Showing papers on "Canonical transformation published in 1974"

Canonical Forms for Symplectic and Hamiltonian Matrices

Abstract: This paper gives a constructive method for finding canonical forms for symplectic and Hamiltonian matrices. No restrictions are made on the eigen values or their multiplicity. Real canonical forms are treated in detail.

Canonical transforms. I. Complex linear transforms

Abstract: Recent work by Moshinsky et al. on the role and applications of canonical transformations in quantum mechanics has focused attention on some complex extensions of linear transformations mapping the position and momentum operators x and p to a pair η and ζ of canonically conjugate, but not necessarily Hermitian, operators. In this paper we show that for a continuum of complex linear canonical transformations, a related Hilbert space of entire analytic functions exists with a scalar product over the complex plane such that the pair η, ζ can be realized in the Schrodinger representation η and −id/dη. We provide a unitary mapping onto the ordinary Hilbert space of square‐integrable functions over the real line through an integral transform. The transform kernels provide a representation of a subsemigroup of SL(2,C). The well‐known Bargmann transform is the special case when η and iζ are the harmonic oscillator raising and lowering operators. The Moshinsky‐Quesne transform is regained in the limit when the canonical transformation becomes real, a case which contains the ordinary Fourier transform. We present a realization of these transforms through hyperdifferential operators.

Canonical transforms. II. Complex radial transforms

Abstract: Continuing the line of development of Paper I [J. Math. Phys. 15, 1295 (1974)], we enlarge the concept of canonical transformations in quantum mechanics in two directions: first, by allowing the definition of a canonical transformation to be made through the preservation of an so(2,1) algebra, rather than the usual Heisenberg algebra, and providing the bridge between the classical and quantum mechanical descriptions, and, second, through the complexification of the transformation group. In this paper we study specifically the transformations which can be interpreted as the radial part of n‐dimensional complex linear transformations in Paper I. We show that we can build Hilbert spaces of analytic functions with a scalar product defined through integration over half the complex plane of a variable which has the meaning of a complex radius. A unitary mapping to the ordinary Hilbert space Lrn−12(0,∞) is provided with a kernel involving a Bessel function. Special cases of this are shown to be the Barut‐Girarde...

Equivalence for Lie Transforms

Abstract: The Lie transform method used in Perturbation Theory is based upon an intrinsic algorithm for transforming functions or vector fields by a transformation close to the identity. It can thus be viewed as a specialization of methods and results of differential geometry as is shown in the first part of this paper. In a second part we answer some of the questions left open in connection with the equivalence of the algorithms proposed by Hori and Deprit. From a formal point of view, the methods are shown to be equivalent for non-canonical as well as canonical transformations and a formula relating directly the two generating functions (or vector fields) is presented (formula (5.17)). On the other hand, the equivalence is shown to hold also in the ring ofp-differentiable functions.

Conversion from material to local coordinates as a canonical transformation

Abstract: It is shown that the conversion from material to local coordinates in continuum mechanics can be considered as a restricted canonical transformation. As a simple example the longitudinal motion of an elastic bar is discussed.

A transformation approach for finding first integrals of motion of dynamical systems

Abstract: A method for finding a first integral of the motion of a system of equations written in Hamiltonian form, for the case in which no “classical integrals” exist, is introduced and derived in this paper. The present method is based on a canonical transformation applied to the state vector of the system with the objective of obtaining a new Hamiltonian that is time-separable in the new state vector.

TheU-matrix as a simple tool to derive canonical transformations

Abstract: A straightforward application of theU-matrix formalism is utilized to derive canonical exponential transformations. It turns out that an extremely compact form for these transformations can be written down. In view of the fact that the transformation used so far, mostly have been found by guessing, a simple calculation prescription for them seems to be of great value. As an application the derivation of the already wellknown transformations of theE-e andT-t Jahn Teller problems as well as that of the Frohlich Hamiltonian is presented. In the Jahn-Teller examples it is further demonstrated, how for each single problem the validity region of the transformation may be examined.

Bose Condensation in Superconductors and Liquid 4 He

Abstract: Calculations on many-body systems containing both composite particles and their constituents are facilitated by the use of representations1–3 in which the composite particles are described in terms of their own dynamic variables. The method3 we shall employ here is based on the familiar idea of the introduction of initially redundant variables corresponding to the composite particles, followed by a canonical transformation which gives these extra modes physical content. This approach is similar to that employed, e.g., in the Bohm—Pines theory of plasma oscillations,4 in which extra plasmon variables were introduced.