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]
Bloch sphere Bloch sphere In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.[1] Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space. The space of pure states of a quantum system is given by the one-dimensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space). In a two-dimensional Hilbert space this is simply the complex projective line, which is a geometrical sphere. The Bloch sphere is a unit 2-sphere, with each pair of antipodal points corresponding to mutually orthogonal state vectors. and Definition[edit] Given an orthonormal basis, any pure state of a two-level quantum system can be written as a superposition of the basis vectors , where the coefficient or amount of each basis vector is a complex number. to be real and non-negative. , meaning . in the following representation: with . or .
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.
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
Quantum spacetime In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some variables that are usually continuous may become discrete. Often only such discrete variables are called "quantized"; usage varies. The idea of quantum spacetime was proposed in the early days of quantum theory by Heisenberg and Ivanenko as a way to eliminate infinities from quantum field theory. The germ of the idea passed from Heisenberg to Rudolf Peierls, who noted that electrons in a magnetic field can be regarded as moving in a quantum space-time, and to Robert Oppenheimer, who carried it to Hartland Snyder, who published the first concrete example.[1] Snyder's Lie algebra was made simple by C. The Lie algebra should be semisimple (Yang, I. for the spatial variables . .
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).
Path integral formulation The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks. For this reason path integrals were used in the study of Brownian motion and diffusion a while before they were introduced in quantum mechanics.[3] These are just three of the paths that contribute to the quantum amplitude for a particle moving from point A at some time t0 to point B at some other time t1. Quantum action principle[edit] But the Hamiltonian in classical mechanics is derived from a Lagrangian, which is a more fundamental quantity considering special relativity. and where and the partial derivative now is with respect to p at fixed q.
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.
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 . . .
Yang–Mills existence and mass gap In mathematical physics, the Yang–Mills existence and mass gap problem is an unsolved problem and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute which has offered a prize of US$1,000,000 to the one who solves it. The problem is phrased as follows: Yang–Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975). In this statement, Yang–Mills theory is the (non-Abelian) quantum field theory underlying the Standard Model of particle physics; is Euclidean 4-space; the mass gap Δ is the mass of the least massive particle predicted by the theory. Background[edit] See also[edit] References[edit] Arthur Jaffe and Edward Witten "Quantum Yang-Mills theory." External links[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:
Scalar field theory In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field. The quanta of the quantized scalar field are spin-zero particles, and as such are bosons. The only fundamental scalar field that has been observed in nature is the Higgs field. However, scalar fields appear in the effective field theory descriptions of many physical phenomena. , has a particularly simple form: it is diagonal, and here we use the + − − − sign convention. Classical scalar field theory[edit] Linear (free) theory[edit] where is known as a Lagrangian density, dD-1 ≝ dx⋅dy⋅dz ≝ dx1⋅dx2⋅dx3 for the three spatial coordinates, is the Kronecker delta function and where for the ρ-th coordinate xρ . . is sometimes known as a mass term, due to its interpretation in the quantized version of this theory in terms of particle mass. is the Laplace operator. The n! . , of