 # 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. 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. Education and career Valentini obtained an undergraduate degree from Cambridge University, then earned his Ph.D. in 1992 with Dennis Sciama at the International School for Advanced Studies (ISAS) in Trieste, Italy. 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. He currently works at the Perimeter Institute for Theoretical Physics. Work Valentini has been working on an extension of the causal interpretation of quantum theory. Quantum equilibrium, locality and uncertainty 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 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 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 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 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 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. This tactical move allowed quantum theory to develop without distractions from interpretational worries. Mathematical description Mathematical background 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 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 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 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, describing what came to be known as the EPR paradox, and several papers by Erwin Schrödinger shortly thereafter. 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"), and argued that the accepted formulation of quantum mechanics must therefore be incomplete. History However, they did not coin the word entanglement, nor did they generalize the special properties of the state they considered. Concept Meaning of entanglement Apparent paradox The hidden variables theory