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, quantum optics, nuclear physics, transport and diffusion phenomena, open quantum systems and information theory, effective quantum gravity and physical vacuum models and theory of superfluidity and Bose–Einstein condensation. Its relativistic version (with D'Alembertian instead of Laplacian and first-order time derivative) was first proposed by G. Rosen. 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:
for the complex-valued function ψ = ψ(x, t) of the particles position vector x = (x, y, z) at time t, and
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.