Exact Solution of the Harmonic Oscillator in Arbitrary Dimensions with Minimal Length Uncertainty Relations

Abstract

We determine the energy eigenvalues and eigenfunctions of the harmonic oscillator where the coordinates and momenta are assumed to obey the modified commutation relations [x_i,p_j]=i hbar[(1+ beta p^2) delta_{ij} + beta' p_i p_j]. These commutation relations are motivated by the fact they lead to the minimal length uncertainty relations which appear in perturbative string theory. Our solutions illustrate how certain features of string theory may manifest themselves in simple quantum mechanical systems through the modification of the canonical commutation relations. We discuss whether such effects are observable in precision measurements on electrons trapped in strong magnetic fields.

15 pages, REVTEX4, new section V added

References in corpus (28)

• Dirichlet-Branes and Ramond-Ramond Charges
• Noncommutative Field Theory
• Hilbert Space Representation of the Minimal Length Uncertainty Relation
• Effective Field Theory, Black Holes, and the Cosmological Constant
• The dS/CFT Correspondence
• Probable Values of the Cosmological Constant in a Holographic Theory
• Nonpointlike Particles in Harmonic Oscillators
• D-branes and Short Distances in String Theory
• Minimal Length Uncertainty Relation and Hydrogen Atom
• Mass, Entropy and Holography in Asymptotically de Sitter Spaces
• UV/IR Relations in AdS Dynamics
• A Small Cosmological Constant from a Large Extra Dimension
• Timelike T-Duality, de Sitter Space, Large $N$ Gauge Theories and Topological Field Theory
• The Effect of the Minimal Length Uncertainty Relation on the Density of States and the Cosmological Constant Problem
• Self-tuning flat domain walls in 5d gravity and string theory
• Inflation as a Probe of Short Distance Physics
• Inflation and the dS/CFT Correspondence
• Cosmological Breaking of Supersymmetry?
• Duality and the Signature of Space-Time
• Perturbation spectrum in inflation with cutoff
• Mode Generating Mechanism in Inflation with Cutoff
• Past electron-positron g-2 experiments yielded sharpest bound on CPT violation for point particles
• String Theory and the Space-Time Uncertainty Principle
• On the Quantization of Nambu Brackets
• Deconstructing de Sitter
• D-Particle Dynamics and The Space-Time Uncertainty Relation
• On the Space-Time Uncertainty Principle and Holography
• Space-Time Uncertainty and Noncommutativity in String Theory

Cited by in corpus (194)

• Minimal Length Scale Scenarios for Quantum Gravity
• The Effect of the Minimal Length Uncertainty Relation on the Density of States and the Cosmological Constant Problem
• Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem
• Generalized Uncertainty Principle: Approaches and Applications
• Short Distance vs. Long Distance Physics: The Classical Limit of the Minimal Length Uncertainty Relation
• Dirac oscillator with nonzero minimal uncertainty in position
• Review on Generalized Uncertainty Principle
• Hydrogen-atom spectrum under a minimal-length hypothesis
• Probing deformed commutators with macroscopic harmonic oscillators
• Minimal Length Uncertainty Relation and gravitational quantum well
• Displacement operator for quantum systems with position-dependent mass
• Composite system in deformed space with minimal length
• Lorentz-covariant deformed algebra with minimal length and application to the 1+1-dimensional Dirac oscillator
• Minimal Length Uncertainty Relation and the Hydrogen Spectrum
• Formulation, Interpretation and Application of non-Commutative Quantum Mechanics
• Gauge Theories under Incorporation of a Generalized Uncertainty Principle
• Covariant realizations of kappa-deformed space
• Regularization of the Singular Inverse Square Potential in Quantum Mechanics with a Minimal length
• Quantum Mechanics and the Generalized Uncertainty Principle
• Statistical physics in deformed spaces with minimal length
• The Big-Bang Singularity in the framework of a Generalized Uncertainty Principle
• Casimir Effect in the Presence of Minimal Lengths
• Quantum Black Hole in the Generalized Uncertainty Principle Framework
• Harmonic oscillator with minimal length uncertainty relations and ladder operators
• Exact solutions of the (2+1) Dimensional Dirac equation in a constant magnetic field in the presence of a minimal length
• One dimensional Coulomb-like problem in deformed space with minimal length
• Effects of quantum gravity on black holes
• 30 years in: Quo vadis generalized uncertainty principle?
• Deformed Heisenberg algebra with minimal length and equivalence principle
• Constraints on the Generalized Uncertainty Principle from Black Hole Thermodynamics
• Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework
• Massive quantum systems as interfaces of quantum mechanics and gravity
• Quantum Dynamics of the Taub Universe in a Generalized Uncertainty Principle framework
• Hydrogen atom in momentum space with a minimal length
• Deformed Symmetry in Snyder Space and Relativistic Particle Dynamics
• More on a SUSYQM approach to the harmonic oscillator with nonzero minimal uncertainties in position and/or momentum
• Singular inverse square potential in arbitrary dimensions with a minimal length: Application to the motion of a dipole in a cosmic string background
• Dilaton Cosmology, Noncommutativity and Generalized Uncertainty Principle
• Scattering problem in deformed space with minimal length
• Generalized kappa-deformed spaces, star-products, and their realizations
• On the Minimal Length Uncertainty Relation and the Foundations of String Theory
• Maths-type q-deformed coherent states for q > 1
• The Most General Form of Deformation of the Heisenberg Algebra from the Generalized Uncertainty Principle
• Generalized uncertainty principle in Bianchi type I quantum cosmology
• Non-commutative SU(N) gauge theories and asymptotic freedom
• Exact solution of the Klein Gordon equation in the presence of a minimal length
• Minimal Uncertainty in Momentum: The Effects of IR Gravity on Quantum Mechanics
• Time Crystals from Minimum Time Uncertainty
• On Quantum Gravity Tests with Composite Particles
• Incorporation of Generalized Uncertainty Principle into Lifshitz Field Theories
• The Path Integral Quantization corresponding to the Deformed Heisenberg Algebra
• Quantum Gravitational Corrections to the Real Klein-Gordon Field in the Presence of a Minimal Length
• Position and Momentum Uncertainties of the Normal and Inverted Harmonic Oscillators under the Minimal Length Uncertainty Relation
• Electroweak Theory with a Minimal Length
• Free particle wavefunction in light of the minimum-length deformed quantum mechanics and some of its phenomenological implications
• On γ-rigid regime of the Bohr-Mottelson Hamiltonian in the presence of a minimal length
• Singular inverse square potential in coordinate space with a minimal length
• Noncommutative Differential Forms on the kappa-deformed Space
• Confined One Dimensional Harmonic Oscillator as a Two-Mode System
• Minimal Length Effects on Chaotic Motion of Particles around Black Hole Horizon
• Corrections to the black body radiation due to minimum-length deformed quantum mechanics
• Kratzer's molecular potential in quantum mechanics with a generalized uncertainty principle
• Deformation of the Wheeler-DeWitt Equation
• Casimir-Polder intermolecular forces in minimal length theories
• Noncommutativity, generalized uncertainty principle and FRW cosmology
• Conformal Invariance in noncommutative geometry and mutually interacting Snyder Particles
• Quantum Wells and the Generalized Uncertainty Principle
• The Soccer-Ball Problem
• On the (2+1)-dimensional Dirac equation in a constant magnetic field with a minimal length uncertainty
• Influence of a Minimal Length on the Creation of Scalar Particles
• Deviation from the Standard Uncertainty Principle and the Dark Energy Problem
• Orbital magnetic moment of the electron in the hydrogen atom in deformed space with minimal length
• Effect of Generalized Uncertainty Principle on Main-Sequence Stars and White Dwarfs
• Deformed Heisenberg algebra and minimal length
• Quantum field theory in generalised Snyder spaces
• Cosmology with minimal length uncertainty relations
• Short distance modification of the quantum virial theorem
• The Minimal Length and the Quantum Partition Functions
• Quantum Gravity, Dynamical Energy-Momentum Space and Vacuum Energy
• The Classical Limit of Minimal Length Uncertainty Relation:Revisit with the Hamilton-Jacobi Method
• Quantum cosmology, minimal length and holography
• Davydov-Chaban Hamiltonian with deformation-dependent mass term for γ = 30°
• Generalized uncertainty principle and the conformally coupled scalar field quantum cosmology
• Dilaton cosmology and the Modified Uncertainty Principle
• Minimal length effect on the broadening of free wave-packets and its physical implications
• Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics
• Exact Shannon entropies for the multidimensional harmonic states
• Exactly solvable problems in the momentum space with a minimum uncertainty in position
• The quantum N-body problem with a minimal length
• AdS/CFT Correspondence Beyond its Supergravity Approximation
• Probing deformed quantum commutators
• Black hole thermodynamics and modified GUP consistent with doubly special relativity
• The Mixmaster Universe in a generalized uncertainty principle framework
• Effect of Minimal lengths on Electron Magnetism
• Heisenberg and entropic uncertainty measures for large-dimensional harmonic systems
• Coulomb potential in one dimension with minimal length: A path integral approach
• Quantum-spacetime effects on nonrelativistic Schrödinger evolution
• Snyder-type spaces, twisted Poincaré algebra and addition of momenta
• Optimal uncertainty relations in a modified Heisenberg algebra
• On deformations of classical mechanics due to Planck-scale physics
• New higher-order generalized uncertainty principle: Applications
• Three dimensional dilatonic gravity's rainbow: exact solutions
• Probing quantum gravity effects with quantum mechanical oscillators
• On entropic uncertainty relations in the presence of a minimal length
• Fermi equation of state with finite temperature corrections in quantum space-times approach: Snyder model vs GUP case
• Spectrum of the hydrogen atom in Snyder space in a semiclassical approximation
• Deformation of Second and Third Quantization
• Unveiling Phase Space Modifications: A Clash of Modified Gravity and the Generalized Uncertainty Principle
• Pauli Oscillator In Noncommutative Space
• Path integral action in the generalized uncertainty principle framework
• Pseudoharmonic oscillator in quantum mechanics with a minimal length
• Formulation of the Spinor Field in the Presence of a Minimal Length Based on the Quesne-Tkachuk Algebra
• A Novel "Magnetic" Field And Its Dual Non-Commutative Phase Space
• Relating the Cosmological Constant and Supersymmetry Breaking in Warped Compactifications of IIB String Theory
• Deformed phase space in a two dimensional minisuperspace model
• Chaotic Motion around a Black Hole under Minimal Length Effects
• Constraining Snyder and GUP models with low-mass stars
• The minimal length and the Shannon entropic uncertainty relation
• Generalized commutation relations and Non linear momenta theories, a close relationship
• Imaginary Interactions with Minimum Length
• Minimal Length Uncertainty Relations and New Shape Invariant Models
• Multi-Dimensional Cosmology and GUP
• Matrix oscillator and Calogero-type models
• Noncommutative oscillators from a Hopf algebra twist deformation. A first principles derivation
• Discreteness of time in the evolution of the universe
• Generalized Extended Uncertainty Principles, Liouville theorem and density of states: Snyder-de Sitter and Yang models
• WKB approximation in deformed space with minimal length
• Some Implications of Two Forms of the Generalized Uncertainty Principle
• 2D Relativistic Oscillators with a Uniform Magnetic Field in Anti-deSitter Space
• On Dunkl-Bose-Einstein Condensation in Harmonic Traps
• Dirac $\delta$-function potential in quasiposition representation of a minimal-length scenario
• Improved constraints on minimum length models with a macroscopic low loss phonon cavity
• Generalized uncertainty principle and thermostatistics: a semiclassical approach
• Generalized uncertainty principle and stochastic gravitational wave background spectrum
• The tunneling radiation of a black hole with a $f(R)$ global monopole under generalized uncertainty principle
• The fragmentation instability of a black hole with $f(R)$ global monopole under GUP
• On the Physics of the Minimal Length: The Question of Gauge Invariance
• Quantum four-body system in D dimensions
• The solitary solutions of nonlinear Klein-Gordon field with minimal length
• Topological Defects in a Deformed Gauge Theory
• Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty
• Collective states of even-even nuclei in gamma-rigid quadrupole Hamiltonian with Minimal Length under the sextic potential
• Noncommutative spaces and covariant formulation of statistical mechanics
• Exact Solutions of D-dimensional Klein-Gordon Oscillator with Snyder-de Sitter Algebra
• Using Neutrinos to test the Time-Energy Uncertainty Relation in an Extreme Regime
• Effective models of quantum gravity induced by Planck scale modifications in the covariant quantum algebra
• Comments on the cosmological constant in generalized uncertainty models
• Multi-Dimensional Cosmology and DSR-GUP
• Exactly solvable dynamical models with a minimal length uncertainty
• Quantum gravitational decoherence of a mechanical oscillator from spacetime fluctuations
• Twist for Snyder space
• A covariant tapestry of linear GUP, metric-affine gravity, their Poincaré algebra and entropy bound
• Relation of deformed nonlinear algebras with linear ones
• Position and momentum uncertainties of a particle in a V-shaped potential under the minimal length uncertainty relation
• Non-Local Deformation of a Supersymmetric Field Theory
• Exact solutions for two-body problems in 1D deformed space with minimal length
• Some consequences of GUP induced ultraviolet wavevector cutoff in one-dimensional Quantum Mechanics
• The generalized uncertainty principle in the presence of extra dimensions
• Solutions of the Bogoliubov-de Gennes equation with position dependent Fermi--velocity and gap profiles
• On Dirac theory in the space with deformed Heisenberg algebra. Exact solutions
• Remarks on generalized uncertainty principle induced from constraint system
• Signature Change by GUP
• A scale at 10 MeV, gravitational topological vacuum, and large extra dimensions
• Towards new relativistic doubly $\kappa$-deformed D=4 quantum phase spaces
• Particle on a torus knot: a Hamiltonian analysis
• Entropy of Vaidya Black Hole on Apparent Horizon with Minimal length Revisited
• Quantized space-time and its influences on some physical problems
• Generalized uncertainty principle and burning stars
• Quantum-gravitational corrections to the hydrogen atom and harmonic oscillator
• The tunneling radiation of Kehagias-Sfetsos black hole under generalized uncertainty principle
• Quantum electrodynamics and the electron self-energy in a deformed space with a minimal length scale
• Bose-Einstein Condensate and Liquid Helium He${}^4$: Implications of GUP and Modified Gravity Correspondence
• Short Distance Modification of a Gravitational System and its Optical Analog
• Particle in a cavity in one-dimensional bandlimited quantum mechanics
• Quantum speed limit as a sensitive probe of Planck-scale effects
• Observable Effects of Quantum Gravity
• Bound states in the continuum are universal under the effect of minimal length
• Minimal Length Effects on Motion of a Particle in Rindler Space
• Generalized Heisenberg relation and Quantum Harmonic Oscillators
• Non-Heisenbergian quantum mechanics
• Hartmann potential with a minimal length and generalized recurrence relations for matrix elements
• Potential Algebra Approach to Quantum Mechanics with Generalized Uncertainty Principle
• Anyons in quantum mechanics with a minimal length
• Minimal length implications on the Hartree-Fock theory
• Deformed Heisenberg Algebra with a minimal length: Application to some molecular potentials
• A note on scattering in deformed space with minimal length
• WKB Approximation for a Deformed Schrodinger-like Equation and its Applications to Quasinormal Modes of Black Holes and Quantum Cosmology
• Einstein crystals in Snyder and Snyder-de Sitter noncommutative backgrounds
• Homogeneous Field and WKB Approximation In Deformed Quantum Mechanics with Minimal Length
• Exact Solutions of the DKP Oscillator in 3D Spaces with Extended Uncertainty Principle
• Exact continuity equation in a space with minimal length
• A relativistic model of the $N$-dimensional singular oscillator
• Thermodynamic properties of ideal Bose gas trapped in different external power-law potentials under generalized uncertainty principle*
• The instability of a black hole with $f(R)$ global monopole under extended uncertainty principle