Scattering theory Top: the real part of a plane wave travelling upwards. Bottom: The real part of the field after inserting in the path of the plane wave a small transparent disk of index of refraction higher than the index of the surrounding medium. This object scatters part of the wave field, although at any individual point, the wave's frequency and wavelength remain intact. In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a rainbow. Since its early statement for radiolocation, the problem has found vast number of applications, such as echolocation, geophysical survey, nondestructive testing, medical imaging and quantum field theory, to name just a few. Conceptual underpinnings[edit] Composite targets and range equations[edit] In theoretical physics[edit] [edit]
Schrödinger picture In physics, the Schrödinger picture (also called the Schrödinger representation[1]) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time.[2][3] This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as active and passive transformations and have the same measurement statistics. In the Schrödinger picture, the state of a system evolves with time. The evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. For time evolution from a state vector at time to a state vector , the time-evolution operator is commonly written , and one has where the exponent is evaluated via its Taylor series. Background[edit] and returns some other ket , or both.
Link between quantum physics and game theory found (Phys.org) —A deep link between two seemingly unconnected areas of modern science has been discovered by researchers from the Universities of Bristol and Geneva. While research tends to become very specialized and entire communities of scientists can work on specific topics with only a little overlap between them, physicist Dr Nicolas Brunner and mathematician Professor Noah Linden worked together to uncover a deep and unexpected connection between their two fields of expertise: game theory and quantum physics. Dr Brunner said: "Once in a while, connections are established between topics which seem, on the face of it, to have nothing in common. Such new links have potential to trigger significant progress and open entirely new avenues for research." Game theory—which is used today in a wide range of areas such as economics, social sciences, biology and philosophy—gives a mathematical framework for describing a situation of conflict or cooperation between intelligent rational players.
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.
Matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. In some contrast to the wave formulation, it produces spectra of energy operators by purely algebraic, ladder operator, methods.[1] Relying on these methods, Pauli derived the hydrogen atom spectrum in 1926,[2] before the development of wave mechanics. Development of matrix mechanics[edit] In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics. Epiphany at Helgoland[edit] In 1925 Werner Heisenberg was working in Göttingen on the problem of calculating the spectral lines of hydrogen. "It was about three o' clock at night when the final result of the calculation lay before me. The Three Fundamental Papers[edit] After Heisenberg returned to Göttingen, he showed Wolfgang Pauli his calculations, commenting at one point:[4] In the paper, Heisenberg formulated quantum theory without sharp electron orbits. W.
Qubit – The Game: Play your way to your next publication! « QuantumBlah Update (22.05.2013): According to the Facebook page of the project, the game (now called “meQuanics”) will be released tomorrow. Admittedly, one of the most common uses of computers is to play computer games, and in no small part did games influence the historic development of computer hardware. For example, the primary purpose of today’s high-end graphics cards is to compute the complex graphics effects of 3D games. Almost as an afterthought, it has been made possible to harness this brute computational power for productive purposes: Using frameworks such as OpenCL or CUDA, graphics cards can provide huge computational speedups in specific areas such as cryptography, molecular dynamics, fluid dynamics and distributed computing. The code on the right shows a much compressed variant of the same underlying circuit, obtained from the code in the middle by a series of circuit-preserving transformations. Tags: game, NII, Qubit, surface codes, topological quantum computation
Quantum statistical mechanics Expectation[edit] From classical probability theory, we know that the expectation of a random variable X is completely determined by its distribution DX by assuming, of course, that the random variable is integrable or that the random variable is non-negative. Similarly, let A be an observable of a quantum mechanical system. uniquely determines A and conversely, is uniquely determined by A. Similarly, the expected value of A is defined in terms of the probability distribution DA by Note that this expectation is relative to the mixed state S which is used in the definition of DA. Remark. One can easily show: Note that if S is a pure state corresponding to the vector ψ, then: Von Neumann entropy[edit] Of particular significance for describing randomness of a state is the von Neumann entropy of S formally defined by Actually, the operator S log2 S is not necessarily trace-class. and we define The convention is that , since an event with probability zero should not contribute to the entropy. and is
Interaction picture Equations that include operators acting at different times, which hold in the interaction picture, don't necessarily hold in the Schrödinger or the Heisenberg picture. This is because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others. Definition[edit] Operators and state vectors in the interaction picture are related by a change of basis (unitary transformation) to those same operators and state vectors in the Schrödinger picture. Any possible choice of parts will yield a valid interaction picture; but in order for the interaction picture to be useful in simplifying the analysis of a problem, the parts will typically be chosen so that H0,S is well understood and exactly solvable, while H1,S contains some harder-to-analyze perturbation to this system. by the corresponding time-evolution operator in the definitions below. State vectors[edit] A state vector in the interaction picture is defined as[4] Operators[edit] References[edit]