Fractional quantum mechanics

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In physics, fractional quantum mechanics is a generalization of standard quantum mechanics, which naturally comes out when the Brownian-like quantum paths substitute with the Lévy-like ones in the Feynman path integral. It has been discovered by Nick Laskin who coined the term fractional quantum mechanics.[1]

Fundamentals[edit]

Standard quantum mechanics can be approached in three different ways: the matrix mechanics, the Schrödinger equation and the Feynman path integral.

The Feynman path integral[2] is the path integral over Brownian-like quantum-mechanical paths. Fractional quantum mechanics has been discovered by Nick Laskin (1999) as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. A path integral over the Lévy-like quantum-mechanical paths results in a generalization of quantum mechanics.[3] If the Feynman path integral leads to the well known Schrödinger equation, then the path integral over Lévy trajectories leads to the fractional Schrödinger equation.[4] The Lévy process is characterized by the Lévy index α, 0 < α ≤ 2. At the special case when α = 2 the Lévy process becomes the process of Brownian motion. The fractional Schrödinger equation includes a space derivative of fractional order α instead of the second order (α = 2) space derivative in the standard Schrödinger equation. Thus, the fractional Schrödinger equation is a fractional differential equation in accordance with modern terminology.[5] This is the main point of the term fractional Schrödinger equation or a more general term fractional quantum mechanics. As mentioned above, at α = 2 the Lévy motion becomes Brownian motion. Thus, fractional quantum mechanics includes standard quantum mechanics as a particular case at α = 2. The quantum-mechanical path integral over the Lévy paths at α = 2 becomes the well-known Feynman path integral and the fractional Schrödinger equation becomes the well-known Schrödinger equation.

Fractional Schrödinger equation[edit]

The fractional Schrödinger equation discovered by Nick Laskin has the following form (see, Refs.[1,3,4])

i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_\alpha (-\hbar
^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t)\,,

using the standard definitions:

Further,

  • Dα is a scale constant with physical dimension [Dα] = [energy]1 − α·[length]α[time]α, at α = 2, D2 =1/2m, where m is a particle mass,
  • the operator (−ħ2Δ)α/2 is the 3-dimensional fractional quantum Riesz derivative defined by (see, Ref.[4]);

(-\hbar ^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar
)^3}\int d^3pe^{i \mathbf{p}\cdot \mathbf{r}/\hbar}|\mathbf{p}|^\alpha \varphi (
\mathbf{p},t),

Here, the wave functions in the position and momentum spaces; \psi(\mathbf{r},t) and  \varphi (\mathbf{p},t) are related each other by the 3-dimensional Fourier transforms:


\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar )^3}\int d^3pe^{i \mathbf{p}\cdot\mathbf{r}/\hbar}\varphi (\mathbf{p},t),\qquad \varphi (\mathbf{p},t)=\int d^3re^{-i
\mathbf{p}\cdot\mathbf{r}/\hbar }\psi (\mathbf{r},t).

The index α in the fractional Schrödinger equation is the Lévy index, 1 < α ≤ 2.

See also[edit]

References[edit]

  1. ^ N. Laskin, (2000), Fractional Quantum Mechanics and Lévy Path Integrals. Physics Letters 268A, 298-304.
  2. ^ R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals ~McGraw-Hill, New York, 1965
  3. ^ N. Laskin, (2000), Fractional Quantum Mechanics, Physical Review E62, 3135-3145. (also available online: http://arxiv.org/abs/0811.1769)
  4. ^ N. Laskin, (2002), Fractional Schrödinger equation, Physical Review E66, 056108 7 pages. (also available online: http://arxiv.org/abs/quant-ph/0206098)
  5. ^ S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications ~Gordon and Breach, Amsterdam, 1993

Richard Herrmann (2011). "9". Fractional Calculus, An Introduction for Physicists. World Scientific. ISBN 981 4340 24 3. 

Further reading[edit]