Fractional Quantum Mechanics

Abstract

A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Lévy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Lévy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schrödinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.

27 pages

References in corpus (1)

• Fractional Quantum Mechanics and Levy Path Integrals

Cited by in corpus (161)

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• Experimental realisations of the fractional Schrödinger equation in the temporal domain
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• From non-Brownian Functionals to a Fractional Schrödinger Equation
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• One-dimensional solitons in fractional Schrödinger equation with a spatially modulated nonlinearity: nonlinear lattice
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• Some Applications of the Fractional Poisson Probability Distribution
• Families of fundamental and multipole solitons in a cubic-quintic nonlinear lattice in fractional dimension
• On the Consistency of the Solutions of the Space Fractional Schrödinger Equation
• Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice
• Metastable soliton necklaces supported by fractional diffraction and competing nonlinearities
• Time Fractional Schrödinger Equation; Fox's H-functions and the Effective Potential
• Definition of the Riesz Derivative and its Application to Space Fractional Quantum Mechanics
• Tunneling in Fractional Quantum Mechanics
• Fractional Derivative as Fractional Power of Derivative
• Fractional Statistical Mechanics
• de Sitter Fractional Quantum Cosmology
• On the nature of the conformable derivative and its applications to physics
• The Most General Form of Deformation of the Heisenberg Algebra from the Generalized Uncertainty Principle
• Nonholonomic Constraints with Fractional Derivatives
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• Comment on "Fractional quantum mechanics" and "Fractional Schroedinger equation"
• The Two-Dimensional Fractional Discrete Nonlinear Schrodinger Equation
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• From Fractional Quantum Mechanics to Quantum Cosmology: An Overture
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• Stable and oscillating solitons of $\mathcal{P}\mathcal{T}$-symmetric couplers with gain and loss in fractional dimension
• Analytical Solution to the Fractional Polytropic Gas Spheres
• From compact localized states to many-body scars in the random quantum comb
• Broadening quantum cosmology with a fractional whirl
• Path Integral Formulation for Lévy Flights - Evaluation of the Propagator for Free, Linear and Harmonic Potentials in the Over- and Underdamped Limits
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• Fractional dark energy
• Fractional Holographic Dark Energy
• Estimated Age of the Universe in Fractional Cosmology
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• Solving fractional Schroedinger-type spectral problems: Cauchy oscillator and Cauchy well
• Conformal Anisotropic Mechanics
• Fractional Quantum Field Theory, Path Integral, and Stochastic Differential Equation for Strongly Interacting Many-Particle Systems
• Tunneling time in space fractional quantum mechanics
• New Integrable Multi-Lévy-Index and Mixed Fractional Nonlinear Soliton Hierarchies
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• Lévy flights in inhomogeneous environments
• Factorization of the Quantum Fractional Oscillator
• Green function of the double fractional Fokker-Planck equation: Path integral and stochastic differential equations
• Localized modes in nonlinear fractional systems with deep lattices
• Fractional Scalar Field Cosmology
• Quantum Cosmology of Fab Four John Theory with Conformable Fractional Derivative
• New scattering features in non-Hermitian space fractional quantum mechanics
• Consistency Problem of the Solutions of the Space Fractional Schrödinger Equation
• Modified Friedmann equations from fractional entropy
• Super Periodic Potential
• Lévy flights confinement in a parabolic potential and fractional quantum oscillator
• Applications of density matrix in the fractional quantum mechanics
• Complex Path Integrals and the Space of Theories
• Unitary evolution for a two-level quantum system in fractional-time scenario
• Conformal and Non-Minimal Couplings in Fractional Cosmology
• Finite Temperature Casimir Effect for a Massless Fractional Klein-Gordon field with Fractional Neumann Conditions
• Fractional Generalization of Kac Integral
• Self-accelerating beam dynamics in the space fractional Schrödinger equation
• Levy path integral approach to the solution of the fractional Schrödinger equation with infinite square well
• Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
• The Fractional Schrodinger Equation with the Generalized Woods-Saxon Potential
• Global well-posedness and long-time behavior of the fractional NLS
• Strichartz Estimates in Spherical Coordinates
• Global well-posedness for the cubic fractional NLS on the unit disk
• Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
• Testing Short Distance Anisotropy in Space
• Lévy-Schrödinger wave packets
• Emergence of Order in Dynamical Phases in Coupled Fractional Gauss Map
• Fractional discrete vortex solitons
• Computational Method for a Fractional Model of the Helium Burning Network
• Fractional Floquet theory
• Fractional quantum mechanics meets quantum gravity phenomenology
• Multiplicity and concentration results for local and fractional NLS equations with critical growth
• Anomalous Cyclic in the Neutrino Oscillations
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• On unique continuation for Schrödinger operators of fractional and higher orders
• Spectral properties of fractional Fokker-Plank operator for the Lévy flight in a harmonic potential
• Orbital stability of standing waves of a class of fractional Schrodinger equations with a general Hartree-type integrand
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• Fractionality and $\mathcal{P}\mathcal{T}$- symmetry in a square lattice
• The second hyperpolarizability of systems described by the space-fractional Schrodinger equation
• The weakly coupled fractional one-dimensional Schrödinger operator with index $1<\alpha \le 2$
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• Dissipative systems fractionally coupled to a bath
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• On the discreet spectrum of fractional quantum hydrogen atom in two dimensions
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• Existence of infinitely many solutions for the fractional Schrödinger- Maxwell equations
• Interplay of fractionality and $\mathcal{P}\mathcal{T}$- symmetry on a 1D lattice
• On a collocation method for the time-fractional convection-diffusion equation with variable coefficients
• Fractional Schrödinger Equation with singular potentials of higher-order
• Propagation dynamics of abruptly autofocusing circular Airy-Gaussian vortex beams in the fractional Schrödinger equation
• Emergent $\Lambda$CDM cosmology from a measure-induced deformation of the Newtonian action
• Fractional nonlinear surface impurity in a 2D lattice
• Local well-posedness of semilinear space-time fractional Schrödinger equation
• Entire sign-changing solutions to the fractional critical Schr{ö}dinger equation
• Principle action for particles faster than light
• Dispersive estimates for time and space fractional Schrödinger equations
• A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation
• A fractional Anderson model
• Static hyperpolarizability of space-fractional quantum systems
• The fractional saturable impurity
• Fractional Green's function for the time-dependent scattering problem in the Space-time-fractional quantum mechanics
• Fractional uncertainty
• Characterization of the Most Probable Transition Paths of Stochastic Dynamical Systems with Stable Lévy Noise
• The fractional nonlinear electrical lattice
• Integral geometry and unique continuation principles
• The fractional nonlinear impurity: A Green function approach
• Analytic functional calculus and Gårding inequality on graded Lie groups with applications to diffusion equations
• Quantum Cosmology in $f(Q)$ theory
• Airy-Gaussian vortex beams in the fractional nonlinear-Schrödinger medium
• Generalized Fractional Derivative, Fractional differential ring
• A conservative difference scheme with optimal pointwise error estimates for two-dimensional space fractional nonlinear Schrödinger equations
• Vortex solitons in fractional nonlinear Schrödinger equation with the cubic-quintic nonlinearity
• Stabilization of single- and multi-peak solitons in the fractional nonlinear Schroedinger equation with a trapping potential
• The Fractional Discrete Nonlinear Schrödinger Equation
• Electron Spin Precession for the Time Fractional Pauli Equation
• Generalised fractional Rabi problem
• Bound state solutions for non-autonomous fractional Schrödinger-Poisson equations with critical exponent
• Bound states in the continuum of fractional Schrödinger equation in the Earth's gravitational field and their effects in the presence of a minimal length: applications to distinguish ultralight particles
• Ground states of nonlinear fractional Schrödinger equation involving critical growth
• Existence of non-trivial solutions for nonlinear fractional Schrödinger-Poisson equations
• Model-Driven Applications of Fractional Derivatives and Integrals
• Propagation dynamics of radially polarized symmetric Airy beams in the fractional Schrödinger equation
• A doubly nonlinear evolution problem involving the fractional p-Laplacian
• Higher topological type semiclassical states for fractional nonlinear elliptic equations
• Anomalous quantum mechanics