1,110 research outputs found
Galilean covariant harmonic oscillator
A Galilean covariant approach to classical mechanics of a single particle is described. Within the proposed formalism, all non-covariant force laws defining acting forces which become to be defined covariantly by some differential equations are rejected. Such an approach leads out of the standard classical mechanics and gives an example of non-Newtonian mechanics. It is shown that the exactly solvable linear system of differential equations defining forces contains the Galilean covariant description of harmonic oscillator as its particular case. Additionally, it is demonstrated that in Galilean covariant classical mechanics the validity of the second Newton law of dynamics implies the Hooke law and vice versa. It is shown that the kinetic and total energies transform differently with respect to the Galilean transformations
Classical confined particles
An alternative picture of classical many body mechanics is proposed. In this picture particles possess individual kinematics but are deprived from individual dynamics. Dynamics exists only for the many particle system as a whole. The theory is complete and allows to determine the trajectories of each particle. It is proposed to use our picture as a classical prototype for a realistic theory of confined particles
Squeezing: the ups and downs
We present an operator theoretic side of the story of squeezed states
regardless the order of squeezing. For low order, that is for displacement
(order 1) and squeeze (order 2) operators, we bring back to consciousness what
is know or rather what has to be known by making the exposition as exhaustive
as possible. For the order 2 (squeeze) we propose an interesting model of the
Segal-Bargmann type. For higher order the impossibility of squeezing in the
traditional sense is proved rigorously. Nevertheless what we offer is the
state-of-the-art concerning the topic.Comment: 21 pages; improved presentation; it has been published by Proceedings
of the Royal Society
Holomorphic Hermite polynomials in two variables
Generalizations of the Hermite polynomials to many variables and/or to the
complex domain have been located in mathematical and physical literature for
some decades. Polynomials traditionally called complex Hermite ones are mostly
understood as polynomials in and which in fact makes them
polynomials in two real variables with complex coefficients. The present paper
proposes to investigate for the first time holomorphic Hermite polynomials in
two variables. Their algebraic and analytic properties are developed here.
While the algebraic properties do not differ too much for those considered so
far, their analytic features are based on a kind of non-rotational
orthogonality invented by van Eijndhoven and Meyers. Inspired by their
invention we merely follow the idea of Bargmann's seminal paper (1961) giving
explicit construction of reproducing kernel Hilbert spaces based on those
polynomials. "Homotopic" behavior of our new formation culminates in comparing
it to the very classical Bargmann space of two variables on one edge and the
aforementioned Hermite polynomials in and on the other. Unlike in
the case of Bargmann's basis our Hermite polynomials are not product ones but
factorize to it when bonded together with the first case of limit properties
leading both to the Bargmann basis and suitable form of the reproducing kernel.
Also in the second limit we recover standard results obeyed by Hermite
polynomials in and
Squeezed States and Hermite polynomials in a Complex Variable
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec
[J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of
coherent states, related to the Hermite polynomials in a complex variable which
are orthogonal with respect to a non-rotationally invariant measure. We
investigate relations between these coherent states and obtain the relationship
between them and the squeezed states of quantum optics. We also obtain a second
realization of the canonical coherent states in the Bargmann space of analytic
functions, in terms of a squeezed basis. All this is done in the flavor of the
classical approach of V. Bargmann [Commun. Pur. Appl. Math. 14, 187 (1961)].Comment: 15 page
- …