Showing papers on "Canonical transformation published in 1975"

Canonical quantization of nonlinear waves

Abstract: By means of a canonical transformation we investigate nonlinear field theories that possess exact classical solutions. This transformation is equivalent to the method of collective coordinates recently applied to the same problem by Gervais and Sakita using functional techniques. It is pointed out that, because of the operator orderings, extra terms occur in the quantized theory which seem to be absent in the straightforward functional approach. Ordinary perturbation treatment of the resulting Hamiltonian reproduces results previously obtained by Goldstone and Jackiw using a different technique.

Nonlinear canonical transformations and their representations in quantum mechanics

Abstract: In the last few years an extensive literature has developed on linear canonical tranformations and their representation in quantum mechanics. Applications of these results have been made to clustering theory in nuclei, problems of accidental degeneracy, etc. In the present paper we wish to turn our attention to nonlinear canonical transformations. We show that by dealing with appropriate functions fα (α=1,...,2n) of xi, pi (i=1,...,n) rather than with these variables themselves, we can in principle set unambiguously the equations that determine the representation in quantum mechanics of the canonical transformation under study. This result holds when the old and new functions fα have the same spectrum. We discuss specific examples when this last condition is satisfied: nonlinear canonical transformations in the radial variable that were obtained from projection of linear ones in higher‐dimensional spaces; canonical transformations that take us from one Hamiltonian to another with the same spectrum, be thi...

Four examples of the inverse method as a canonical transformation

Abstract: The Toda lattice, the nonlinear Schrodinger equation, the sine−Gordon equation, and the Korteweg−de Vries equation are four nonlinear equations of physical importance which have recently been solved by the inverse method. For these examples, this method of solution is interpreted as a canonical transformation from the initial Hamiltonian dynamics to an ’’action−angle’’ form. This canonical structure clarifies the independence of an infinite number of constants of the motion and indicates the special nature of the solution by the inverse method.

First principles derivation of the effective valence shell Hamiltonian for large molecules

Abstract: We construct an effective Hamiltonian for the valence shell of a large electronic system. The procedure begins by classifying the complete one electron space according to core, valence, and excited orbitals. An N−electron subspace TN spanned by Slater determinants with a fixed set of Nc core orbitals and different sets of Nv = (N − Nc) valence orbitals is defined. A canonical transformation on the Coulomb Hamiltonian is used to eliminate the interaction between TN and its orthogonal complement, ?N, thereby defining an effective N−electron Hamiltonian. This effective Hamiltonian is expanded in a cluster development of linked one−, two−, three−,⋅⋅⋅body operators in terms of which the conditions of having vanishing matrix elements between TN and ?N can be explicitly formulated. Then starting with this effective N−electron Hamiltonian we construct an equivalent Hamiltonian to operate in the space of antisymmetrized products of valence orbitals. To within a constant (times the identity on the valence space) th...

A Canonical Transformation for the Exponential Lattice

Abstract: A canonical transformation which gives the relation between two solutions of an exponential lattice is presented. Using this relation a new solution can be obtained from a known solution. It is thus a discrete version of the Backlund transformation.

Canonical transformation and accidental degeneracy. III. A unified approach to the problem

Abstract: We continue the discussion of the groups of canonical transformations responsible for accidental degeneracy in quantum mechanical problems. A general unified treatment is provided for a wide class of two−dimensional physical systems, having an energy spectrum which is a linear combination of two quantum numbers. The general method involves the use of both nonorthonormal and orthonormal sets of states to construct groups of complex or real canonical transformations, mapping the problem under consideration onto the two−dimensional isotropic harmonic oscillator. The group responsible for the accidental degeneracy is then quite obviously SU (2). The problem of an isotropic oscillator in a sector π/q (q integer) was discussed previously using a nonorthonormal basis. In the present paper we carry the analysis in an orthonormal basis to establish the general procedure mentioned above. We also analyze in detail the Calogero problem for three particles which has a spectrum of the type given above, and obtain expli...

Plasma frequency of the electron gas in layered structures

Abstract: The plasma frequency of a complete degenerate electron gas in the layered model of Visscher and Falicov is calculated by means of both the Bohm-Pines canonical transformation method and the equation-of-motion method in the RPA. The dispersion equation for plasmons is obtained in a finite system ofn planes with both the cyclic condition and free ends. It is shown that the thermodynamic limit (n→∞) of the plasma frequency is independent of the boundary conditions. The previous results obtained by various authors in different ways are shown to be certain limits of our result.

Canonical transforms. III. Configuration and phase descriptions of quantum systems possessing an sl (2,R) dynamical algebra

Abstract: The purpose of this article is to present a detailed analysis on the quantum mechnical level of the canonical transformation between coordinate‐momentum and number‐phase descriptions for systems possessing an sl (2,R) dynamical algebra, specifically, the radial harmonic oscillator and pseudo‐Coulomb systems. The former one includes the attractive and repulsive oscillators and the free particle, each with an additional ’’centrifugal’’ force, while the latter includes the bound, free and threshold states with an added ’’centrifugal’’ force. This is implemented as a unitary mapping—canonical transform—between the usual Hilbert space L2 of quantum mechanics and a new set of Hilbert spaces on the circle whose coordinate has the meaning of a phase variable. Moreover, the UIR’s D+k of the universal covering group of SL (2,R) realized on the former space are mapped unitarily onto the latter.

Plasma oscillations in a one-dimensional many-fermion system

Abstract: Following the Shevchik technique, a model Hamiltonian of collective oscillations (plasmons) in a one-dimensional system of complete degenerate fermions is obtained in terms of the Tomonaga boson operators. This Hamiltonian is diagonalized by means of the Mattis and Lieb canonical transformation and the plasma frequency is derived. The equation-of-motion method is applied in the RPA in order to include the coupling between the collective and individual degrees of freedom. The generalization to finite temperatures is performed and connection with the Tomonaga model is discussed.

Canonical transformations and path integrals

Abstract: A limited class of canonical transformations is introduced into the Lagrangian path integral method of quantization. Path integral quantization in different representations is discussed and a simple example is given.

New method for solving the many-body boson problem.—II

Abstract: We develop in this paper a new method for solving the many-body boson system based on a generalizedN-particle ground-state trial wave function and the principle of canonical transformation. Explicit construction of the matrix representations and an iteration method are given. Some mathematical theorems and the range of validity pertaining to the theory are also presented.

Hamiltonian formulation of the classical two‐charge problem in straight‐line approximation

Abstract: The Newtonian equations of motion expressing the interaction of electric charges correct up to terms of order e2, the product of the charges, is cast into Hamiltonian form. Lorentz transformations are canonically represented to order e2, but as anticipated by the zero‐interaction theorem, there is no canonical transformation from the canonical variables to the charges’ physical positions.

Canonical Forms for General Polynomials

Abstract: by means of a homogeneous linear transformation (see [3; p. 90] or [6; p. 329] or [10; p. 73]). L. Cremona has called \" fascio sizigetico \" (syzygetic pencil) the pencil of plane curves defined by (i) with respect to a variable A (see [2; pp. 274-279] or [8; pp. 196198]). The aim of this paper is to give an extension of the forementioned result to general homogeneous polynomials of arbitrary degree s 3* 3, in any number of variables, defined over any field T whose characteristic does not divide s. The method consists in showing that to prove the existence of the desired canonical form for a given general polynomial is equivalent to proving that a certain field extension is algebraic. Then a well-known jacobian criterion for algebraic extensions can be applied, thus reducing the problem to the calculation of a determinant. All this is expounded in §2, where some other theorems are also stated and proved, which develop some algebro-geometric consequences of our line of thought. Namely they express in technical form a sort of \" linear rigidity \" of general forms, and general syzygetic forms (see (iii), §2) as well. Then in §3 details are given of one possible way to carry out the calculations of jacobians which are necessary to complete the basic proof. In the last section the relationship between syzygetic forms and general forms is viewed as a morphism of schemes, and several open problems are formulated about that morphism. As a further comment we may mention that this paper came out of an attempt to generalize the proof given in [4; p. 118] of (a weak form of) the classical result on ternary cubic forms. We were successful in adapting that method to (general) quaternary cubics, thereby proving that they can be reduced to the form

Nonrelativistic field theory for strong coupling

Abstract: We study the Hamiltonian for a nonrelativistic nucleon with Yukawa coupling to a scalar meson field. The nucleon self-energy is logarithmically divergent. It is known that one can isolate the divergent part by a canonical transformation, but this technique is feasible only for simple field theories. We study the theory by a systematic strong coupling in inverse powers of the coupling constant e. The leading term in the energy is of order e$sup 4$ and corresponds to classical particlelike solutions. The terms of order e$sup 2$ represent a generalized meson-pair theory. The zero-point energy shift yields the logarithmically divergent energy. The residual Hamiltonian contains scattering resonances, and techniques are developed to study the resonances.

On the theory of the Coupling Between Radial Oscillations and H.F. Phase and Energy of Particles in Cyclotrons

Abstract: The motion of accelerated particles is described by a Hamilton function in which the acceleration is taken into account by a timedependent potential function. By canonical transformations four variables are defined of which one set describes energy and H.F. phase and the other set the radial oscillations of the particles. In the new Hamilton function the coupling between these two sets of variables is clearly demonstrated.

The averaged Hamiltonian of a time-dependent nearly multiple-periodic system

Abstract: A discussion is presented of the canonical averaging of time-independent, nearly multiple-periodic systems having Hamiltonians of the form H=H0(Jalpha )+ lambda H1(walpha ,Jalpha ;qk,pk)+ lambda 2H2(walpha ,Jalpha ;qk,pk)+... where H1, Hs,... are periodic functions of the angles walpha . A perturbation procedure is given for constructing a direct canonical transformation converting the Hamiltonian into a new one, independent of the proper angles walpha , irrespective of whether the system is non-degenerate or intrinsically degenerate. In each case constants of motion to all orders of the perturbation theory exist, corresponding to each proper angle walpha .

A canonical transformation near a boundary point

Abstract: : A local homogeneous canonical transformation is constructed which straightens a curved boundary and freezes the coefficients of the principal part of a pseudo-differential operator in the neighborhood of a non-glancing ray. (Author)