Showing papers on "Canonical transformation published in 1976"

The spectrum of Jacobi matrices

Abstract: Let L be a periodic symmetric tridiagonal matrix of size N; "periodic" means that L has one extra-entry in the upper right corner and by symmetry in the lower left one. Let b i be the diagonal and ai the subdiagonal entries. The present paper deals with the space ~ ' of such matrices with a given spectrum. On Jr' there is a natural class of commuting flows (isospectral deformations), which derive from Hamiltonian mechanics. When the given spectrum is non-degenerate, there are N 1 independent flows except for some degeneracies on some lower dimensional submanifolds. Each of these flows has in general N 1 integrals in involution, so that generically the solutions are quasi-periodic, their orbits are dense on a N 1 dimensional torus and there exists a canonical transformation to a set of action-angle variables. However there is much more involved, because these tori are algebraic surfaces and their periods can be expressed in terms of hyperelliptic functions; this is to say each such torus is a Jacobi variety. The transformation from the original variables (a~, bi) to a set of separation variables (/~i, vi) is of rational character. The "posi t ion" components p~ of these variables are provided by the spectrum of the matrix L, from which the first row and the first column has been removed. They define a local system of coordinates on the torus. Another system of coordinates t~ is provided by the group action of R N1 on the torus, such that the flows appear as linear motions on the torus. The Jacobi transformation maps the local system of coordinates (#1 . . . . . #N-l) into the global one (q . . . . . tN_l). The inverse map can be expressed in terms of the flows above and can be explicited in terms of quotients of theta functions invoking the theory of the Jacobi inversion problem. As a bonus, this yields explicit solutions to the differential equation defined by the isospectral flows, in terms of Abelian and theta functions. Finally, ~t' can be foliated by N-1-d imens iona l tori, each of which can be labelled by a modulus; this modulus is defined as the product of the non-diagonal

Non linear canonical transformations and their representations in quantum mechanics

Abstract: In the present paper we analyze the representations in quantum mechanics of classical canonical transformations that are non-bijective, 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 K−1 into one of frequency k−1 where k, K are relatively prime integers. For the particular case k = 1, the mapping of the original phase space (x,p) onto the new one \((\bar x,\bar p)\)is K to 1 and the equivalent points in (x,p) are related by a cyclic group CK of linear canonical transformations. When formulating this problem in Bargmann Hilbert space the canonical transformation can be related with the conformal transformation w = zK which again is K to 1 and where a group CK 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 component function, in the new Bargmann Hilbert space we must take a K component one. In this way we can map in a one to one fashion the states and operators in the old and new Bargmann Hilbert spaces. When translating these results to ordinary Hilbert space we get in an ambiguous way the quantization of the observables appearing in the equations that determine the representation of the classical canonical transformation relating oscillators of frequencies K−1 and k−1. Furthermore we also get the solutions of these equations, and the resulting representation is unitary.

Renormalised canonical perturbation theory for stochastic propagators

Abstract: A canonical transformation which removes the coherent oscillatory motion of a particle in a stochastic potential (the renormalised oscillation-centre transformation) is constructed by a new classical perturbation method using Lie operators and Green function techniques. A frequency and wavevector dependent particle-wave collision operator is calculated explicitly for stationary, homogeneous electrostatic turbulence in the short wavelength limit. The width of the resonance is proportional to the one-third power of the quasilinear diffusion coefficient, in agreement with Dupree's result (1966). However the k dependence is quite different from that expected from a simple Wiener process model. At large k spatial diffusion dominates over velocity diffusion in sharp contrast with previous theories.

Collective motions in nuclei and the spectrum generating algebras T5 × SO(3), GL(3,R), and CM(3)

Abstract: Cusson's classical treatment of the collective rotations of a discrete system of N particles is extended to the full quantum mechanical system by means of a straightforward generalization of Villars' canonical transformation. In this manner, Bohr's collective Hamiltonian, with various values for the rotational mass, is microscopically derived. The nature and criteria for the existence of various collective flows in a many-body system are also given. The collective parts of the Hamiltonian are then separately expressed in original particle coordinates and momenta and in this manner the possibility of microscopic calculations for the collective motions is suggested. Finally appropriate microscopic Hamiltonians for the S.G.A.'s T5 × SO(3), GL(3,R), and CM(3) are determined.

Oscillation-center formulation of the classical theory of induced scattering in plasma

Abstract: Particle trapping in the beat potential produced by the mixing of two coherent waves lies outside the domain of the conventional weak‐turbulence theory of induced wave–particle scattering in plasma. A reformulation of the theory is presented which is applicable to this extended domain. The formulation is based upon a canonical transformation to ’’oscillation‐center variables’’ and in particular affords a simpler method for deriving the ’’linear’’ matrix elements of weak‐turbulence theory. This method is first illustrated in the problem of the nonlinear Landau damping of longitudinal plasma modes. The power of the method is then usefully exploited in two more difficult situations, namely, the cases of electrostatic waves in the presence of a uniform magnetic field, and electromagnetic waves in an arbitrarily relativistic plasma. The application of the transformation to the extended strong‐interaction regime is demonstrated.

Theory of Canonical Transformations for Nonlinear Evolution Equations. II

Abstract: For nonlinear evolution equations, a canonical transformation which keeps the Hamiltonian form invariant is investigated. It is shown that the so-called Backlund transformation is the canonical transformation of this type. Group property of the canonical transformation and relations between infinitesimal canonical transformations and conservation laws are also in· vestigated. Sine-Gordon equation, Korteweg-de Vries equation and modified Korteweg-de Vries equation are considered as examples.

Excitation spectra of atoms and small molecules using effective valence shell Hamiltonians generated by canonical transformations

Abstract: The canonical transformation–cluster expansion formalism is used to generate the effective valence shell Hamiltonian for carbon. Hydrogenlike orbitals defined by an effective nuclear charge parameter Z are used to span the core (K shell), valence (L shell), and excited (3⩽n⩽9) spaces. The effective Hamiltonian containing one‐ and two‐body interactions is diagonalized on the Nv‐particle valence space to yield the low‐lying excitation spectrum. Considering alternative approximations to carry out the calculations, we indicate the importance of including the two‐body pair potential as well as the single particle operators in the generator of the canonical transformation. Upon doing this, good agreement with experiment is obtained for the lowest valence shell transitions over a wide range of Z. For certain physically reasonable Z the entire valence shell experimental excitation spectrum can be accurately reproduced. In contrast, a ’’zeroth order’’ effective Hamiltonian using only the ’’charge cloud’’ of the co...

Canonical transformations and radiative effects in electron-electron interactions

Abstract: Canonical transformation techniques are used together with variational and perturbative methods to describe the effects of the coupling between electrons and the radiation field in the Coulomb gauge. A simple transformation, equivalent to the use of Glauber's coherent states (1963) is treated exactly, and the 'direct' part of the currently used Breit correction (1932) to the electron-electron interaction in relativistic Hartree-Fock theory is derived variationally. More general transformations leading to electron self-energy terms as well as electron-electron interaction corrections are discussed and compared to the procedure used by Mittleman (1972). The case where the unperturbed Hamiltonian is of Hartree-Fock type gives the generalized Breit interaction.

On the multipole expansion in the theory of optical activity

Abstract: The different Hamiltonians for a molecule interacting with the electromagnetic field, which have been used in the literature on the optical activity of isotropic media, are shown to be related by a canonical transformation and hence to be equivalent. Because of its greater simplicity, the use of the multipole Hamiltonian in the theory of optical activity is advocated.

The dissipation of restricted rotation

Abstract: The decay of a classical hindered rotor over the whole domain of motion from almost free overall rotation to harmonic vibration in a well is examined. The decay is described by a modification of a projection operator formalism involving the separation of dynamical and lattice time scales, and depends on a canonical transformation into a hindered rotating frame that rotates and oscillates in accord with the unperturbed equation of motion of two coaxial dipoles.

Interplay of Pairing and Intrinsic Modes of Excitation in Nuclei. I

Abstract: A new method treating the interplay of pairing and intrinsic modes of excitation is proposed. In this method, the pairing mode associated with the J=O-coupled nucleon pairs is represented by pairing bosons and the intrinsic mode characterized by the seniority quantum number is explicitly treated by ideal quasi-particle operators. We obtain a closed expression of the single-nucleon operator in terms of pairing bosons and ideal quasi-particles. As a simple illustration, the superconducting system is treated by introducing the coherent state of pairing bosons and the relation to the Bogoliubov transformation is discussed. The relation between this method and the· canonical transformation method with auxiliary variables is also clarified.

Bäcklund Transformation as a Continuous Two-Parametric Mapping

Abstract: The Backlund transformation, known as the self-mapping on the sine-Gordon equation is found to be a special case of continuous two-parametric transformations. It is a kind of extended canonical transformations defined in the double-fold integral dynamics. This fact motivates to explore into the physical meanings of that transformation.