arXiv subjects
Publications and source records attributed to E. I. Jafarov.
The two-dimensional relativistic configurational $\vec r$-space is proposed and the exactly solvable finite-difference model of the harmonic oscillator in this space is constructed. The wave functions of the stationary states and the corresponding energy spectrum are found for the model under consideration. It is shown that, they have correct non-relativistic limit. Explicit form of the plane momentum operator is determined.
In the beginning of the 1950's, Wigner introduced a fundamental deformation from the canonical quantum mechanical harmonic oscillator, which is nowadays sometimes called a Wigner quantum oscillator or a parabose oscillator. Also, in quantum mechanics the so-called Wigner distribution is considered to be the closest quantum analogue of the classical probability distribution over the phase space. In this article, we consider which definition for such distribution function could be used in the case of non-canonical quantum mechanics. We then explicitly compute two different expressions for this distribution function for the case of the parabose oscillator. Both expressions turn out to be multiple sums involving (generalized) Laguerre polynomials. Plots then show that the Wigner distribution function for the ground state of the parabose oscillator is similar in behaviour to the Wigner distribution function of the first excited state of the canonical quantum oscillator.
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.
The SU(1,1) coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU(1,1) coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr-Sommerfeld quantization rule yields the exact expression for the energy spectrum.
Exactly solvable $N$-dimensional model of the quantum isotropic singular oscillator in the relativistic configurational $\vec r_N$-space is proposed. It is shown that through the simple substitutions the finite-difference equation for the $N$-dimensional singular oscillator can be reduced to the similar finite-difference equation for the relativistic isotropic three-dimensional singular oscillator. We have found the radial wavefunctions and energy spectrum of the problem and constructed a dynamical symmetry algebra.
An exact approach for the factorization of the relativistic linear singular oscillator is proposed. This model is expressed by the finite-difference Schrödinger-like equation. We have found finite-difference raising and lowering operators, which are with the Hamiltonian operator form the close Lie algebra of the $SU(1, 1)$ group.
Exactly solvable model of the quantum isotropic three-dimensional singular oscillator in the relativistic configurational $\vec r$-space is proposed. We have found the radial wavefunctions, which are expressed through the continuous dual Hahn polynomials and energy spectrum for the model under consideration. It is shown that they have the correct non-relativistic limits.