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. 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][16]} and theory of superfluidity and Bose–Einstein condensation.^[17] Its relativistic version (with D'Alembertian instead of Laplacian and first-order time derivative) was first proposed by G. Rosen.^[18] 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:

$i{\frac {\partial \psi }{\partial t}}+\Delta \psi +\psi \ln |\psi |^{2}=0.$

for the complex-valued function ψ = ψ(x, t) of the particles position vector x = (x, y, z) at time t, and

${\displaystyle \Delta \psi ={\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}\,}$

is the Laplacian of ψ in Cartesian coordinates.

The 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

• Nonlinear Schrödinger equation

References

1. I. Bialynicki-Birula and J. Mycielski, Annals of Physics 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 of Physics 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. 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.
17. 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.
18. G. Rosen, Phys. Rev. 183 (1969) 1186.

External links

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