474 research outputs found
Deformed Heisenberg algebra and minimal length
A one-dimensional deformed Heisenberg algebra is studied. We
answer the question: For what function of deformation 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
Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces
We introduce two new families of quasi-exactly solvable (QES) extensions of
the oscillator in a -dimensional constant-curvature space. For the first
three members of each family, we obtain closed-form expressions of the energies
and wavefunctions for some allowed values of the potential parameters using the
Bethe ansatz method. We prove that the first member of each family has a hidden
sl(2,) symmetry and is connected with a QES equation of the first
or second type, respectively. One-dimensional results are also derived from the
-dimensional ones with , thereby getting QES extensions of the
Mathews-Lakshmanan nonlinear oscillator.Comment: 30 pages, 8 figures, published versio
Spectrum generating algebras for position-dependent mass oscillator Schrodinger equations
The interest of quadratic algebras for position-dependent mass Schr\"odinger
equations is highlighted by constructing spectrum generating algebras for a
class of d-dimensional radial harmonic oscillators with and a
specific mass choice depending on some positive parameter . Via some
minor changes, the one-dimensional oscillator on the line with the same kind of
mass is included in this class. The existence of a single unitary irreducible
representation belonging to the positive-discrete series type for and
of two of them for d=1 is proved. The transition to the constant-mass limit
is studied and deformed su(1,1) generators are constructed.
These operators are finally used to generate all the bound-state wavefunctions
by an algebraic procedure.Comment: 21 pages, no figure, 2 misprints corrected; published versio
Extending Romanovski polynomials in quantum mechanics
Some extensions of the (third-class) Romanovski polynomials (also called
Romanovski/pseudo-Jacobi polynomials), which appear in bound-state
wavefunctions of rationally-extended Scarf II and Rosen-Morse I potentials, are
considered. For the former potentials, the generalized polynomials satisfy a
finite orthogonality relation, while for the latter an infinite set of
relations among polynomials with degree-dependent parameters is obtained. Both
types of relations are counterparts of those known for conventional
polynomials. In the absence of any direct information on the zeros of the
Romanovski polynomials present in denominators, the regularity of the
constructed potentials is checked by taking advantage of the disconjugacy
properties of second-order differential equations of Schr\"odinger type. It is
also shown that on going from Scarf I to Scarf II or from Rosen-Morse II to
Rosen-Morse I potentials, the variety of rational extensions is narrowed down
from types I, II, and III to type III only.Comment: 25 pages, no figure, small changes, 3 additional references,
published versio
Application of nonlinear deformation algebra to a physical system with P\"oschl-Teller potential
We comment on a recent paper by Chen, Liu, and Ge (J. Phys. A: Math. Gen. 31
(1998) 6473), wherein a nonlinear deformation of su(1,1) involving two
deforming functions is realized in the exactly solvable quantum-mechanical
problem with P\" oschl-Teller potential, and is used to derive the well-known
su(1,1) spectrum-generating algebra of this problem. We show that one of the
defining relations of the nonlinear algebra, presented by the authors, is only
valid in the limiting case of an infinite square well, and we determine the
correct relation in the general case. We also use it to establish the correct
link with su(1,1), as well as to provide an algebraic derivation of the
eigenfunction normalization constant.Comment: 9 pages, LaTeX, no figure
Generalized Continuity Equation and Modified Normalization in PT-Symmetric Quantum Mechanics
The continuity equation relating the change in time of the position
probability density to the gradient of the probability current density is
generalized to PT-symmetric quantum mechanics. The normalization condition of
eigenfunctions is modified in accordance with this new conservation law and
illustrated with some detailed examples.Comment: 16 pages, amssy
New approach to (quasi)-exactly solvable Schrodinger equations with a position-dependent effective mass
By using the point canonical transformation approach in a manner distinct
from previous ones, we generate some new exactly solvable or quasi-exactly
solvable potentials for the one-dimensional Schr\"odinger equation with a
position-dependent effective mass. In the latter case, SUSYQM techniques
provide us with some additional new potentials.Comment: 11 pages, no figur
The Localization of -Wave and Quantum Effective Potential of a Quasi-Free Particle with Position-Dependent Mass
The properties of the s-wave for a quasi-free particle with
position-dependent mass(PDM) have been discussed in details. Differed from the
system with constant mass in which the localization of the s-wave for the free
quantum particle around the origin only occurs in two dimensions, the
quasi-free particle with PDM can experience attractive forces in dimensions
except D=1 when its mass function satisfies some conditions. The effective mass
of a particle varying with its position can induce effective interaction which
may be attractive in some cases. The analytical expressions of the
eigenfunctions and the corresponding probability densities for the s-waves of
the two- and three-dimensional systems with a special PDM are given, and the
existences of localization around the origin for these systems are shown.Comment: 12pages, 8 figure
PT-symmetric square well and the associated SUSY hierarchies
The PT-symmetric square well problem is considered in a SUSY framework. When
the coupling strength lies below the critical value
where PT symmetry becomes spontaneously broken, we find a hierarchy of SUSY
partner potentials, depicting an unbroken SUSY situation and reducing to the
family of -like potentials in the limit. For above
, there is a rich diversity of SUSY hierarchies, including
some with PT-symmetry breaking and some with partial PT-symmetry restoration.Comment: LaTeX, 18 pages, no figure; broken PT-symmetry case added (Sec. 6
Complexified PSUSY and SSUSY interpretations of some PT-symmetric Hamiltonians possessing two series of real energy eigenvalues
We analyze a set of three PT-symmetric complex potentials, namely harmonic
oscillator, generalized Poschl-Teller and Scarf II, all of which reveal a
double series of energy levels along with the corresponding superpotential.
Inspired by the fact that two superpotentials reside naturally in order-two
parasupersymmetry (PSUSY) and second-derivative supersymmetry (SSUSY) schemes,
we complexify their frameworks to successfully account for the three
potentials.Comment: LaTeX2e, 28 pages, no figure
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