Variance gamma process: Difference between revisions

Revision as of 09:42, 3 August 2013

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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

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

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

i ℏ \frac{\partial ψ (r, t)}{\partial t} = D_{α} (- ℏ^{2} Δ)^{α / 2} ψ (r, t) + V (r, t) ψ (r, t),

using the standard definitions:

r is the 3-dimensional position vector,
ħ is the reduced Planck constant,
ψ(r, t) is the wavefunction, which is the quantum mechanical probability amplitude for the particle to have a given position r at any given time t,
V(r, t) is a potential energy,
Δ = ∂²/∂r² is the Laplace operator.

Further,

D_α is a scale constant with physical dimension [D_α] = [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]);

(- ℏ^{2} Δ)^{α / 2} ψ (r, t) = \frac{1}{(2 π ℏ)^{3}} \int d^{3} p e^{i p \cdot r / ℏ} | p |^{α} φ (p, t),

Here, the wave functions in the position and momentum spaces; $ψ (r, t)$ and $φ (p, t)$ are related each other by the 3-dimensional Fourier transforms:

ψ (r, t) = \frac{1}{(2 π ℏ)^{3}} \int d^{3} p e^{i p \cdot r / ℏ} φ (p, t), φ (p, t) = \int d^{3} r e^{- i p \cdot r / ℏ} ψ (r, t) .

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

References

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

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+{{Quantum mechanics|cTopic=Advanced topics}}
+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''.<ref>N. Laskin, (2000), [http://dx.doi.org/10.1016/S0375-9601(00)00201-2 Fractional Quantum Mechanics and Lévy Path Integrals. ''Physics Letters'' 268A, 298-304].</ref>
+==Fundamentals==
+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]]<ref>R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals ~McGraw-Hill, New York, 1965 </ref> 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]].<ref>N. Laskin, (2000), [http://pre.aps.org/abstract/PRE/v62/i3/p3135_1 Fractional Quantum Mechanics, ''Physical Review'' E62, 3135-3145]. '' (also available online: http://arxiv.org/abs/0811.1769)</ref> If the [[Feynman path integral]] leads to the well known [[Schrödinger equation]], then the path integral over [[Paul Lévy (mathematician)|Lévy]] trajectories leads to the [[fractional Schrödinger equation]].<ref>N. Laskin, (2002), [http://pre.aps.org/abstract/PRE/v66/i5/e056108 Fractional Schrödinger equation, ''Physical Review'' E66, 056108 7 pages]. '' (also available online: http://arxiv.org/abs/quant-ph/0206098)</ref> The [[Lévy process]] is characterized
+by the Lévy index ''α'', 0&nbsp;<&nbsp;''α''&nbsp;≤&nbsp;2. At the special case when ''α''&nbsp;=&nbsp;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''&nbsp;=&nbsp;2) space derivative in the standard Schrödinger equation. Thus, the fractional Schrödinger equation is a [[fractional differential equation]] in accordance with modern terminology.<ref>S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional
+Integrals and Derivatives, Theory and Applications ~Gordon
+and Breach, Amsterdam, 1993 </ref> This is the main point of the term [[fractional Schrödinger equation]] or a more general term ''fractional quantum mechanics''. As mentioned above, at ''α''&nbsp;=&nbsp;2 the Lévy motion becomes [[Brownian motion]]. Thus, fractional quantum mechanics includes standard quantum mechanics as a particular case at ''α''&nbsp;=&nbsp;2. The quantum-mechanical path integral over the Lévy paths at ''α''&nbsp;=&nbsp;2 becomes the well-known [[Feynman path integral]] and the [[fractional Schrödinger equation]] becomes the well-known [[Schrödinger equation]].
+=== Fractional Schrödinger equation ===
+The [[fractional Schrödinger equation]] discovered by [[Nick Laskin]] has the following form (see, Refs.[1,3,4])
+: <math>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)\,,</math>
+using the standard definitions:
+*'''r''' is the 3-dimensional [[position vector]],
+*''ħ'' is the reduced [[Planck constant]],
+*''ψ''('''r''', ''t'') is the [[wavefunction]], which is the quantum mechanical probability amplitude for the particle to have a given position '''r''' at any given time ''t'',
+*''V''('''r''', ''t'') is a [[potential energy]],
+*Δ = &part;<sup>2</sup>/&part;'''r'''<sup>2</sup> is the [[Laplace operator]].
+Further,
+*''D<sub>α</sub>'' is a scale constant with [[dimensional analysis|physical dimension]] [D<sub>α</sub>] = [energy]<sup>1 − ''α''</sup>·[length]<sup>''α''</sup>[time]<sup>−''α''</sup>, at ''α''&nbsp;=&nbsp;2, ''D''<sub>2</sub> =1/2''m'', where ''m'' is a particle mass,
+*the operator (−''ħ''<sup>2</sup>Δ)<sup>''α''/2</sup> is the 3-dimensional fractional quantum Riesz derivative defined by (see, Ref.[4]);
+::<math>
+(-\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),
+</math>
+Here, the wave functions in the [[position and momentum space]]s; <math>\psi(\mathbf{r},t)</math> and <math> \varphi (\mathbf{p},t)</math> are related each other by the 3-dimensional [[Fourier transform]]s:
+:<math>
+\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).
+</math>
+The index ''α'' in the fractional Schrödinger equation is the Lévy index, 1&nbsp;<&nbsp;''α''&nbsp;≤&nbsp;2.
+== See also ==
+*[[Quantum mechanics]]
+*[[Matrix mechanics]]
+*[[Fractional calculus]]
+*[[Fractional dynamics]]
+*[[Fractional Schrödinger equation]]
+*[[Non-linear Schrödinger equation]]
+*[[Path integral formulation]]
+*[[Relation between Schrödinger's equation and the path integral formulation of quantum mechanics]]
+*[[Lévy process]]
+== References ==
+<references/>
+{{refbegin}}
+<!---NO DELETING--->{{cite book|title=Fractional Calculus, An Introduction for Physicists|author=Richard Herrmann|year=2011|publisher=World Scientific|chapter=9|isbn=981 4340 24 3}}
+{{refend}}
+== Further reading ==
+* [http://iopscience.iop.org/0305-4470/25/19/026 L.P.G. do Amaral, E.C. Marino, Canonical quantization of theories containing fractional powers of the d’Alembertian operator. J. Phys. A Math. Gen. 25 (1992) 5183-5261]
+* [http://prb.aps.org/abstract/PRB/v42/i18/p11751_1 Xing-Fei He, Fractional dimensionality and fractional derivative spectra of interband optical transitions. Phys. Rev. B, 42 (1990) 11751-11756.]
+* [http://pre.aps.org/abstract/PRE/v80/i2/e022103 A. Iomin, Fractional-time quantum dynamics. Phys. Rev. E 80, (2009) 022103.]
+* [http://iopscience.iop.org/0305-4470/34/49/321 A. Matos-Abiague, Deformation of quantum mechanics in fractional-dimensional space. J. Phys. A: Math. Gen. 34 (2001) 11059–11068.]
+* [http://chaos.aip.org/resource/1/chaoeh/v10/i4/p780_s1?isAuthorized=no N. Laskin, Fractals and quantum mechanics. Chaos 10(2000) 780-790]
+* [http://jmp.aip.org/jmapaq/v45/i8/p3339_s1?isAuthorized=no M. Naber, Time fractional Schrodinger equation. J. Math. Phys. 45 (2004) 3339-3352.] [http://arxiv.org/abs/math-ph/0410028 arXiv:math-ph/0410028]
+* [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVM-4RPM7RC-1&_user=10&_coverDate=04%2F21%2F2008&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1400457817&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=fd7815c9c95482bc8873a50573068777 V.E. Tarasov, Fractional Heisenberg equation. Phys. Lett. A 372 (2008) 2984-2988.]
+* [http://jmp.aip.org/jmapaq/v49/i10/p102112_s1?bypassSSO=1 V.E. Tarasov, Weyl quantization of fractional derivatives. J. Math. Phys. 49 (2008) 102112.]
+* [http://jmp.aip.org/jmapaq/v48/i4/p043502_s1?isAuthorized=no  S. Wang, M. Xu, Generalized fractional Schrödinger equation with space-time fractional derivatives J. Math. Phys. 48 (2007) 043502 ]
+* [http://iopscience.iop.org/1751-8121/44/18/185303 E Capelas de Oliveira and Jayme Vaz Jr, "Tunneling in Fractional Quantum Mechanics" Journal of Physics A Volume 44 (2011) 185303.]
+{{Physics-footer}}
+[[Category:Concepts in physics]]
+[[Category:Fractals]]
+[[Category:Quantum mechanics| ]]

Variance gamma process: Difference between revisions

Revision as of 09:42, 3 August 2013

Contents

Fundamentals

Fractional Schrödinger equation

See also

References

Further reading

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