Lévy flights over quantum paths

Laskin, Nick

doi:10.1016/j.cnsns.2006.01.001

Cited by 51 publications

(25 citation statements)

References 16 publications

(30 reference statements)

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“…It was shown in [2], [19] that the integral in Eq. (118) can be expressed in terms of the H 1,1 2,2 -function,…”

Section: Fox H -Function Representation For a Free Particle Space-timmentioning

confidence: 99%

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Time fractional quantum mechanics

Laskin

2017

Chaos, Solitons & Fractals

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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. PACS number(s): 05.40.Fb, 03.65.Sq

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“…It was shown in [2], [19] that the integral in Eq. (118) can be expressed in terms of the H 1,1 2,2 -function,…”

Section: Fox H -Function Representation For a Free Particle Space-timmentioning

confidence: 99%

“…(118) for a free particle fractional quantum kernel K (0) α,1 (x, t). It has been shown in [19] that at α = 2 Eq. (119) goes into Eq.…”

Section: Fox H -Function Representation For a Free Particle Space-timmentioning

confidence: 99%

Time fractional quantum mechanics

Laskin

2017

Chaos, Solitons & Fractals

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“…31 It is also useful in evaluating the Green's function for fractional diffusion processes 32 and in obtaining the probability density for fractional Brownian motion.…”

Section: ͑49͒mentioning

confidence: 99%

A scaling method and its applications to problems in fractional dimensional space

Muslih

Agrawal

2009

Journal of Mathematical Physics

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A scaling method is proposed to find ͑1͒ the volume and the surface area of a generalized hypersphere in a fractional dimensional space and ͑2͒ the solid angle at a point for the same space. It is demonstrated that the total dimension of the fractional space can be obtained by summing the dimension of the fractional line element along each axis. The regularization condition is defined for functions depending on more than one variable. This condition is applied ͑1͒ to find a closed form expression for the fractional Gaussian integral, ͑2͒ to establish a relationship between a fractional dimensional space and a fractional integral, ͑3͒ to develop the Bochner theorem, and ͑4͒ to obtain an expression for the fractional integral of the Mittag-Leffler function. Some possible extensions of this work are also discussed.

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“…A 2007 work used a different method of analysis to claim solutions for the linear, delta function, and Coulomb potentials in one dimension [10]. Laskin also recently built on the same claimed solution to derive properties of the quantum kernel [11]. The purpose of this work is to point out that of the many purported exact solutions presented in the literature, only the one for the delta function potential is correct.…”

Section: Introductionmentioning

confidence: 98%

On the nonlocality of the fractional Schrödinger equation

Jeng

Hawkins

et al. 2010

Journal of Mathematical Physics

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A number of papers over the past eight years have claimed to solve the fractional Schrödinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrödinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported groundstate, which is based on a piecewise approach, is definitely not a solution of the fractional Schrödinger equation for general fractional parameters α. On a more positive note, we present a solution to the fractional Schrödinger equation for the one-dimensional harmonic oscillator with α = 1.

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Lévy flights over quantum paths

Cited by 51 publications

References 16 publications

Time fractional quantum mechanics

Time fractional quantum mechanics

A scaling method and its applications to problems in fractional dimensional space

On the nonlocality of the fractional Schrödinger equation

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