Deformed su(1,1) Algebra as a Model for Quantum Oscillators

The Lie algebra $\mathfrak{su}(1,1)$ can be deformed by a reflection operator, in such a way that the positive discrete series representations of $\mathfrak{su}(1,1)$ can be extended to representations of this deformed algebra $\mathfrak{su}(1,1)_\gamma$. Just as the positive discrete series representations of $\mathfrak{su}(1,1)$ can be used to model a quantum oscillator with Meixner-Pollaczek polynomials as wave functions, the corresponding representations of $\mathfrak{su}(1,1)_\gamma$ 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 $N\rightarrow \infty$ limit of the bivariate Krawtchouk polynomials as a product of one-variable Hermite polynomials

The su(2)\mathfrak{su}(2) Krawtchouk oscillator model under the CP{\cal C}{\cal P} deformed symmetry

We define a new algebra, which can formally be considered as a ${\cal C}{\cal P}$ deformed $\mathfrak{su}(2)$ 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 $p=1/2$, $K_{2n}(k;1/2,2j)$ and $K_{2n}(k-1;1/2,2j-2)$. The dynamical symmetry of the model is the newly introduced $\mathfrak{su}(2)_{{\cal C}{\cal P}}$ 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 $\pm\sqrt{k}$, 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 $q$-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 $h \to 0$ ($q \to 1$), 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 $q \ll 1$, 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 $q$ and $n$. 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 $a$ and $b$, but the Laguerre polynomials depend only on the parameter $a$. 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