Logarithmic Schrödinger equation

In theoretical physics, the logarithmic Schrödinger equation (sometimes abbreviated as LNSE or LogSE) is one of the nonlinear modifications of Schrödinger's equation, first proposed by Gerald H. Rosen in its relativistic version (with D'Alembertian instead of Laplacian and first-order time derivative) in 1969.[1] It is a classical wave equation with applications to extensions of quantum mechanics,[2][3][4] quantum optics,[5] nuclear physics,[6][7] transport and diffusion phenomena,[8][9] open quantum systems and information theory,[10][11] [12][13][14][15] effective quantum gravity and physical vacuum models[16][17][18][19] and theory of superfluidity and Bose–Einstein condensation.[20][21] It is an example of an integrable model.

## The equation

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The logarithmic Schrödinger equation is a partial differential equation. In mathematics and mathematical physics one often uses its dimensionless form:  i ∂ ψ ∂ t + ∇ 2 ψ + ψ ln ⁡ | ψ | 2 = 0. {\displaystyle i{\frac {\partial \psi }{\partial t}}+\nabla ^{2}\psi +\psi \ln |\psi |^{2}=0.} for the complex-valued function ψ = ψ(x, t) of the particles position vector x = (x, y, z) at time t, and  ∇ 2 ψ = ∂ 2 ψ ∂ x 2 + ∂ 2 ψ ∂ y 2 + ∂ 2 ψ ∂ z 2 {\displaystyle \nabla ^{2}\psi ={\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}} is the Laplacian of ψ in Cartesian coordinates. The logarithmic term  ψ ln ⁡ | ψ | 2 {\displaystyle \psi \ln |\psi |^{2}} has been shown indispensable in determining the speed of sound scales as the cubic root of pressure for Helium-4 at very low temperatures.[22] This logarithmic term is also needed for cold sodium atoms.[23] In spite of the logarithmic term, it has been shown in the case of central potentials, that even for non-zero angular momentum, the LogSE retains certain symmetries similar to those found in its linear counterpart, making it potentially applicable to atomic and nuclear systems.[24]

The relativistic version of this equation can be obtained by replacing the derivative operator with the D'Alembertian, similarly to the Klein–Gordon equation. Soliton-like solutions known as Gaussons figure prominently as analytical solutions to this equation for a number of cases.

## See also

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* Galaxy rotation curve
* Nonlinear Schrödinger equation
* Superfluid Helium-4
* Superfluid vacuum theory

## References

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1. ^ Rosen, Gerald (1969). "Dilatation Covariance and Exact Solutions in Local Relativistic Field Theories". Physical Review. 183 (5): 1186–1188\. Bibcode:1969PhRv..183.1186R. doi:10.1103/PhysRev.183.1186. ISSN 0031-899X.
2. ^ Bialynicki-Birula, Iwo; Mycielski, Jerzy (1976). "Nonlinear wave mechanics". Annals of Physics. 100 (1–2): 62–93\. Bibcode:1976AnPhy.100...62B. doi:10.1016/0003-4916(76)90057-9. ISSN 0003-4916.1–2&rft.pages=62-93&rft.date=1976&rft.issn=0003-4916&rft_id=info:doi/10.1016/0003-4916(76)90057-9&rft_id=info:bibcode/1976AnPhy.100...62B&rft.aulast=Bialynicki-Birula&rft.aufirst=Iwo&rft.au=Mycielski,+Jerzy&rfr_id=info:sid/en.wikipedia.org:Logarithmic+Schrödinger+equation" class="Z3988">
3. ^ Białynicki-Birula, Iwo; Mycielski, Jerzy (1975). "Uncertainty relations for information entropy in wave mechanics". Communications in Mathematical Physics. 44 (2): 129–132\. Bibcode:1975CMaPh..44..129B. doi:10.1007/BF01608825. ISSN 0010-3616. S2CID 122277352. Archived from the original on 2021-02-08. Retrieved 2024-12-24.
4. ^ Bialynicki-Birula, Iwo; Mycielski, Jerzy (1979). "Gaussons: Solitons of the Logarithmic Schrödinger Equation". Physica Scripta. 20 (3–4): 539–544\. Bibcode:1979PhyS...20..539B. doi:10.1088/0031-8949/20/3-4/033. ISSN 0031-8949. S2CID 250833292.3–4&rft.pages=539-544&rft.date=1979&rft_id=info:doi/10.1088/0031-8949/20/3-4/033&rft_id=https://api.semanticscholar.org/CorpusID:250833292#id-name=S2CID&rft.issn=0031-8949&rft_id=info:bibcode/1979PhyS...20..539B&rft.aulast=Bialynicki-Birula&rft.aufirst=Iwo&rft.au=Mycielski,+Jerzy&rfr_id=info:sid/en.wikipedia.org:Logarithmic+Schrödinger+equation" class="Z3988">
5. ^ Buljan, H.; Šiber, A.; Soljačić, M.; Schwartz, T.; Segev, M.; Christodoulides, D. N. (2003). "Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media". Physical Review E. 68 (3): 036607. Bibcode:2003PhRvE..68c6607B. doi:10.1103/PhysRevE.68.036607. ISSN 1063-651X. PMID 14524912. S2CID 831827. Archived from the original on 2020-03-05. Retrieved 2019-12-10.
6. ^ Hefter, Ernst F. (1985). "Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics". Physical Review A. 32 (2): 1201–1204\. Bibcode:1985PhRvA..32.1201H. doi:10.1103/PhysRevA.32.1201. ISSN 0556-2791. PMID 9896178.
7. ^ Kartavenko, V. G.; Gridnev, K. A.; Greiner, W. (1998). "Nonlinear Effects in Nuclear Cluster Problem". International Journal of Modern Physics E. 07 (2): 287–299\. arXiv:nucl-th/9907015. Bibcode:1998IJMPE...7..287K. doi:10.1142/S0218301398000129. ISSN 0218-3013. S2CID 19009168.
8. ^ Martino, S. De; Falanga, M; Godano, C; Lauro, G (2003). "Logarithmic Schrödinger-like equation as a model for magma transport". Europhysics Letters. 63 (3): 472–475\. Bibcode:2003EL.....63..472D. doi:10.1209/epl/i2003-00547-6. ISSN 0295-5075. S2CID 250736155.
9. ^ Hansson, T.; Anderson, D.; Lisak, M. (2009). "Propagation of partially coherent solitons in saturable logarithmic media: A comparative analysis". Physical Review A. 80 (3): 033819. Bibcode:2009PhRvA..80c3819H. doi:10.1103/PhysRevA.80.033819. ISSN 1050-2947.
10. ^ Yasue, Kunio (1978). "Quantum mechanics of nonconservative systems". Annals of Physics. 114 (1–2): 479–496\. Bibcode:1978AnPhy.114..479Y. doi:10.1016/0003-4916(78)90279-8. ISSN 0003-4916.1–2&rft.pages=479-496&rft.date=1978&rft.issn=0003-4916&rft_id=info:doi/10.1016/0003-4916(78)90279-8&rft_id=info:bibcode/1978AnPhy.114..479Y&rft.aulast=Yasue&rft.aufirst=Kunio&rfr_id=info:sid/en.wikipedia.org:Logarithmic+Schrödinger+equation" class="Z3988">
11. ^ Lemos, Nivaldo A. (1980). "Dissipative forces and the algebra of operators in stochastic quantum mechanics". Physics Letters A. 78 (3): 239–241\. Bibcode:1980PhLA...78..239L. doi:10.1016/0375-9601(80)90080-8. ISSN 0375-9601.
12. ^ Brasher, James D. (1991). "Nonlinear wave mechanics, information theory, and thermodynamics". International Journal of Theoretical Physics. 30 (7): 979–984\. Bibcode:1991IJTP...30..979B. doi:10.1007/BF00673990. ISSN 0020-7748. S2CID 120250281.
13. ^ Schuch, Dieter (1997). "Nonunitary connection between explicitly time-dependent and nonlinear approaches for the description of dissipative quantum systems". Physical Review A. 55 (2): 935–940\. Bibcode:1997PhRvA..55..935S. doi:10.1103/PhysRevA.55.935. ISSN 1050-2947.
14. ^ M. P. Davidson, Nuov. Cim. B 116 (2001) 1291.
15. ^ López, José L. (2004). "Nonlinear Ginzburg-Landau-type approach to quantum dissipation". Physical Review E. 69 (2): 026110. Bibcode:2004PhRvE..69b6110L. doi:10.1103/PhysRevE.69.026110. ISSN 1539-3755. PMID 14995523.
16. ^ Zloshchastiev, K. G. (2010). "Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences". Gravitation and Cosmology. 16 (4): 288–297\. arXiv:0906.4282. Bibcode:2010GrCo...16..288Z. doi:10.1134/S0202289310040067. ISSN 0202-2893. S2CID 119187916.
17. ^ Zloshchastiev, Konstantin G. (2011). "Vacuum Cherenkov effect in logarithmic nonlinear quantum theory". Physics Letters A. 375 (24): 2305–2308\. arXiv:1003.0657. Bibcode:2011PhLA..375.2305Z. doi:10.1016/j.physleta.2011.05.012. ISSN 0375-9601. S2CID 118152360.
18. ^ Zloshchastiev, K.G. (2011). "Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory". Acta Physica Polonica B. 42 (2): 261–292\. arXiv:0912.4139. Bibcode:2011AcPPB..42..261Z. doi:10.5506/APhysPolB.42.261. ISSN 0587-4254. S2CID 118152708.
19. ^ Scott, T.C.; Zhang, Xiangdong; Mann, Robert; Fee, G.J. (2016). "Canonical reduction for dilatonic gravity in 3 + 1 dimensions". Physical Review D. 93 (8): 084017. arXiv:1605.03431. Bibcode:2016PhRvD..93h4017S. doi:10.1103/PhysRevD.93.084017.
20. ^ Avdeenkov, Alexander V; Zloshchastiev, Konstantin G (2011). "Quantum Bose liquids with logarithmic nonlinearity: self-sustainability and emergence of spatial extent". Journal of Physics B: Atomic, Molecular and Optical Physics. 44 (19): 195303. arXiv:1108.0847. Bibcode:2011JPhB...44s5303A. doi:10.1088/0953-4075/44/19/195303. ISSN 0953-4075. S2CID 119248001.
21. ^ Zloshchastiev, Konstantin G. (2019). "Temperature-driven dynamics of quantum liquids: Logarithmic nonlinearity, phase structure and rising force". Int. J. Mod. Phys. B. 33 (17): 1950184. arXiv:2001.04688. Bibcode:2019IJMPB..3350184Z. doi:10.1142/S0217979219501844. S2CID 199674799.
22. ^ Scott, T. C.; Zloshchastiev, K. G. (2019). "Resolving the puzzle of sound propagation in liquid helium at low temperatures". Low Temperature Physics. 45 (12): 1231–1236\. arXiv:2006.08981. Bibcode:2019LTP....45.1231S. doi:10.1063/10.0000200. S2CID 213962795.
23. ^ Zloshchastiev, Konstantin (2022). "Resolving the puzzle of sound propagation in a dilute Bose–Einstein condensate". International Journal of Modern Physics B. 36 (20): 2250121. arXiv:2211.10570. Bibcode:2022IJMPB..3650121Z. doi:10.1142/S0217979222501211. S2CID 249262506.
24. ^ Shertzer, J.; Scott, T.C. (2020). "Solution of the 3D logarithmic Schrödinger equation with a central potential". J. Phys. Commun. 4 (6): 065004. Bibcode:2020JPhCo...4f5004S. doi:10.1088/2399-6528/ab941d.

## External links

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* Weisstein, Eric W. "SchroedingerEquation". MathWorld.