theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

, the logarithmic Schrödinger equation (sometimes abbreviated as LNSE or LogSE) is one of the

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...

modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...

quantum optics Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...

nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...

, transport and

diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemica ...

phenomena, open quantum systems and information theory, effective quantum gravity and physical

vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...

models and theory of

superfluidity Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...

and

Bose–Einstein condensation Bose–Einstein may refer to: * Bose–Einstein condensate ** Bose–Einstein condensation (network theory) * Bose–Einstein correlations * Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describe ...

. Its relativistic version (with

D'Alembertian In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...

instead of Laplacian and first-order time derivative) was first proposed by Gerald 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 Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...

one often uses its

dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...

form:

i \frac + \nabla^2 \psi + \psi \ln , \psi, ^2 = 0.

for the complex-valued function of the particles position vector at time , and

\nabla^2 \psi = \frac + \frac + \frac

is the Laplacian of in Cartesian coordinates. The logarithmic term

\psi \ln , \psi, ^2

has been shown indispensable in determining the speed of sound scales as the cubic root of pressure for

Helium-4 Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consis ...

at very low temperatures. This logarithmic term is also needed for cold sodium atoms. 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. The relativistic version of this equation can be obtained by replacing the derivative operator with the

, similarly to the

Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant ...

. Soliton-like solutions known as Gaussons figure prominently as analytical solutions to this equation for a number of cases.

References

External links

* {{DEFAULTSORT:Logarithmic Schrodinger Equation Theoretical physics Schrödinger equation

The equation

See also

References

External links