Logarithmic Schrödinger equation

The logarithmic Schrödinger equation is a 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] 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.

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

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