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Schrödinger equation

Schrödinger equation
In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of some physical system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger.[1] In classical mechanics, the equation of motion is Newton's second law, and equivalent formulations are the Euler–Lagrange equations and Hamilton's equations. All of these formulations are used to solve for the motion of a mechanical system and mathematically predict what the system will do at any time beyond the initial settings and configuration of the system. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but (in general) a linear partial differential equation. The concept of a state vector is a fundamental postulate of quantum mechanics.

Antony Valentini Antony Valentini is a theoretical physicist and a professor at Clemson University. He is known for his work on the foundations of quantum physics.[1] Education and career[edit] Valentini obtained an undergraduate degree from Cambridge University, then earned his Ph.D. in 1992[2] with Dennis Sciama at the International School for Advanced Studies (ISAS) in Trieste, Italy.[1][3] In 1999, after seven years in Italy, he took up a post-doc grant to work at the Imperial College with Lee Smolin and Christopher Isham.[1] He currently works at the Perimeter Institute for Theoretical Physics. Work[edit] Valentini has been working on an extension of the causal interpretation of quantum theory. Quantum equilibrium, locality and uncertainty[edit] In 1991, Valentini provided indications for deriving the quantum equilibrium hypothesis which states that in the frame work of the pilot wave theory. may be accounted for by a H-theorem constructed in analogy to the Boltzmann H-theorem of statistical mechanics.

Klein–Gordon equation The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger equation. It is the equation of motion of a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It cannot be straightforwardly interpreted as a Schrödinger equation for a quantum state, because it is second order in time and because it does not admit a positive definite conserved probability density. Still, with the appropriate interpretation, it does describe the quantum amplitude for finding a point particle in various places, the relativistic wavefunction, but the particle propagates both forwards and backwards in time. Statement[edit] The Klein–Gordon equation is This is often abbreviated as where and is the d'Alembert operator, defined by The equation is most often written in natural units: The form is determined by requiring that plane wave solutions of the equation: which is the homogeneous screened Poisson equation.

Pilot wave In theoretical physics, the pilot wave theory was the first known example of a hidden variable theory, presented by Louis de Broglie in 1927. Its more modern version, the Bohm interpretation, remains a controversial attempt to interpret quantum mechanics as a deterministic theory, avoiding troublesome notions such as instantaneous wavefunction collapse and the paradox of Schrödinger's cat. The pilot wave theory[edit] The pilot wave theory is one of several interpretations of quantum mechanics. It uses the same mathematics as other interpretations of quantum mechanics; consequently, it is also supported by the current experimental evidence to the same extent as the other interpretations. Principles[edit] The pilot wave theory is a hidden variable theory. the theory has realism (meaning that its concepts exist independently of the observer);the theory has determinism. The positions and momenta of the particles are considered to be the hidden variables. Consequences[edit] where is known. . and .

Airy function This article is about the Airy special function. For the Airy stress function employed in solid mechanics, see Stress functions. In the physical sciences, the Airy function Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–92). The function Ai(x) and the related function Bi(x), which is also called the Airy function, but sometimes referred to as the Bairy function, are solutions to the differential equation known as the Airy equation or the Stokes equation. The Airy function is the solution to Schrödinger's equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field. Definitions[edit] Plot of Ai(x) in red and Bi(x) in blue For real values of x, the Airy function of the first kind can be defined by the improper Riemann integral: which converges because the positive and negative parts of the rapid oscillations tend to cancel one another out (as can be checked by integration by parts).

Wave function collapse When the Copenhagen interpretation was first expressed, Niels Bohr postulated wave function collapse to cut the quantum world from the classical.[5] This tactical move allowed quantum theory to develop without distractions from interpretational worries. Mathematical description[edit] Mathematical background[edit] The quantum state of a physical system is described by a wave function (in turn – an element of a projective Hilbert space). The kets Where represents the Kronecker delta. An observable (i.e. measurable parameter of the system) is associated with each eigenbasis, with each quantum alternative having a specific value or eigenvalue, ei, of the observable. The coefficients c1, c2, c3... are the probability amplitudes corresponding to each basis . For simplicity in the following, all wave functions are assumed to be normalized; the total probability of measuring all possible states is unity: The process of collapse[edit] of that observable. , the wave function collapses from the full . .

Euler–Bernoulli beam theory This vibrating glass beam may be modeled as a cantilever beam with acceleration, variable linear density, variable section modulus, some kind of dissipation, springy end loading, and possibly a point mass at the free end. Additional analysis tools have been developed such as plate theory and finite element analysis, but the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering. History[edit] Schematic of cross-section of a bent beam showing the neutral axis. Prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue that Leonardo da Vinci was the first to make the crucial observations. The Bernoulli beam is named after Jacob Bernoulli, who made the significant discoveries. Static beam equation[edit] Bending of an Euler–Bernoulli beam. The curve describes the deflection of the beam in the direction at some position , or other variables. Note that where , and and . . .

Quantum entanglement Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently – instead, a quantum state may be given for the system as a whole. Such phenomena were the subject of a 1935 paper by Albert Einstein, Boris Podolsky and Nathan Rosen,[1] describing what came to be known as the EPR paradox, and several papers by Erwin Schrödinger shortly thereafter.[2][3] Einstein and others considered such behavior to be impossible, as it violated the local realist view of causality (Einstein referred to it as "spooky action at a distance"),[4] and argued that the accepted formulation of quantum mechanics must therefore be incomplete. History[edit] However, they did not coin the word entanglement, nor did they generalize the special properties of the state they considered. Concept[edit] Meaning of entanglement[edit] Apparent paradox[edit] The hidden variables theory[edit]

Dispersive partial differential equation Dispersive partial differential equation From Wikipedia, the free encyclopedia Jump to: navigation , search In mathematics , a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive . In this context, dispersion means that waves of different wavelength propagate at different phase velocities . Examples [ edit ] Linear equations [ edit ] Nonlinear equations [ edit ] See also [ edit ] External links [ edit ] The Dispersive PDE Wiki . Retrieved from " Categories : Navigation menu Personal tools Namespaces Variants Views Actions Navigation Interaction Toolbox Print/export Languages Edit links This page was last modified on 21 May 2012 at 20:15.

Interpretations of quantum mechanics An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations. There exist a number of contending schools of thought, differing over whether quantum mechanics can be understood to be deterministic, which elements of quantum mechanics can be considered "real", and other matters. This question is of special interest to philosophers of physics, as physicists continue to show a strong interest in the subject. They usually consider an interpretation of quantum mechanics as an interpretation of the mathematical formalism of quantum mechanics, specifying the physical meaning of the mathematical entities of the theory. History of interpretations[edit] Main quantum mechanics interpreters Nature of interpretation[edit] Two qualities vary among interpretations:

Yves Couder In the first decades of the 20th century, physicists hotly debated how to make sense of the strange phenomena of quantum mechanics, such as the tendency of subatomic particles to behave like both particles and waves. One early theory, called pilot-wave theory, proposed that moving particles are borne along on some type of quantum wave, like driftwood on the tide. But this theory ultimately gave way to the so-called Copenhagen interpretation, which gets rid of the carrier wave, but with it the intuitive notion that a moving particle follows a definite path through space. Recently, Yves Couder, a physicist at Université Paris Diderot, has conducted a series of experiments in which millimeter-scale fluid droplets, bouncing up and down on a vibrated fluid bath, are guided by the waves that they themselves produce. The wave-particle duality is best illustrated by a canonical experiment in quantum mechanics that’s generally referred to as the two-slit, or two-hole, experiment. Scaling up

Derek Leinweber Centre for the Subatomic Structure of Matter (CSSM) and Department of Physics, University of Adelaide, 5005 Australia Copyright © 2003, 2004 This page provides a collection of the most recent visualizations of Quantum Chromodynamics (QCD), the underlying theory of the strong interactions. As a key component of the Standard Model of the Universe, QCD describes the interactions between quarks and gluons as they compose particles such as the proton or neutron. State of the art order a4-improved lattice operators are used in creating the animations, including the three-loop improved lattice gauge action and the five-loop improved lattice field strength tensor. The animaton at right was featured in Prof. Frank Wilczek's 2004 Nobel Prize Lecture. Contributions from Sundance Bilson-Thompson on improved operator construction and Ben Lasscock and James Zanotti on the vacuum response to static quarks, are gratefully acknowledged. For copyright information, please contact Derek Leinweber.

Guhr's research Quantum Chaos and Random Matrix Models Random matrices provide powerful models for a rich variety of complex systems. Here is a brief overview. A good example is the atomic nucleus shown to the right. In several different branches of physics, we study chaotically coupled systems or, equivalently, systems with symmetry breaking. Disordered systems are a wide field for applications of Random Matrix Models. local collaborator: Johan Grönqvist (PhD-student) external collaborators: Professor Achim Richter and his group at Technical University of Darmstadt, Professor Hans-Jürgen Stöckmann at University of Marburg. Study of SubAtomic Interactions through Lattice Quantum Chromo Dynamics on Mare Nostrum (SAIL) | Annual Report 2008 Abstract Quantum Chromodynamics (QCD) is the underlying theory governing the interaction between quarks and gluons, the strong force, and therefore, responsible for all the states of matter in the Universe. Analytical solutions of QCD in the low energy regime cannot be obtained due to the complexity of the quark-gluon dynamics. The only known non-perturbative method that systematically implements QCD from first principles is its formulation on a discretized space-time, lattice QCD. Results obtained The computing resources awarded to the project were invested to explore hadron-hadron scattering at different values of the light quark masses and at different lattice spacings. Images Needs of computation for different physics problems: mnad1_jpeg.jpg From quarks to stars. Publications or reports William Detmold, Martin J.