932 research outputs found
Effect of the generalized uncertainty principle on Galilean and Lorentz transformations
Generalized Uncertainty Principle (GUP) was obtained in string theory and
quantum gravity and suggested the existence of a fundamental minimal length
which, as was established, can be obtained within the deformed Heisenberg
algebra. We use the deformed commutation relations or in classical case
(studied in this paper) the deformed Poisson brackets, which are invariant with
respect to the translation in configurational space. We have found
transformations relating coordinates and times of moving and rest frames of
reference in the space with GUP in the first order over parameter of
deformation. For the non-relativistic case we find the deformed Galilean
transformation which is similar to the Lorentz one written for Euclidean space
with signature . The role of the speed of light here plays some
velocity related to the parameter of deformation, which as we estimate is
many order of magnitude larger than the speed of light . The coordinates of the rest and moving frames of reference for
relativistic particle in the space with GUP satisfy the Lorentz transformation
with some effective speed of light. We estimate that the relative deviation of
this effective speed of light from is . The influence of GUP on the motion of particle and the
Lorentz transformation in the first order over parameter of deformation is
hidden in relativistic effects.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1301.189
Supersymmetric Method for Constructing Quasi-Exactly Solvable Potentials
We propose a new method for constructing the quasi-exactly solvable (QES)
potentials with two known eigenstates using supersymmetric quantum mechanics.
General expression for QES potentials with explicitly known energy levels and
wave functions of ground state and excited state are obtained. Examples of new
QES potentials are considered.Comment: 11 pages, latex, to appear in Cond. Matt. Phys. (Lviv) (Proceedings
of INTAS-Ukraine Workshop on Condensed Matter Physics, May, Lviv, 1998
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 , nor symmetry between the and cases, both features being connected with
supersymmetry or, equivalently, the 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 states corresponding to small, intermediate and
very large 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
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 , .
We establish that whenever there is a nonzero minimal uncertainty in momentum,
i.e., for , the correction to the harmonic oscillator eigenvalues
due to the electric field is level dependent. In the opposite case, i.e., for
, 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 . Then we consider the problem of a
-dimensional harmonic oscillator in the case of isotropic nonzero minimal
uncertainties in the position coordinates, depending on two parameters ,
. 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 -dimensional
problems.Comment: 26 pages, no figure, new section 2.4 + small changes, accepted in J.
Phys. A, Special issue on Supersymmetric Quantum Mechanic
Quantum brachistochrone problem for spin-1 in a magnetic field
We study quantum brachistochrone problem for the spin-1 system in a magnetic
field of a constant absolute value. Such system gives us a possibility to
examine in detail the statement of papers [A. Carlini {\it et al.}, Phys. Rev.
Lett. {\bf 96}, 060503 (2006), D. C. Brody, D. W. Hook, J. Phys. A {\bf 39},
L167, (2006)] that {\it the state vectors realizing the evolution with the
minimal time of passage evolve along the subspace spanned by the initial and
final state vectors.} Using explicit example we show the existence of quantum
brachistochrone with minimal possible time, but the state vector of which,
during the evolution {\em leaves} the subspace spanned by the initial and final
state vectors. This is the result of the choice of more constrained Hamiltonian
then assumed in the general quantum brachistochrone problem, but what is worth
noting, despite that such evolution is more complicated it is still time
optimal. This might be important for experiment, where general Hamiltonian with
the all allowed parameters is difficult to implement, but constrained one
depending on magnetic field can be realized. However for pre-constrained
Hamiltonian not all final states are accessible. Present result does not
contradict general statement of the quantum brachistochrone problem, but gives
new insight how time optimal passage can be realized.Comment: 7 pages, no figure
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