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][2][3] quantum optics,^[4] nuclear physics,^[5][6] transport and diffusion phenomena,^[7][8] open quantum systems and information theory,^{[9][10] [11][12][13][14]} effective quantum gravity and physical vacuum models^{[15][16][17][18]} and theory of superfluidity and Bose–Einstein condensation.^[19][20] Its relativistic version (with D'Alembertian instead of Laplacian and first-order time derivative) was first proposed by Gerald Rosen.^[21] 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:

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

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

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

is the Laplacian of ψ in Cartesian coordinates. The logarithmic term ${\displaystyle \psi \ln |\psi |^{2}}$ 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] 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.^[23]

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[]

• Nonlinear Schrödinger equation
• Superfluid Helium-4
• Superfluid vacuum theory

References[]

External links[]

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