Singular inverse square potential in coordinate space with a minimal length

Abstract

The problem of a particle of mass m in the field of the inverse square potential is studied in quantum mechanics with a generalized uncertainty principle, characterized by the existence of a minimal length. Using the coordinate representation, for a specific form of the generalized uncertainty relation, we solve the deformed Schrödinger equation analytically in terms of confluent Heun functions. We explicitly show the regularizing effect of the minimal length on the singularity of the potential. We discuss the problem of bound states in detail and we derive an expression for the energy spectrum in a natural way from the square integrability condition; the results are in complete agreement with the literature.

arXiv commentPublished in Annals of Physics

References in corpus (52)

• Hilbert Space Representation of the Minimal Length Uncertainty Relation
• Discreteness of area and volume in quantum gravity
• Quantum gravity and minimum length
• Discreteness of Space from the Generalized Uncertainty Principle
• The algebraic structure of the generalized uncertainty principle
• Nonpointlike Particles in Harmonic Oscillators
• Minimal Length Uncertainty Relation and Ultraviolet Regularisation
• Minimal Length Uncertainty Relation and Hydrogen Atom
• Uncertainty Relation in Quantum Mechanics with Quantum Group Symmetry
• Exact Solution of the Harmonic Oscillator in Arbitrary Dimensions with Minimal Length Uncertainty Relations
• Discreteness of Space from GUP II: Relativistic Wave Equations
• A Higher Order GUP with Minimal Length Uncertainty and Maximal Momentum
• Dirac oscillator with nonzero minimal uncertainty in position
• Minimal length, maximal momentum and Hilbert space representation of quantum mechanics
• The effects of minimal length and maximal momentum on the transition rate of ultra cold neutrons in gravitational field
• On Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta
• Singular Potentials and Limit Cycles
• Hydrogen-atom spectrum under a minimal-length hypothesis
• Renormalization in Quantum Mechanics
• Renormalization of the Inverse Square Potential
• Minimal Length Uncertainty Relation and gravitational quantum well
• Near-Horizon Conformal Structure of Black Holes
• Anomalies in Quantum Mechanics: the 1/r^2 Potential
• Regularization of the Singular Inverse Square Potential in Quantum Mechanics with a Minimal length
• Perturbation hydrogen-atom spectrum in deformed space with minimal length
• Further Evidence for the Conformal Structure of a Schwarzschild Black Hole in an Algebraic Approach
• The Casimir Effect in the Presence of a Minimal Length
• Statistical physics in deformed spaces with minimal length
• Minimal Length and Bouncing Particle Spectrum
• Horizon States for AdS Black Holes
• Exact solutions of the (2+1) Dimensional Dirac equation in a constant magnetic field in the presence of a minimal length
• The singular inverse square potential, limit cycles and self-adjoint extensions
• One dimensional Coulomb-like problem in deformed space with minimal length
• Corrections to the ns-levels of hydrogen atom in deformed space with minimal length
• Lorentz-covariant deformed algebra with minimal length
• The Renormalization Group Limit Cycle for the 1/r^2 Potential
• Constraining the generalized uncertainty principle with cold atoms
• Hydrogen atom in momentum space with a minimal length
• On Noncommutative Geometric Regularisation
• Singular inverse square potential in arbitrary dimensions with a minimal length: Application to the motion of a dipole in a cosmic string background
• A Neutral Atom and a Charged Wire: From Elastic scattering to Absorption
• On the limit cycle for the 1/r^2 potential in momentum space
• Exact solution of the Klein Gordon equation in the presence of a minimal length
• Kratzer's molecular potential in quantum mechanics with a generalized uncertainty principle
• Casimir-Polder intermolecular forces in minimal length theories
• Charged particle in the field an electric quadrupole in two dimensions
• One-dimensional hydrogen atom with minimal length uncertainty and maximal momentum
• A note on the one-dimensional hydrogen atom with minimal length uncertainty
• Singular inverse-square potential: renormalization and self-adjoint extensions for medium to weak coupling
• Exactly solvable problems in the momentum space with a minimum uncertainty in position
• Pseudoharmonic oscillator in quantum mechanics with a minimal length
• Renormalization of the strongly attractive inverse square potential: Taming the singularity

Cited by in corpus (5)

• Heun Functions and Some of Their Applications in Physics
• 30 years in: Quo vadis generalized uncertainty principle?
• On the attractive inverse-square potential in the induced electric dipole system under the influence of the harmonic oscillator
• Modified attractive inverse-square potential in the induced electric dipole system
• Regularization of ${\delta }^{\prime }$ potential in general case of deformed space with minimal length