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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>
[[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]]

Revision as of 05:36, 16 October 2012

In theoretical physics, the logarithmic Schrödinger equation (sometimes abbreviated as LNSE or LogSE) is one of the nonlinear modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of quantum mechanics,[1] quantum optics,[2] nuclear physics,[3][4] transport and diffusion phenomena,[5][6] open quantum systems and information theory,[7][8][9][10][11][12] effective quantum gravity and physical vacuum models[13][14][15] and theory of superfluidity and Bose–Einstein condensation.[16] Its relativistic version (with D'Alembertian instead of Laplacian and first-order time derivative) was first proposed by G. Rosen.[17] 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:

i∂ψ∂t+Δψ+ψln⁡|ψ|2=0.

for the complex-valued function ψ=ψ(x,t). Here Δ is the Laplacian with respect to the vector x.

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

See also

References

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External links



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  1. ↑ I. Bialynicki-Birula and J. Mycielski, Annals Phys. 100, 62 (1976); Commun. Math. Phys. 44, 129 (1975); Phys. Scripta 20, 539 (1979).
  2. ↑ H. Buljan, A. Šiber, M. Soljačić, T. Schwartz, M. Segev, and D. N. Christodoulides, Phys. Rev. E 68, 036607 (2003).
  3. ↑ E. F. Hefter, Phys. Rev. A 32, 1201 (1985).
  4. ↑ V. G. Kartavenko, K. A. Gridnev and W. Greiner, Int. J. Mod. Phys. E 7 (1998) 287.
  5. ↑ 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.
  6. ↑ T. Hansson, D. Anderson, and M. Lisak, Phys. Rev. A 80, 033819 (2009).
  7. ↑ K. Yasue, Quantum mechanics of nonconservative systems, Annals Phys. 114 (1978) 479.
  8. ↑ N. A. Lemos, Phys. Lett. A 78 (1980) 239.
  9. ↑ J. D. Brasher, Nonlinear wave mechanics, information theory, and thermodynamics, Int. J. Theor. Phys. 30 (1991) 979.
  10. ↑ D. Schuch, Phys. Rev. A 55, 935 (1997).
  11. ↑ M. P. Davidson, Nuov. Cim. B 116 (2001) 1291.
  12. ↑ J. L. Lopez, Phys. Rev. E. 69 (2004) 026110.
  13. ↑ K. G. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences, Grav. Cosmol. 16 (2010) 288–297 ArXiv:0906.4282.
  14. ↑ K. G. Zloshchastiev, Vacuum Cherenkov effect in logarithmic nonlinear quantum theory, Phys. Lett. A 375 (2011) 2305–2308 ArXiv:1003.0657.
  15. ↑ 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 ArXiv:0912.4139.
  16. ↑ 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 ArXiv:1108.0847.
  17. ↑ G. Rosen, Phys. Rev. 183 (1969) 1186.
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