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-In [[theoretical physics]], the '''logarithmic Schrödinger equation''' (sometimes abbreviated as '''LNSE''' or '''LogSE''') is one of the [[nonlinear]] modifications of [[Schrödinger equation|Schrödinger's equation]]. It is a classical wave equation with applications to extensions of [[quantum mechanics]],<ref>I. Bialynicki-Birula and J. Mycielski, Annals Phys. 100, 62 (1976); Commun. Math. Phys.  44, 129 (1975); Phys. Scripta 20, 539 (1979).</ref>
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-The logarithmic Schrödinger equation is the [[partial differential equation]]. In [[mathematics]] and [[mathematical physics]] one often uses its [[dimensionless]] form:
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en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Dependency

2012-10-16T04:36:26Z

Repairing links to disambiguation pages - You can help! - Dependency

← Older revision		Revision as of 06:36, 16 October 2012
Line 1:		Line 1:
	~~Cabinetmaker Hosea Reister from Canada~~, ~~usually spends time~~ with ~~passions for example r~~/~~c helicopters~~, ~~ganhando dinheiro na internet~~ and ~~sketching~~. ~~Has~~ of ~~late finished a trip to Old City~~ of ~~Zamosc~~.<br><br>~~Here is my site;~~ [http://~~comoficarrico~~.~~comoganhardinheiro101~~.~~com~~/ ~~como ficar rico~~]		In [[theoretical physics]], the '''logarithmic Schrödinger equation''' (sometimes abbreviated as '''LNSE''' or '''LogSE''') is one of the [[nonlinear]] modifications of [[Schrödinger equation\|Schrödinger's equation]]. It is a classical wave equation with applications to extensions of [[quantum mechanics]],<ref>I. Bialynicki-Birula and J. Mycielski, Annals Phys. 100, 62 (1976); Commun. Math. Phys. 44, 129 (1975); Phys. Scripta 20, 539 (1979).</ref>
			[[quantum optics]],<ref>H. Buljan, A. Šiber, M. Soljačić, T. Schwartz, M. Segev, and D. N. Christodoulides, Phys. Rev. E 68, 036607 (2003).</ref>
			[[nuclear physics]],<ref>E. F. Hefter, Phys. Rev. A 32, 1201 (1985).</ref><ref>V. G. Kartavenko, K. A. Gridnev and W. Greiner, Int. J. Mod. Phys. E 7 (1998) 287.</ref>
			[[transport]] and [[diffusion]] phenomena,<ref>S. De Martino, M. Falanga, C. Godano and G. Lauro, Europhys. Lett. 63, 472 (2003); S. De Martino and G. Lauro, in: Proceed. 12th Conference on WASCOM, 2003.</ref><ref>T. Hansson, D. Anderson, and M. Lisak, Phys. Rev. A 80, 033819 (2009).</ref>
			open [[quantum]] systems and [[information theory]],<ref>K. Yasue, ''Quantum mechanics of nonconservative systems'', Annals Phys. 114 (1978) 479.</ref><ref>N. A. Lemos, Phys. Lett. A 78 (1980) 239.</ref><ref>J. D. Brasher, ''Nonlinear wave mechanics, information theory, and thermodynamics'', Int. J. Theor. Phys. 30 (1991) 979.</ref><ref>D. Schuch, Phys. Rev. A 55, 935 (1997).</ref><ref>M. P. Davidson, Nuov. Cim. B 116 (2001) 1291.</ref><ref>J. L. Lopez, Phys. Rev. E. 69 (2004) 026110.</ref>
			effective [[quantum gravity]] and physical [[vacuum]] models<ref>K. G. Zloshchastiev, ''Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences'', Grav. Cosmol. 16 (2010) 288–297 [http://arxiv.org/abs/0906.4282 ArXiv:0906.4282].</ref><ref>K. G. Zloshchastiev, ''Vacuum Cherenkov effect in logarithmic nonlinear quantum theory'', Phys. Lett. A 375 (2011) 2305–2308 [http://arxiv.org/abs/1003.0657 ArXiv:1003.0657].</ref><ref>K. G. Zloshchastiev, ''Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory'', Acta Phys. Polon. B 42 (2011) 261–292 [http://arxiv.org/abs/0912.4139 ArXiv:0912.4139].</ref>
			and
			theory of [[superfluidity]] and [[Bose–Einstein condensation]].<ref>A. V. Avdeenkov and K.G. Zloshchastiev, ''Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent'', J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 195303 [http://arxiv.org/abs/1108.0847 ArXiv:1108.0847].</ref>
			Its relativistic version (with [[D'Alembert_operator\|D'Alembertian]] instead of [[Laplacian]] and first-order time derivative) was first proposed by G. Rosen.<ref>G. Rosen, Phys. Rev. '''183''' (1969) 1186.</ref>
			It is an example of an [[integrable model]].

			== The equation ==

			The logarithmic Schrödinger equation is the [[partial differential equation]]. In [[mathematics]] and [[mathematical physics]] one often uses its [[dimensionless]] form:

			:<math> i \frac{\partial \psi}{\partial t} + \Delta \psi + \psi \ln \|\psi\|^2 = 0. </math>

			for the [[complex number\|complex-valued]] function <math>\psi=\psi (\mathrm{\mathbf{x}},t)</math>. Here <math>\Delta\,</math> is the [[Laplacian]] with respect to the [[Euclidean vector\|vector]] <math>\mathrm{\mathbf{x}}</math>.

			The relativistic version of this equation can be obtained by replacing the
			derivative operator with the [[D'Alembert_operator\|D'Alembertian]], similarly to the [[Klein-Gordon equation]].

			==See also==
			* [[Nonlinear Schrödinger equation]]

			==References==
			{{reflist\|2}}

			==External links==
			* {{MathWorld\|id=SchroedingerEquation\|title=SchroedingerEquation}}

			{{DEFAULTSORT:Logarithmic Schrodinger Equation}}
			[[Category:Theoretical physics]]
			[[Category:Quantum mechanics]]
			[[Category:Schrödinger equation]]

en>FrescoBot: Bot: link syntax/spacing

2012-07-20T08:32:41Z

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