More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum

We continue our previous application of supersymmetric quantum mechanical methods to eigenvalue problems in the context of some deformed canonical commutation relations leading to nonzero minimal uncertainties in position and/or momentum. Here we determine for the first time the spectrum and the eigenvectors of a one-dimensional harmonic oscillator in the presence of a uniform electric field in terms of the deforming parameters $\alpha$, $\beta$. We establish that whenever there is a nonzero minimal uncertainty in momentum, i.e., for $\alpha \ne 0$, the correction to the harmonic oscillator eigenvalues due to the electric field is level dependent. In the opposite case, i.e., for $\alpha = 0$, we recover the conventional quantum mechanical picture of an overall energy-spectrum shift even when there is a nonzero minimum uncertainty in position, i.e., for $\beta \ne 0$. Then we consider the problem of a $D$-dimensional harmonic oscillator in the case of isotropic nonzero minimal uncertainties in the position coordinates, depending on two parameters $\beta$, $\beta'$. We extend our methods to deal with the corresponding radial equation in the momentum representation and rederive in a simple way both the spectrum and the momentum radial wave functions previously found by solving the differential equation. This opens the way to solving new $D$-dimensional problems.Comment: 26 pages, no figure, new section 2.4 + small changes, accepted in J. Phys. A, Special issue on Supersymmetric Quantum Mechanic

Dirac oscillator with nonzero minimal uncertainty in position

In the context of some deformed canonical commutation relations leading to isotropic nonzero minimal uncertainties in the position coordinates, a Dirac equation is exactly solved for the first time, namely that corresponding to the Dirac oscillator. Supersymmetric quantum mechanical and shape-invariance methods are used to derive both the energy spectrum and wavefunctions in the momentum representation. As for the conventional Dirac oscillator, there are neither negative-energy states for $E=-1$, nor symmetry between the $l = j - {1/2}$ and $l = j + {1/2}$ cases, both features being connected with supersymmetry or, equivalently, the $\omega \to - \omega$ transformation. In contrast with the conventional case, however, the energy spectrum does not present any degeneracy pattern apart from that associated with the rotational symmetry. More unexpectedly, deformation leads to a difference in behaviour between the $l = j - {1/2}$ states corresponding to small, intermediate and very large $j$ values in the sense that only for the first ones supersymmetry remains unbroken, while for the second ones no bound state does exist.Comment: 28 pages, no figure, submitted to JP

Lorentz-covariant deformed algebra with minimal length and application to the 1+1-dimensional Dirac oscillator

The $D$-dimensional $(\beta, \beta')$-two-parameter deformed algebra introduced by Kempf is generalized to a Lorentz-covariant algebra describing a ($D+1$)-dimensional quantized space-time. In the D=3 and $\beta=0$ case, the latter reproduces Snyder algebra. The deformed Poincar\'e transformations leaving the algebra invariant are identified. It is shown that there exists a nonzero minimal uncertainty in position (minimal length). The Dirac oscillator in a 1+1-dimensional space-time described by such an algebra is studied in the case where $\beta'=0$. Extending supersymmetric quantum mechanical and shape-invariance methods to energy-dependent Hamiltonians provides exact bound-state energies and wavefunctions. Physically acceptable states exist for $\beta < 1/(m^2 c^2)$. A new interesting outcome is that, in contrast with the conventional Dirac oscillator, the energy spectrum is bounded.Comment: 20 pages, no figure, some very small changes, published versio

Maths-type q-deformed coherent states for q > 1

Maths-type q-deformed coherent states with $q > 1$ allow a resolution of unity in the form of an ordinary integral. They are sub-Poissonian and squeezed. They may be associated with a harmonic oscillator with minimal uncertainties in both position and momentum and are intelligent coherent states for the corresponding deformed Heisenberg algebra.Comment: LaTeX2e, 15 pages + 3 eps figures, figures replace

WKB approximation in deformed space with minimal length

The WKB approximation for deformed space with minimal length is considered. The Bohr-Sommerfeld quantization rule is obtained. A new interesting feature in presence of deformation is that the WKB approximation is valid for intermediate quantum numbers and can be invalid for small as well as very large quantum numbers. The correctness of the rule is verified by comparing obtained results with exact expressions for corresponding spectra.Comment: 13 pages Now it is avaible at http://stacks.iop.org/0305-4470/39/37

Composite system in deformed space with minimal length

For composite systems made of $N$ different particles living in a space characterized by the same deformed Heisenberg algebra, but with different deformation parameters, we define the total momentum and the center-of-mass position to first order in the deformation parameters. Such operators satisfy the deformed algebra with new effective deformation parameters. As a consequence, a two-particle system can be reduced to a one-particle problem for the internal motion. As an example, the correction to the hydrogen atom $n$S energy levels is re-evaluated. Comparison with high-precision experimental data leads to an upper bound of the minimal length for the electron equal to $3.3\times 10^{-18} {\rm m}$. The effective Hamiltonian describing the center-of-mass motion of a macroscopic body in an external potential is also found. For such a motion, the effective deformation parameter is substantially reduced due to a factor $1/N^2$. This explains the strangely small result previously obtained for the minimal length from a comparison with the observed precession of the perihelion of Mercury. From our study, an upper bound of the minimal length for quarks equal to $2.4\times 10^{-17}{\rm m}$ is deduced, which appears close to that obtained for electrons.Comment: 22 pages, no figure; small additions in Secs. I, III and V

Deformed Heisenberg algebra and minimal length

A one-dimensional deformed Heisenberg algebra $[X,P]=if(P)$ is studied. We answer the question: For what function of deformation $f(P)$ there exists a nonzero minimal uncertainty in position (minimal length). We also find an explicit expression for the minimal length in the case of arbitrary function of deformation.Comment: to be published in JP

Lorentz-covariant deformed algebra with minimal length

The $D$-dimensional two-parameter deformed algebra with minimal length introduced by Kempf is generalized to a Lorentz-covariant algebra describing a ($D+1$)-dimensional quantized space-time. For D=3, it includes Snyder algebra as a special case. The deformed Poincar\'e transformations leaving the algebra invariant are identified. Uncertainty relations are studied. In the case of D=1 and one nonvanishing parameter, the bound-state energy spectrum and wavefunctions of the Dirac oscillator are exactly obtained.Comment: 8 pages, no figure, presented at XV International Colloquium on Integrable Systems and Quantum Symmetries (ISQS-15), Prague, June 15-17, 200

One dimensional Coulomb-like problem in deformed space with minimal length

Spectrum and eigenfunctions in the momentum representation for 1D Coulomb potential with deformed Heisenberg algebra leading to minimal length are found exactly. It is shown that correction due to the deformation is proportional to square root of the deformation parameter. We obtain the same spectrum using Bohr-Sommerfeld quantization condition.Comment: 11 pages, typos corrected, references adde