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Nick Laskin

Publications and source records attributed to Nick Laskin.

13 recordsLinked to original sources
arXiv2025

Quantum Well in Fractional Quantum Mechanics

Within the framework of fractional quantum mechanics, an exact solution has been found for the energy spectrum of a quantum particle confined in a quantum well - a symmetric one-dimensional finite potential well. A simple graphical algorithm is described for obtaining the number of discrete levels in a quantum well and their associated energy values. The presented results open up new possibilities for emulating fractional quantum mechanics using quantum wells.

quant-ph↗
arXiv2023

Fractional non-homogeneous counting process

A new fractional non-homogeneous counting process has been introduced and developed using the Kilbas and Saigo three-parameter generalization of the Mittag-Leffler function. The probability distribution function of this process reproduces for certain set of the fractality parameters the famous Poisson and fractional Poisson probability distributions as well as the probability distribution function of a counting process displaying stretched exponential interarrival times distribution. Applications of the developed fractional non-homogeneous counting process cover fractional compound process, the generalization of combinatorial polynomials and numbers, statistics of the fractional non-homogeneous counting process and a new representation of the Kilbas and Saigo function.

math.PR↗
arXiv2023

Stretched Coherent States

The fundamental properties of recently introduced stretched coherent states are investigated. It has been shown that stretched coherent states retain the fundamental properties of standard coherent states and generalize the resolution of unity, or completeness condition, and the probability distribution that $n$ photons are in a stretched coherent state. The stretched displacement and stretched squeezing operators are introduced and the multiplication law for stretched displacement operator is established. The results of the action of the stretched displacement and stretched squeezing operators on the vacuum and the Fock states are presented. Stretched squeezed stretched coherent states and stretched squeezed stretched displaced number states are introduced and their properties are studied. The inner product of two quantum mechanical vectors was defined in terms of their stretched coherent state representations, and functional Hilbert space was introduced.

quant-ph↗
arXiv2023

A New Family of Fractional Counting Probability Distributions

A new family of fractional counting processes based on a three-parameter generalized Mittag-Leffler function was introduced and studied. As applications we develop a fractional generalized compound process, introduce and develop fractional generalized Bell polynomials and numbers, fractional generalized Stirling numbers of the second kind, and a new family of quantum coherent states. Stretched quantum coherent states, which are a generalization of the famous Schr\"odinger-Glauber coherent states, were also introduced and studied. In particular cases, the presented results reproduce known equations for Poisson and fractional Poisson probability distributions, Bell numbers and fractional Bell numbers, Stirling numbers and fractional Stirling numbers of the second kind, as well as for known quantum coherent states.

math.PR↗
arXiv2017

Time Fractional Quantum Mechanics

A time fractional quantum framework has been introduced into quantum mechanics. A new version of the space-time fractional Schrödinger equation has been launched. The introduced space-time fractional Schrödinger equation has a new scale parameter, which is a fractional generalization of Planck's constant in quantum physics. It has been shown that the presence of a time fractional time derivative in the space-time fractional Schrödinger equation significantly impacts quantum mechanical fundamentals. Time fractional quantum mechanical operators of coordinate, momentum and angular momentum were defined and their commutation relationships were established. The pseudo-Hamilton operator was introduced and its Hermicity has been proven. Two new functions related to the Mittag-Leffler function have been introduced to solve the space-time fractional Schrödinger equation. Energy of a time fractional quantum system has been defined and calculated in terms of the newly introduced functions. It has been found that in the framework of time fractional quantum mechanics there are no stationary states, and the eigenvalues of the pseudo-Hamilton operator are not the energy levels of the time fractional quantum system. A free particle solution to the space-time fractional Schrödinger equation was found. A free particle space-time fractional quantum mechanical kernel has been found and expressed in terms of the H-function. Renormalization properties of a free particle solution and the space-time fractional quantum kernel were established. Some particular cases of time fractional quantum mechanics have been analyzed and discussed.

physics.gen-ph↗
arXiv2014

New Pricing Framework: Options and Bonds

A unified analytical pricing framework with involvement of the shot noise random process has been introduced and elaborated. Two exactly solvable new models have been developed. The first model has been designed to value options. It is assumed that asset price stochastic dynamics follows a Geometric Shot Noise motion. A new arbitrage-free integro-differential option pricing equation has been found and solved. The put-call parity has been proved and the Greeks have been calculated. Three additional new Greeks associated with market model parameters have been introduced and evaluated. It has been shown that in diffusion approximation the developed option pricing model incorporates the well-known Black-Scholes equation and its solution. The stochastic dynamic origin of the Black-Scholes volatility has been uncovered. The new option pricing model has been generalized based on asset price dynamics modeled by the superposition of Geometric Brownian motion and Geometric Shot Noise. To model stochastic dynamics of a short term interest rate, the second model has been introduced and developed based on Langevin type equation with shot noise. A new bond pricing formula has been obtained. It has been shown that in diffusion approximation the developed bond pricing formula goes into the well-known Vasicek solution. The stochastic dynamic origin of the long-term mean and instantaneous volatility of the Vasicek model has been uncovered. A generalized bond pricing model has been introduced and developed based on short term interest rate stochastic dynamics modeled by superposition of a standard Wiener process and shot noise. Despite the non-Gaussianity of probability distributions involved, all newly elaborated models have the same degree of analytical tractability as the Black-Scholes model and the Vasicek model.

q-fin.PR↗
arXiv2013

Fractional Classical Mechanics

Fractional classical mechanics has been introduced and developed as a classical counterpart of the fractional quantum mechanics. Lagrange, Hamilton and Hamilton-Jacobi frameworks have been implemented for the fractional classical mechanics. The Lagrangian of fractional classical mechanics has been introduced, and equation of motion has been obtained. Fractional oscillator model has been launched and solved in 1D case. A new equation for the period of oscillations of fractional classical oscillator has been found. The interplay between the energy dependency of the period of classical oscillations and the non-equidistant distribution of the energy levels for fractional quantum oscillator has been discussed. We discuss as well, the relationships between new equations of fractional classical mechanics and the well-known fundamental equations of classical mechanics.

math-ph↗
arXiv2011

Some Applications of the Fractional Poisson Probability Distribution

Physical and mathematical applications of fractional Poisson probability distribution have been presented. As a physical application, a new family of quantum coherent states has been introduced and studied. As mathematical applications, we have discovered and developed the fractional generalization of Bell polynomials, Bell numbers, and Stirling numbers. Appearance of fractional Bell polynomials is natural if one evaluates the diagonal matrix element of the evolution operator in the basis of newly introduced quantum coherent states. Fractional Stirling numbers of the second kind have been applied to evaluate skewness and kurtosis of the fractional Poisson probability distribution function. A new representation of Bernoulli numbers in terms of fractional Stirling numbers of the second kind has been obtained. A representation of Schlafli polynomials in terms of fractional Stirling numbers of the second kind has been found. A new representations of Mittag-Leffler function involving fractional Bell polynomials and fractional Stirling numbers of the second kind have been discovered. Fractional Stirling numbers of the first kind have been introduced and studied. Two new polynomial sequences associated with fractional Poisson probability distribution have been launched and explored. The relationship between new polynomials and the orthogonal Charlier polynomials has also been investigated. In the limit case when the fractional Poisson probability distribution becomes the Poisson probability distribution, all of the above listed developments and implementations turn into the well-known results of quantum optics, the theory of combinatorial numbers and the theory of orthogonal polynomials of discrete variable.

math-ph↗
arXiv2011

Exciton-Phonon Dynamics with Long-Range Interaction

Exciton-phonon dynamics on a 1D lattice with long-range exciton-exciton interaction have been introduced and elaborated. Long-range interaction leads to a nonlocal integral term in the motion equation of the exciton subsystem if we go from discrete to continuous space. In some particular cases for power-law interaction, the integral term can be expressed through a fractional order spatial derivative. A system of two coupled equations has been obtained, one is a fractional differential equation for the exciton subsystem, the other is a standard differential equation for the phonon subsystem. These two equations present a new fundamental framework to study nonlinear dynamics with long-range interaction. New approaches to model the impact of long-range interaction on nonlinear dynamics are: fractional generalization of Zakharov system, Hilbert-Zakharov system, Hilbert-Ginzburg-Landau equation and nonlinear Hilbert-Schrodinger equation. Nonlinear fractional Schrodinger equation and fractional Ginzburg-Landau equation are also part of this framework.

math-ph↗
arXiv2010

Principles of Fractional Quantum Mechanics

A review of fundamentals and physical applications of fractional quantum mechanics has been presented. Fundamentals cover fractional Schrödinger equation, quantum Riesz fractional derivative, path integral approach to fractional quantum mechanics, hermiticity of the Hamilton operator, parity conservation law and the current density. Applications of fractional quantum mechanics cover dynamics of a free particle, new representation for a free particle quantum mechanical kernel, infinite potential well, bound state in δ-potential well, linear potential, fractional Bohr atom and fractional oscillator. We also review fundamentals of the Lévy path integral approach to fractional statistical mechanics.

math-ph↗
arXiv2008

Fractional Quantum Mechanics

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.

math-ph↗
arXiv2005

Levy Flights over Quantum Paths

An impact of integration over the paths of the Levy flights on the quantum mechanical kernel has been studied. Analytical expression for a free particle kernel has been obtained in terms of the Fox H-function. A new equation for the kernel of a partical in the box has been found. New general results include the well known quantum formulae for a free particle kernel and particle in box kernel.

quant-ph↗