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Option pricing

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	~~Hi therе!~~ :) ~~Mу name~~ is ~~Santiago~~, I'm a ~~stuԁent studying Αthletics~~ and ~~Physical Education from Northampton~~, ~~United States~~.<br><br>~~Feel free~~ to ~~visit my homepage~~; ~~Watch~~ The ~~Amazing Spider~~-~~Man 2 Full Movie~~ [[http://~~theamazingspider~~-~~man2movie~~.~~blogspot~~.com/ ~~theamazingspider~~-~~man2movie~~.~~blogspot~~.~~com~~]]		{{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 < ''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.<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'' = 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])

			: <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 = ∂<sup>2</sup>/∂'''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'' = 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 < ''α'' ≤ 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\| ]]

en>Rlendog: /* Moments */ fix formula

2012-04-23T16:29:12Z

Moments: fix formula

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Hi therе! :) Mу name is Santiago, I'm a stuԁent studying Αthletics and Physical Education from Northampton, United States.<br><br>Feel free to visit my homepage; Watch The Amazing Spider-Man 2 Full Movie [[http://theamazingspider-man2movie.blogspot.com/ theamazingspider-man2movie.blogspot.com]]

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