in the classical calculus of variations, showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original Hamiltonian to a locally concave-convex Hamiltonian.
34 citations
TL;DR: In this paper, the authors introduce the concept of generalized canonical transformations as symplectomorphisms of the extended phase space and prove that any such transformation factorizes in a standard canonical transformation times another one that changes only the time variable.
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.
27 citations
TL;DR: In this article, the effects of coupling to a harmonic oscillator on the quantum tunneling of a macroscopic motion are studied through the influence functional formalism of Feynman's path integral method for the general coupling form factor.
Abstract: The effects of coupling to a harmonic oscillator on the quantum tunneling of a macroscopic motion are studied through the influence functional formalism of Feynman's path integral method for the general coupling form factor. As an example, we consider the model in which the potential barrier is parabolic and the coupling Hamiltonian is linear in both coordinates of the macroscopic motion and of the intrinsic harmonic oscillator. The results are then compared with the exact solution obtained through the canonical transformation into normal coordinates in the limiting cases when the normal coordinates reduce to the original coordinates. We found that: (1) In the adiabatic case, i.e., when the recurrence time ..pi../..omega.. of the oscillator is much shorter than the transmission time through the macroscopic potential barrier, the effect of oscillator coupling can be well represented by an effective potential. The coupling enhances the tunneling probability on the whole. (2) There exists a critical energy, above which the tunneling probability is reduced because of the linear oscillator coupling. In the weak coupling limit and when ..omega -->..0, the critical energy becomes -infinity, so that the coupling to the oscillator always reduces the tunneling probability.
11 citations
TL;DR: In this paper, the authors generalized the Wu and Austin expression by improving their approximations with the superconducting canonical transformation and obtained the relaxation time depending exponentially on the inverse temperature.
Abstract: Following the conjecture of Frohlich that the Bose condensation of phonons is the dominant biological effect, Wu and Austin derived from the Frohlich Hamiltonian an expression for the relaxation time which is inversely proportional to the temperature by using the technique of Green's function. In this paper, we have generalized the Wu and Austin expression by improving their approximations with the superconducting canonical transformation and obtained the relaxation time depending exponentially on the inverse temperature. Moreover, we have checked our expression by using a different method based on the “continued fraction expansion” and also employed this technique to derive the dielectric constant for this system.
10 citations
TL;DR: In this paper, a canonical transformation leading to the nonsecular part of time-independent perturbation calculus is proposed, which is used to derive expressions for effective Shen-Walls Hamiltonians, taken in the two-level approximation and on the inclusion of non-Hamiltonian terms into the dynamics of the system, lead to generalized Maxwell-Bloch equations.
Abstract: A new method is proposed which involves a canonical transformation leading to the nonsecular part of time-independent perturbation calculus. The method is used to derive expressions for effective Shen-Walls Hamiltonians which, taken in the two-level approximation and on the inclusion of non-Hamiltonian terms into the dynamics of the system, lead to generalized Maxwell-Bloch equations. The rotating-wave approximation is written anew within the framework of our formalism.
10 citations
TL;DR: A Backlund transformation for a generalised Calogero-Moser system, in which the particles posses extra, internal, degrees of freedom, is given in this paper, which is shown to be a canonical transformation and its generating function is given explicitly.
9 citations
TL;DR: In this article, the dynamical effective interactions between two swift ions due to surface-plasmon exchange are calculated by a canonical transformation, and the alignment effect discovered in the bulk case is also found as a function of the distance from the surface.
Abstract: The dynamical effective interactions between two swift ions due to surface-plasmon exchange are calculated by a canonical transformation. The alignment effect discovered in the bulk case is also found as a function of the distance from the surface. We can show that another kind of force appears which tends to make the ion pair stand up perpendicular to the surface.
5 citations
TL;DR: In this article, the relation between unitary transformations and canonical transformations of the p, q labels was studied. But the relation was not explored in the context of the Wigner transform.
Abstract: Quantum mechanical operators can be associated with functions of p, q through the Weyl or Wigner transform. In this paper we develop alternative associations through the use of unitary transformations, and study the relation between unitary transformations and canonical transformations of the p, q labels.
TL;DR: In this paper, the authors prove the existence of 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.
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.
TL;DR: In this article, the theory of the Brownian motion of deformable particles is presented taking into account the interaction between the rotational motion of the particle and vibrations of its form.
Abstract: The theory of the Brownian motion of deformable particles is presented taking into account the interaction between the rotational motion of the particle and vibrations of its form. The changes of the form of the Brownian particle are described by means of the coordinates used in the collective model of the atomic nuclei. The Einstein-Smoluchowski kinetic equation has been obtained in these coordinates. A canonical transformation of the variables is carried out into the space of the Eulerian angles and two deformation parameters. An approximate solution of the kinetic equation has been found using the new variables for particles which have an axially symmetrical shape in equilibrium. Simple time correlation functions of dynamic variables of the particle are calculated.
TL;DR: In this paper, it was shown that if isotropically averaged polarisation and magnetisation fields are introduced, the minimal-coupling and multipolar Lagrangians are identical, provided also the potentials used are those of the Coulomb gauge.
Abstract: In the minimal-coupling Lagrangian for the interaction of the electromagnetic field and nonrelativistic charged particles, the charge and current densities are coupled to scalar and vector potentials. In the multipolar Lagrangian, on the other hand, the aggregate or particles is particularly (through sometimes completely) described by polarisation and magnetisation fields and these are coupled to the electric and magnetic induction fields. It is shown that if isotropically averaged polarisation and magnetisation fields are introduced, the minimal-coupling and multipolar Lagrangians are identical, provided also the potentials used are those of the Coulomb gauge. The associated canonical transformation of the Hamiltonian is the identity transformation. Thus the minimal-coupling Hamiltonian can be written directly in multipolar form, without any change in the canonical dynamical variables.
TL;DR: In this article, the solutions of the parabolic equation in the canonical transformation method were obtained for the field in an anisotropic stratified medium and the perturbation theory for the calculation of the wave reflection and transmission coefficients was developed.
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.
TL;DR: In this paper, a theory of action-angle variables for coupled oscillator systems is developed which involves solving the Schrodinger equation using a basis of WKB eigenfunctions, then using the logarithm of the resulting wave function to define the generator for the canonical transformation which determines the action-angles variables.
01 Jan 1983
TL;DR: In this article, it was shown that the two-dimensional Ising model is equivalent to a system of locally coupled fermions, and that the fermionic degrees of freedom can also be rewritten in terms of fermanionic degrees.
Abstract: In 1964, Schultz, Mattis, and Lieb (SML) [1] showed that the two dimensional Ising model is equivalent to a system of locally coupled fermions. After a canonical transformation, these fermions become noninteracting, and so SML were able to construct a simple, elegant, and exact solution. Recently, Fradkin, Susskind, and I (FSS) [2] showed that the three dimensional Z(2) gauge theory [3] could also be rewritten in terms of locally coupled fermionic degrees of freedom. Unfortunately, the coupling turned out to be quartic, and so we were unable to solve the theory.
TL;DR: In this paper, the motion of fluid particles in shear flows perturbed by unamplifying traveling sinusoidal waves is considered, and the analogy between the formulation of the problem in terms of the stream function and the Hamiltonian dynamics is pointed out.
Abstract: The motion of fluid particles in shear flows perturbed by unamplifying traveling sinusoidal waves is considered. The analogy between the formulation of the problem in terms of the stream function and the Hamiltonian dynamics is pointed out. An exact analytical solution is obtained for a perturbed mixing-layer type flow. For a perturbed wake type flow, an approximate analytical solution is obtained by employing the method of canonical transformation due to Hori.
TL;DR: In this article, the Jahn-Teller problem of a system of an electronic doublet having spin-orbit splitting in first order and interacting with a doubly degenerate vibration is considered.
Abstract: The Jahn-Teller problem of a system of an electronic doublet having spin-orbit splitting in first order and interacting with a doubly degenerate vibration is considered. A canonical transformation and variational approach is used to determine the ground vibronic Kramers' doublets. The variational wave functions are used to calculate the zero-field splitting and ESR g -values, which are compared with the experimental results on cobaltocene.