38 research outputs found
Shape invariant hypergeometric type operators with application to quantum mechanics
A hypergeometric type equation satisfying certain conditions defines either a
finite or an infinite system of orthogonal polynomials. The associated special
functions are eigenfunctions of some shape invariant operators. These operators
can be analysed together and the mathematical formalism we use can be extended
in order to define other shape invariant operators. All the considered shape
invariant operators are directly related to Schrodinger type equations.Comment: More applications available at http://fpcm5.fizica.unibuc.ro/~ncotfa
Icosahedral multi-component model sets
A quasiperiodic packing Q of interpenetrating copies of C, most of them only
partially occupied, can be defined in terms of the strip projection method for
any icosahedral cluster C. We show that in the case when the coordinates of the
vectors of C belong to the quadratic field Q[\sqrt{5}] the dimension of the
superspace can be reduced, namely, Q can be re-defined as a multi-component
model set by using a 6-dimensional superspace.Comment: 7 pages, LaTeX2e in IOP styl
On the linear representations of the symmetry groups of single-wall carbon nanotubes
The positions of atoms forming a carbon nanotube are usually described by
using a system of generators of the symmetry group. Each atomic position
corresponds to an element of the set Z x {0,1,...,n} x {0,1}, where n depends
on the considered nanotube. We obtain an alternate rather different description
by starting from a three-axes description of the honeycomb lattice. In our
mathematical model, which is a factor space defined by an equivalence relation
in the set {(v_0,v_1,v_2)\in Z^3 | v_0+v_1+v_2\in {0,1}}, the neighbours of an
atomic position can be described in a simpler way, and the mathematical objects
with geometric or physical significance have a simpler and more symmetric form.
We present some results concerning the linear representations of single-wall
carbon nanotubes in order to illustrate the proposed approach.Comment: Major change of content. More details will be available at
http://fpcm5.fizica.unibuc.ro/~ncotfa
Symmetry properties of Penrose type tilings
The Penrose tiling is directly related to the atomic structure of certain
decagonal quasicrystals and, despite its aperiodicity, is highly symmetric. It
is known that the numbers 1, , , , ..., where
, are scaling factors of the Penrose tiling. We show that
the set of scaling factors is much larger, and for most of them the number of
the corresponding inflation centers is infinite.Comment: Paper submitted to Phil. Mag. (for Proceedings of Quasicrystals: The
Silver Jubilee, Tel Aviv, 14-19 October, 2007
Properties of finite Gaussians and the discrete-continuous transition
Weyl's formulation of quantum mechanics opened the possibility of studying
the dynamics of quantum systems both in infinite-dimensional and
finite-dimensional systems. Based on Weyl's approach, generalized by Schwinger,
a self-consistent theoretical framework describing physical systems
characterised by a finite-dimensional space of states has been created. The
used mathematical formalism is further developed by adding finite-dimensional
versions of some notions and results from the continuous case. Discrete
versions of the continuous Gaussian functions have been defined by using the
Jacobi theta functions. We continue the investigation of the properties of
these finite Gaussians by following the analogy with the continuous case. We
study the uncertainty relation of finite Gaussian states, the form of the
associated Wigner quasi-distribution and the evolution under free-particle and
quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis
is put on the recovery of the known continuous-limit results when the dimension
of the system increases.Comment: 21 pages, 4 figure
Gazeau-Klauder type coherent states for hypergeometric type operators
The hypergeometric type operators are shape invariant, and a factorization
into a product of first order differential operators can be explicitly
described in the general case. Some additional shape invariant operators
depending on several parameters are defined in a natural way by starting from
this general factorization. The mathematical properties of the eigenfunctions
and eigenvalues of the operators thus obtained depend on the values of the
involved parameters. We study the parameter dependence of orthogonality, square
integrability and of the monotony of eigenvalue sequence. The obtained results
allow us to define certain systems of Gazeau-Klauder coherent states and to
describe some of their properties. Our systematic study recovers a number of
well-known results in a natural unified way and also leads to new findings.Comment: An error occurring in Theorem 12 and Theorem 13 has been correcte
A search on the Nikiforov-Uvarov formalism
An alternative treatment is proposed for the calculations carried out within
the frame of Nikiforov-Uvarov method, which removes a drawback in the original
theory and by pass some difficulties in solving the Schrodinger equation. The
present procedure is illustrated with the example of orthogonal polynomials.
The relativistic extension of the formalism is discussed.Comment: 10 page
On the Spectrum of Field Quadratures for a Finite Number of Photons
The spectrum and eigenstates of any field quadrature operator restricted to a
finite number of photons are studied, in terms of the Hermite polynomials.
By (naturally) defining \textit{approximate} eigenstates, which represent
highly localized wavefunctions with up to photons, one can arrive at an
appropriate notion of limit for the spectrum of the quadrature as goes to
infinity, in the sense that the limit coincides with the spectrum of the
infinite-dimensional quadrature operator. In particular, this notion allows the
spectra of truncated phase operators to tend to the complete unit circle, as
one would expect. A regular structure for the zeros of the Christoffel-Darboux
kernel is also shown.Comment: 16 pages, 11 figure
Weak mutually unbiased bases
Quantum systems with variables in are considered. The
properties of lines in the phase space of
these systems, are studied. Weak mutually unbiased bases in these systems are
defined as bases for which the overlap of any two vectors in two different
bases, is equal to or alternatively to one of the
(where is a divisor of apart from ). They are designed for the
geometry of the phase space, in the sense
that there is a duality between the weak mutually unbiased bases and the
maximal lines through the origin. In the special case of prime , there are
no divisors of apart from and the weak mutually unbiased bases are
mutually unbiased bases