Phase space. Phase space of a dynamic system with focal instability, showing one phase space trajectory A plot of position and momentum variables as a function of time is sometimes called a phase plot or a phase diagram. Phase diagram, however, is more usually reserved in the physical sciences for a diagram showing the various regions of stability of the thermodynamic phases of a chemical system, which consists of pressure, temperature, and composition. In classical mechanics, any choice of generalized coordinates q i for the position (i.e. coordinates on configuration space) defines conjugate generalized momenta pi which together define co-ordinates on phase space. More abstractly, in classical mechanics phase space is the cotangent space of configuration space, and in this interpretation the procedure above expresses that a choice of local coordinates on configuration space induces a choice of natural local Darboux coordinates for the standard symplectic structure on a cotangent space.
Phase diagram. Overview[edit] Common components of a phase diagram are lines of equilibrium or phase boundaries, which refer to lines that mark conditions under which multiple phases can coexist at equilibrium. Phase transitions occur along lines of equilibrium. Triple points are points on phase diagrams where lines of equilibrium intersect. Triple points mark conditions at which three different phases can coexist. For example, the water phase diagram has a triple point corresponding to the single temperature and pressure at which solid, liquid, and gaseous water can coexist in a stable equilibrium. Types of phase diagrams[edit] 2D phase diagrams[edit] The curves on the phase diagram show the points where the free energy (and other derived properties) becomes non-analytic: their derivatives with respect to the coordinates (temperature and pressure in this example) change discontinuously (abruptly). Other thermodynamic properties Temperature vs. specific entropy phase diagram for water/steam.
See also[edit] Chaos theory. A double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
Chaotic behavior can be observed in many natural systems, such as weather and climate.[6][7] This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Introduction[edit] Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic dynamics[edit] The map defined by x → 4 x (1 – x) and y → x + y mod 1 displays sensitivity to initial conditions. In common usage, "chaos" means "a state of disorder".[9] However, in chaos theory, the term is defined more precisely. Where , and , is: . The Structure of Scientific Revolutions. The Structure of Scientific Revolutions is a 1962 book about the history of science by Thomas S. Kuhn. Its publication was a landmark event in the history, philosophy, and sociology of scientific knowledge and triggered an ongoing worldwide assessment and reaction in—and beyond—those scholarly communities.
Kuhn challenged the then prevailing view of progress in "normal science. " Normal scientific progress was viewed as "development-by-accumulation" of accepted facts and theories. Kuhn argued for an episodic model in which periods of such conceptual continuity in normal science were interrupted by periods of revolutionary science. The discovery of "anomalies" during revolutions in science leads to new paradigms. New paradigms then ask new questions of old data, move beyond the mere "puzzle-solving" of the previous paradigm, change the rules of the game and the "map" directing new research.[1] History[edit] Synopsis[edit] Basic approach[edit] Historical examples[edit] Coherence[edit] Wave function. However, complex numbers are not necessarily used in all treatments. Louis de Broglie in his later years proposed a real-valued wave function connected to the complex wave function by a proportionality constant and developed the de Broglie–Bohm theory.
The unit of measurement for ψ depends on the system. For one particle in three dimensions, its units are [length]−3/2. These unusual units are required so that an integral of |ψ|2 over a region of three-dimensional space is a unitless probability (the probability that the particle is in that region). Historical background[edit] In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra.
Wave functions and function spaces[edit] Functional analysis is commonly used to formulate the wave function with a necessary mathematical precision; usually they are quadratically integrable functions (at least locally) because it is compatible with the Hilbert space formalism mentioned below. Requirements[edit] 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). This can be expressed in Dirac or bra–ket notation as a vector: 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] . . Quantum state. Black hole information paradox. Artist's representation of a black hole Principles in action There are two main principles in play: Quantum determinism means that given a present wave function, its future changes are uniquely determined by the evolution operator.Reversibility refers to the fact that the evolution operator has an inverse, meaning that the past wave functions are similarly unique.
The combination of the two means that information must always be preserved. Starting in the mid-1970s, Stephen Hawking and Jacob Bekenstein put forward theoretical arguments based on general relativity and quantum field theory that appeared to be inconsistent with information conservation. Hawking radiation The Penrose diagram of a black hole which forms, and then completely evaporates away. Hawking remained convinced that the equations of black-hole thermodynamics together with the no-hair theorem led to the conclusion that quantum information may be destroyed.
There are various ideas about how the paradox is solved. See also. Causal dynamical triangulation. Causal dynamical triangulation (abbreviated as CDT) invented by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, and popularized by Fotini Markopoulou and Lee Smolin, is an approach to quantum gravity that like loop quantum gravity is background independent. This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves.
The Loops '05 conference, hosted by many loop quantum gravity theorists, included several presentations which discussed CDT in great depth, and revealed it to be a pivotal insight for theorists. It has sparked considerable interest as it appears to have a good semi-classical description. At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time.
Introduction[edit] Derivation[edit] Advantages and Disadvantages[edit] Related theories[edit] See also[edit] References[edit] Black hole complementarity. Black hole complementarity is a conjectured solution to the black hole information paradox, proposed by Leonard Susskind and Larus Thorlacius,[1] and Gerard 't Hooft.[2] Leonard Susskind[3] proposed a radical resolution to this problem by claiming that the information is both reflected at the event horizon and passes through the event horizon and can't escape, with the catch being no observer can confirm both stories simultaneously.
According to an external observer, the infinite time dilation at the horizon itself makes it appear as if it takes an infinite amount of time to reach the horizon. He also postulated a stretched horizon, which is a membrane hovering about a Planck length outside the event horizon and which is both physical and hot. According to the external observer, infalling information heats up the stretched horizon, which then reradiates it as Hawking radiation, with the entire evolution being unitary.
Global symmetries don't exist in quantum gravity[why?]. Quantum gravity. Quantum gravity (QG) is a field of theoretical physics that seeks to describe the force of gravity according to the principles of quantum mechanics. Although a quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, difficulties arise when one attempts to apply the usual prescriptions of quantum field theory to the force of gravity.[3] From a technical point of view, the problem is that the theory one gets in this way is not renormalizable and therefore cannot be used to make meaningful physical predictions.
As a result, theorists have taken up more radical approaches to the problem of quantum gravity, the most popular approaches being string theory and loop quantum gravity.[4] Strictly speaking, the aim of quantum gravity is only to describe the quantum behavior of the gravitational field and should not be confused with the objective of unifying all fundamental interactions into a single mathematical framework. Induced gravity. Induced gravity (or emergent gravity) is an idea in quantum gravity that space-time background emerges as a mean field approximation of underlying microscopic degrees of freedom, similar to the fluid mechanics approximation of Bose–Einstein condensates. The concept was originally proposed by Andrei Sakharov in 1967. Sakharov observed that many condensed matter systems give rise to emergent phenomena which are identical to general relativity.
For example, crystal defects can look like curvature and torsion in an Einstein–Cartan spacetime. This allows one to create a theory of gravity with torsion from a World Crystal model of spacetime[1] in which the lattice spacing is of the order of a Planck length. Sakharov's idea was to start with an arbitrary background pseudo-Riemannian manifold (in modern treatments, possibly with torsion) and introduce quantum fields (matter) on it but not introduce any gravitational dynamics explicitly. Some argue[who?] See also[edit] References[edit] Inhomogeneous electromagnetic wave equation. In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves, due to nonzero source charges and currents.
The source terms in the wave equations makes the partial differential equations inhomogeneous. Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. The equations follow from Maxwell's equations. Maxwell's equations[edit] For reference, Maxwell's equations are summarized below in SI units and Gaussian units. Where ε0 is the vacuum permittivity and μ0 is the vacuum permeability. SI units[edit] E and B fields[edit] Similarly substituting Gauss's law for magnetism into the curl of Ampère's circuital law (with Maxwell's additional time-dependent term), and using the curl of the curl identity, gives the wave equation for the magnetic field B: A and φ potential fields[edit] with and.
Higgs boson. The Higgs boson is named after Peter Higgs, one of six physicists who, in 1964, proposed the mechanism that suggested the existence of such a particle. Although Higgs's name has come to be associated with this theory, several researchers between about 1960 and 1972 each independently developed different parts of it.
In mainstream media the Higgs boson has often been called the "God particle", from a 1993 book on the topic; the nickname is strongly disliked by many physicists, including Higgs, who regard it as inappropriate sensationalism.[17][18] In 2013 two of the original researchers, Peter Higgs and François Englert, were awarded the Nobel Prize in Physics for their work and prediction[19] (Englert's co-researcher Robert Brout had died in 2011).
A non-technical summary[edit] "Higgs" terminology[edit] Overview[edit] If this field did exist, this would be a monumental discovery for science and human knowledge, and is expected to open doorways to new knowledge in many fields. History[edit] Boson. In quantum mechanics, a boson (/ˈboʊsɒn/,[1] /ˈboʊzɒn/[2]) is a particle that follows Bose–Einstein statistics. Bosons make up one of the two classes of particles, the other being fermions.[3] The name boson was coined by Paul Dirac[4] to commemorate the contribution of the Indian physicist Satyendra Nath Bose[5][6] in developing, with Einstein, Bose–Einstein statistics—which theorizes the characteristics of elementary particles.[7] Examples of bosons include fundamental particles such as photons, gluons, and W and Z bosons (the four force-carrying gauge bosons of the Standard Model), the recently discovered Higgs boson, and the hypothetical graviton of quantum gravity; composite particles (e.g. mesons and stable nuclei of even mass number such as deuterium (with one proton and one neutron, mass number = 2), helium-4, or lead-208[Note 1]); and some quasiparticles (e.g.
Cooper pairs, plasmons, and phonons).[8]:130 Types[edit] Properties[edit] Elementary bosons[edit] Composite bosons[edit] Supersymmetry. Electromagnetic radiation. The electromagnetic waves that compose electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This diagram shows a plane linearly polarized EMR wave propagating from left to right. The electric field is in a vertical plane and the magnetic field in a horizontal plane.
The two types of fields in EMR waves are always in phase with each other with a fixed ratio of electric to magnetic field intensity. Electromagnetic radiation (EM radiation or EMR) is a form of radiant energy, propagating through space via electromagnetic waves and/or particles called photons. In a vacuum, it propagates at a characteristic speed, the speed of light, normally in straight lines. In classical physics, EMR is considered to be produced when charged particles are accelerated by forces acting on them. The photon is the quantum of the electromagnetic interaction, and is the basic "unit" or constituent of all forms of EMR.
Physics[edit] Light. Photon. Wave–particle duality. Electromagnetic spectrum. Spacetime. Complementarity (physics) Quantum nonlocality. Black body. Absolute zero. Thermal radiation.