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,
,
,
transport and
diffusion phenomena,
open
quantum
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
systems and
information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
,
effective
quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
and physical
vacuum models
and theory of
superfluidity and
Bose–Einstein condensation.
Its relativistic version (with
D'Alembertian instead of
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
and first-order time derivative) was first proposed by
Gerald Rosen
Gerald Ellis Rosen (born October 26, 1951) is a former United States district judge of the United States District Court for the Eastern District of Michigan.
Professional career
Prior to taking the bench, Rosen was a senior partner in the law ...
.
It is an example of an
integrable model.
The equation
The logarithmic Schrödinger equation is the
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
. In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
mathematical physics one often uses its
dimensionless form:
for the
complex-valued
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
function of the particles
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s ...
at time , and
is the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
of in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
. The logarithmic term
has been shown indispensable in determining the speed of sound scales as the cubic root of pressure for
Helium-4 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
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 helium-4 is the superfluid form of helium-4, an isotope of the element helium. A superfluid is a state of matter in which matter behaves like a fluid with zero viscosity. The substance, which looks like a normal liquid, flows without ...
*
Superfluid vacuum theory
Superfluid vacuum theory (SVT), sometimes known as the BEC vacuum theory, is an approach in theoretical physics and quantum mechanics where the fundamental physical vacuum (non-removable background) is viewed as superfluid or as a Bose–Einstei ...
References
External links
*
{{DEFAULTSORT:Logarithmic Schrodinger Equation
Theoretical physics
Schrödinger equation