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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, first proposed by Gerald H. Rosen in its relativistic version (with D'Alembertian instead of Laplacian and first-order time derivative) in 1969. 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. It is an example of an integrable model. The equation The logarithmic Schrödinger equation is a partial differential equation. In mathematics and mathematical physics one often uses its dimensionless 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 = \fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 List of natural phenomena, natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 considered as a superfluid or as a Bose–Einstein condensate (BEC). The microscopic structure of this physical vacuum is currently unknown and is a subject of intensive studies in SVT. An ultimate goal of this research is to develop scientific models that unify quantum mechanics (which describes three of the four known fundamental interactions) with gravity, making SVT a derivative of quantum gravity and describes all known interactions in the Universe, at both microscopic and astronomic scales, as different manifestations of the same entity, superfluid vacuum. History The concept of a luminiferous aether as a medium sustaining electromagnetic waves was discarded after the advent of the special theory of relativity, as the presence of the concept alongside special relativity res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superfluid Helium-4
Superfluid helium-4 (helium II or He-II) is the superfluid form of helium-4, the most common isotope of the element helium. The substance, which resembles other liquids such as helium I (conventional, non-superfluid liquid helium), flows without friction past any surface, which allows it to continue to circulate over obstructions and through pores in containers which hold it, subject only to its own inertia. The formation of the superfluid is a manifestation of the formation of a Bose–Einstein condensate of helium atoms. This condensation occurs in liquid helium-4 at a far higher temperature (2.17 K) than it does in helium-3 (2.5 mK) because each atom of helium-4 is a boson particle, by virtue of its zero Spin (physics), spin. Helium-3, however, is a fermion particle, which can form bosons only by pairing with itself at much lower temperatures, in a weaker process that is similar to the electron pairing in superconductivity. History Known as a major facet in the stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Schrödinger Equation
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers, planar waveguides and hot rubidium vapors and to Bose–Einstein condensates confined to highly anisotropic, cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galaxy Rotation Curve
The rotation curve of a disc galaxy (also called a velocity curve) is a plot of the orbital speeds of visible stars or gas in that galaxy versus their radial distance from that galaxy's centre. It is typically rendered graphically as a plot, and the data observed from each side of a spiral galaxy are generally asymmetric, so that data from each side are averaged to create the curve. A significant discrepancy exists between the experimental curves observed, and a curve derived by applying gravity theory to the matter observed in a galaxy. Theories involving dark matter are the main postulated solutions to account for the variance. The rotational/orbital speeds of galaxies/stars do not follow the rules found in other orbital systems such as stars/planets and planets/moons that have most of their mass at the centre. Stars revolve around their galaxy's centre at equal or increasing speed over a large range of distances. In contrast, the orbital velocities of planets in planetary s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gausson (physics)
The Gausson is a soliton which is the solution of the logarithmic Schrödinger equation, which describes a quantum particle in a possible nonlinear quantum mechanics. The logarithmic Schrödinger equation preserves the dimensional homogeneity of the equation, i.e. the product of the independent solutions in one dimension remain the solution in multiple dimensions. While the nonlinearity alone cannot cause the quantum entanglement between dimensions, the logarithmic Schrödinger equation can be solved by the separation of variables. Let the nonlinear Logarithmic Schrödinger equation in one dimension will be given by (\hbar = 1, unit mass m=1): :i = -\frac \frac- a \ln , \psi, ^2\psi Let assume the Galilean invariance i.e. :\frac\psi(x,t)=e^\psi(x-k t) Substituting :\fracy=x-k t The first equation can be written as : -\frac \left \right2 \psi -a \ln , \psi, ^2 \psi=\left( E + \frac \right) \psi Substituting additionally :\frac\Psi(y)=e^\psi(y) and assuming :\Psi(y)=N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 named after Oskar Klein and Walter Gordon. It is second-order in space and time and manifestly Lorentz-covariant. It is a differential equation version of the relativistic energy–momentum relation E^2 = (pc)^2 + \left(m_0c^2\right)^2\,. Statement The Klein–Gordon equation can be written in different ways. The equation itself usually refers to the position space form, where it can be written in terms of separated space and time components \ \left(\ t, \mathbf\ \right)\ or by combining them into a four-vector \ x^\mu = \left(\ c\ t, \mathbf\ \right) ~. By Fourier transforming the field into momentum space, the solution is usually written in terms of a superposition of plane waves whose energy and momentum obey the energy-momentum dispersion relation from special relativity. Here, the Klein– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 consists of two protons and two neutrons. Helium-4 makes up about one quarter of the ordinary matter in the universe by mass, with almost all of the rest being hydrogen. While nuclear fusion in stars also produces helium-4, most of the helium-4 in the Sun and in the universe is thought to have been produced during the Big Bang, known as " primordial helium". However, primordial helium-4 is largely absent from the Earth, having escaped during the high-temperature phase of Earth's formation. On Earth, most naturally occurring helium-4 is produced by the alpha decay of heavy elements in the Earth's crust, after the planet cooled and solidified. When liquid helium-4 is cooled to below , it becomes a superfluid, with properties very different from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular oriented lines, called '' coordinate lines'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the '' origin'' and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame. Similarly, the position of any point in three-dimensional space can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension . These coordinates are the signed distances from the point to mutually perpendicular fixed h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Position Vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and its direction represents the angular orientation with respect to given reference axes. Usually denoted x, r, or s, it corresponds to the straight line segment from ''O'' to ''P''. In other words, it is the displacement or translation that maps the origin to ''P'': :\mathbf=\overrightarrow. The term position vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus. Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces and affine spaces of any dimension.Keller, F. J., Gettys, W. E. et al. (1993), p. 28–29. Relative position The relative position of a point ''Q'' with respect to point ''P'' is the Euclidean vector res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
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 form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
