97 research outputs found
Deformed su(1,1) Algebra as a Model for Quantum Oscillators
The Lie algebra can be deformed by a reflection
operator, in such a way that the positive discrete series representations of
can be extended to representations of this deformed
algebra . Just as the positive discrete series
representations of can be used to model a quantum
oscillator with Meixner-Pollaczek polynomials as wave functions, the
corresponding representations of can be utilized to
construct models of a quantum oscillator. In this case, the wave functions are
expressed in terms of continuous dual Hahn polynomials. We study some
properties of these wave functions, and illustrate some features in plots. We
also discuss some interesting limits and special cases of the obtained
oscillator models
A superintegrable finite oscillator in two dimensions with SU(2) symmetry
A superintegrable finite model of the quantum isotropic oscillator in two
dimensions is introduced. It is defined on a uniform lattice of triangular
shape. The constants of the motion for the model form an SU(2) symmetry
algebra. It is found that the dynamical difference eigenvalue equation can be
written in terms of creation and annihilation operators. The wavefunctions of
the Hamiltonian are expressed in terms of two known families of bivariate
Krawtchouk polynomials; those of Rahman and those of Tratnik. These polynomials
form bases for SU(2) irreducible representations. It is further shown that the
pair of eigenvalue equations for each of these families are related to each
other by an SU(2) automorphism. A finite model of the anisotropic oscillator
that has wavefunctions expressed in terms of the same Rahman polynomials is
also introduced. In the continuum limit, when the number of grid points goes to
infinity, standard two-dimensional harmonic oscillators are obtained. The
analysis provides the limit of the bivariate Krawtchouk
polynomials as a product of one-variable Hermite polynomials
The Krawtchouk oscillator model under the deformed symmetry
We define a new algebra, which can formally be considered as a deformed Lie algebra. Then, we present a one-dimensional
quantum oscillator model, of which the wavefunctions of even and odd states are
expressed by Krawtchouk polynomials with fixed , and
. The dynamical symmetry of the model is the newly
introduced algebra. The model itself
gives rise to a finite and discrete spectrum for all physical operators (such
as position and momentum). Among the set of finite oscillator models it is
unique in the sense that any specific limit reducing it to a known oscillator
models does not exist.Comment: Contribution to the 30th International Colloquium on Group
Theoretical Methods in Physics (Ghent, Belgium, 2014). To be published in
Journal of Physics: Conference Serie
A finite oscillator model related to sl(2|1)
We investigate a new model for the finite one-dimensional quantum oscillator
based upon the Lie superalgebra sl(2|1). In this setting, it is natural to
present the position and momentum operators of the oscillator as odd elements
of the Lie superalgebra. The model involves a parameter p (0<p<1) and an
integer representation label j. In the (2j+1)-dimensional representations W_j
of sl(2|1), the Hamiltonian has the usual equidistant spectrum. The spectrum of
the position operator is discrete and turns out to be of the form
, where k=0,1,...,j. We construct the discrete position wave
functions, which are given in terms of certain Krawtchouk polynomials. These
wave functions have appealing properties, as can already be seen from their
plots. The model is sufficiently simple, in the sense that the corresponding
discrete Fourier transform (relating position wave functions to momentum wave
functions) can be constructed explicitly
The Wigner function of a q-deformed harmonic oscillator model
The phase space representation for a q-deformed model of the quantum harmonic
oscillator is constructed. We have found explicit expressions for both the
Wigner and Husimi distribution functions for the stationary states of the
-oscillator model under consideration. The Wigner function is expressed as a
basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is
shown that, in the limit case (), both the Wigner and Husimi
distribution functions reduce correctly to their well-known non-relativistic
analogues. Surprisingly, examination of both distribution functions in the
q-deformed model shows that, when , their behaviour in the phase space
is similar to the ground state of the ordinary quantum oscillator, but with a
displacement towards negative values of the momentum. We have also computed the
mean values of the position and momentum using the Wigner function. Unlike the
ordinary case, the mean value of the momentum is not zero and it depends on
and . The ground-state like behaviour of the distribution functions for
excited states in the q-deformed model opens quite new perspectives for further
experimental measurements of quantum systems in the phase space.Comment: 16 pages, 24 EPS figures, uses IOP style LaTeX, some misprints are
correctd and journal-reference is adde
Exact solution of the position-dependent mass Schr\"odinger equation with the completely positive oscillator-shaped quantum well potential
Two exactly-solvable confined models of the completely positive
oscillator-shaped quantum well are proposed. Exact solutions of the
position-dependent mass Schr\"odinger equation corresponding to the proposed
quantum well potentials are presented. It is shown that the discrete energy
spectrum expressions of both models depend on certain positive confinement
parameters. The spectrum exhibits positive equidistant behavior for the model
confined only with one infinitely high wall and non-equidistant behavior for
the model confined with the infinitely high wall from both sides. Wavefunctions
of the stationary states of the models under construction are expressed through
the Laguerre and Jacobi polynomials. In general, the Jacobi polynomials
appearing in wavefunctions depend on parameters and , but the Laguerre
polynomials depend only on the parameter . Some limits and special cases of
the constructed models are discussed.Comment: 20 pages, 4 figure
The Wigner function of a semiconfined harmonic oscillator model with a position-dependent effective mass
We develop a phase-space representation concept in terms of the Wigner
function for a quantum harmonic oscillator model that exhibits the
semiconfinement effect through its mass varying with the position. The new
method is applied for the analytical computation of the Wigner distribution
function for such a semiconfinement quantum system. The method allows for
suppression of the divergence of the integrand in the definition of the quantum
distribution function and leads to the computation of its analytical
expressions for the stationary states of the semiconfined oscillator model.
Both cases of the presence and absence of the applied external homogeneous
field for this quantum system are studied. Obtained exact expressions of the
Wigner distribution function are expressed through the Bessel function of the
first kind and Laguerre polynomials. Further, some of the special cases and
limits are discussed in detail.Comment: 10 pages, 9 figure
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