3,561 research outputs found
Quantum Systems and Alternative Unitary Descriptions
Motivated by the existence of bi-Hamiltonian classical systems and the
correspondence principle, in this paper we analyze the problem of finding
Hermitian scalar products which turn a given flow on a Hilbert space into a
unitary one. We show how different invariant Hermitian scalar products give
rise to different descriptions of a quantum system in the Ehrenfest and
Heisenberg picture.Comment: 18 page
Symplectic Structures and Quantum Mechanics
Canonical coordinates for the Schr\"odinger equation are introduced, making
more transparent its Hamiltonian structure. It is shown that the Schr\"odinger
equation, considered as a classical field theory, shares with Liouville
completely integrable field theories the existence of a {\sl recursion
operator} which allows for the infinitely many conserved functionals pairwise
commuting with respect to the corresponding Poisson bracket. The approach may
provide a good starting point to get a clear interpretation of Quantum
Mechanics in the general setting, provided by Stone-von Neumann theorem, of
Symplectic Mechanics. It may give new tools to solve in the general case the
inverse problem of quantum mechanics whose solution is given up to now only for
one-dimensional systems by the Gel'fand-Levitan-Marchenko formula.Comment: 11 pages, LaTex fil
Remarks on Nambu-Poisson and Nambu-Jacobi brackets
It is shown that Nambu-Poisson and Nambu-Jacobi brackets can be defined
inductively: a n-bracket, n>2, is Nambu-Poisson (resp. Nambu-Jacobi) if and
only if fixing an argument we get a (n-1)-Nambu-Poisson (resp. Nambu-Jacobi)
bracket. As a by-product we get relatively simple proofs of Darboux-type
theorems for these structures.Comment: Latex, 13 page
Highlights of Symmetry Groups
The concepts of symmetry and symmetry groups are at the heart of several
developments in modern theoretical and mathematical physics. The present paper
is devoted to a number of selected topics within this framework: Euclidean and
rotation groups; the properties of fullerenes in physical chemistry; Galilei,
Lorentz and Poincare groups; conformal transformations and the Laplace
equation; quantum groups and Sklyanin algebras. For example, graphite can be
vaporized by laser irradiation, producing a remarkably stable cluster
consisting of 60 carbon atoms. The corresponding theoretical model considers a
truncated icosahedron, i.e. a polygon with 60 vertices and 32 faces, 12 of
which are pentagonal and 20 hexagonal. The Carbon 60 molecule obtained when a
carbon atom is placed at each vertex of this structure has all valences
satisfied by two single bonds and one double bond. In other words, a structure
in which a pentagon is completely surrounded by hexagons is stable. Thus, a
cage in which all 12 pentagons are completely surrounded by hexagons has
optimum stability. On a more formal side, the exactly solvable models of
quantum and statistical physics can be studied with the help of the quantum
inverse problem method. The problem of enumerating the discrete quantum systems
which can be solved by the quantum inverse problem method reduces to the
problem of enumerating the operator-valued functions that satisfy an equation
involving a fixed solution of the quantum Yang--Baxter equation. Two basic
equations exist which provide a systematic procedure for obtaining completely
integrable lattice approximations to various continuous completely integrable
systems. This analysis leads in turn to the discovery of Sklyanin algebras.Comment: Plain Tex with one figur
Reduction and unfolding: the Kepler problem
In this paper we show, in a systematic way, how to relate the Kepler problem
to the isotropic harmonic oscillator. Unlike previous approaches, our
constructions are carried over in the Lagrangian formalism dealing with second
order vector fields. We therefore provide a tangent bundle version of the
Kustaahneimo-Stiefel map.Comment: latex2e, 28 pages; misprints correcte
On Filippov algebroids and multiplicative Nambu-Poisson structures
We discuss relations between linear Nambu-Poisson structures and Filippov
algebras and define Filippov algebroids which are n-ary generalizations of Lie
algebroids. We also prove results describing multiplicative Nambu- Poisson
structures on Lie groups. In particular, we show that simple Lie groups do not
admit multiplicative Nambu-Poisson structures of order n>2.Comment: Latex, 22 pages, to appear in Diff. Geom. App
Experimental and Theoretical Studies in Planetary Aeronomy Quarterly Progress Report, 24 May - 31 Aug. 1968
Determining absorption and photoionization cross sections of planetary gases - planetary aeronomy researc
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