Quantum mechanics

Uncertainty principle

Introduction to... Mathematical formulation of... This box:      Werner Heisenberg and Erwin Schrödinger, founders of Quantum Mechanics. ... In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ... This box:      Werner Heisenberg and Erwin Schrödinger, founders of Quantum Mechanics. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... Background Classical mechanics Interference · Bra-ket notation Hamiltonian Fundamental concepts Quantum state · Wave function Superposition · Entanglement Measurement · Uncertainty Exclusion · Duality Decoherence · Ehrenfest theorem · Tunneling Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... For other uses, see Interference (disambiguation). ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... Probability densities for the electron at different quantum numbers (l) In quantum mechanics, the quantum state of a system is a set of numbers that fully describe a quantum system. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... Quantum superposition is the application of the superposition principle to quantum mechanics. ... It has been suggested that Quantum coherence be merged into this article or section. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... In physics and chemistry, wave-particle duality is a conceptualization that all objects in our universe exhibit properties of both waves and of particles. ... In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. ... The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. ... In quantum mechanics, quantum tunnelling is a micro and nanoscopic phenomenon in which a particle violates principles of classical mechanics by penetrating or passing through a potential barrier or impedance higher than the kinetic energy of the particle. ... Experiments Double-slit experiment Davisson–Germer experiment Stern–Gerlach experiment Bell's inequality experiment Popper's experiment Schrödinger's cat Formulations Schrödinger picture Heisenberg picture Interaction picture Matrix mechanics Sum over histories Equations Schrödinger equation Pauli equation Klein-Gordon equation Dirac equation Interpretations Copenhagen · Ensemble Hidden variable theory · Transactional Many-worlds · Consistent histories Quantum logic Advanced topics Quantum field theory Quantum gravity Theory of everything Scientists Planck · Einstein · Schrödinger · Heisenberg· Jordan · Bohr · Pauli · Dirac · Born · de Broglie · von Neumann · Feynman · Bohm · Everett · Hawking · others

This box: view • talk • edit

Fig. 1: The wavefunctions of an electron in a hydrogen atom possessing definite energy (increasing downward: n = 1, 2, 3, ...) and angular momentum (increasing across: s, p, d,...). Brighter areas correspond to higher probability density for a position measurement. Wavefunctions like these are directly comparable to Chladni's figures of acoustic modes of vibration in classical physics and are indeed modes of oscillation as well: they possess a sharp energy and thus a keen frequency. The angular momentum and energy are quantized, and only take on discrete values like those shown (as is the case for resonant frequencies in acoustics).

In physics, quantum mechanics is the study of the relationship between quanta and elementary particles. Among other relationships the valence shell electrons and photons are quantized. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics. Quantum theory generalizes all classical theories, including mechanics, electromagnetism (except general relativity), and provides accurate descriptions for many previously unexplained phenomena such as black body radiation and stable electron orbits.The effects of quantum mechanics are typically not observable on macroscopic scales, but become evident at the atomic and subatomic level. Slit experiment redirects here. ... In quantum mechanics, the Stern–Gerlach experiment, named after Otto Stern and Walther Gerlach, is a celebrated experiment in 1920 on deflection of particles, often used to illustrate basic principles of quantum mechanics. ... In quantum mechanics, Bells Theorem states that a Bell inequality must be obeyed under any local hidden variable theory but can in certain circumstances be violated under quantum mechanics (QM). ... Poppers experiment is an experiment proposed by the 20th century philosopher of science Karl Popper, to test the standard interpretation (the Copenhagen interpretation) of Quantum mechanics. ... Schrödingers Cat: When the nucleus (bottom left) decays, the Geiger counter (bottom centre) may sense it and trigger the release of the gas. ... Heisenbergs form for the equations of motion We have seen that in Schrödingers scheme the dynamical variables of the system remain fixed during a period of undisturbed motion. ... The Heisenberg Picture of quantum mechanics is also known as Matrix mechanics. ... In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the Schrödinger picture and the Heisenberg picture. ... Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ... This article or section is in need of attention from an expert on the subject. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The Pauli equation is a Schrödinger equation which handles spin. ... The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the Schrödinger equation. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ... Early twentieth century studies of the physics of very small-scale phenomena led to the Copenhagen interpretation. ... The Ensemble Interpretation, or Statistical Interpretation of Quantum Mechanics, is an interpretation that can be viewed as a minimalist interpretation. ... In physics, the hidden variable theory is espoused by a minority of physicists who argue that the statistical nature of quantum mechanics indicates that QM is incomplete. ... The transactional interpretation of quantum mechanics (TIQM) by Professor John Cramer is an unusual interpretation of quantum mechanics that describes quantum interactions in terms of a standing wave formed by retarded (forward in time) and advanced (backward in time) waves. ... The many-worlds interpretation or MWI (also known as relative state formulation, theory of the universal wavefunction, many-universes interpretation, Oxford interpretation or many worlds), is an interpretation of quantum mechanics that claims to resolve all the paradoxes of quantum theory by allowing every possible outcome to every event to... In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology. ... In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. ... Quantum field theory (QFT) is the quantum theory of fields. ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... This page discusses Theories of Everything in physics. ... “Planck” redirects here. ... “Einstein” redirects here. ... Schrödinger in 1933, when he was awarded the Nobel Prize in Physics Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Niels Henrik David Bohr (October 7, 1885 – November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ... This article is about the Austrian-Swiss physicist. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Max Born (December 11, 1882 – January 5, 1970) was a German physicist and mathematician. ... Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892–March 19, 1987), was a French physicist and Nobel Prize laureate. ... For other persons named John Neumann, see John Neumann (disambiguation). ... This article is about the physicist. ... David Bohm. ... Hugh Everett III (November 11, 1930 – July 19, 1982) was an American physicist who first proposed the many-worlds interpretation(MWI) of quantum physics, which he called his relative state formulation. ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ... Image File history File links HAtomOrbitals. ... Image File history File links HAtomOrbitals. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ... For other uses, see Electron (disambiguation). ... Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. ... This gyroscope remains upright while spinning due to its angular momentum. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... Ernst Florenz Friedrich Chladni (November 30, 1756 - April 3, 1827) was a German physicist. ... Acoustics is the branch of physics concerned with the study of sound (mechanical waves in gases, liquids, and solids). ... Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... For other uses, see Frequency (disambiguation). ... This gyroscope remains upright while spinning due to its angular momentum. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... This article is about resonance in physics. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In physics, a quantum (plural: quanta) is an indivisible entity of energy. ... In particle physics, an elementary particle is a particle of which other, larger particles are composed. ... The valence shell is the outermost shell of an atom, which contains the electrons most likely to account for the nature of any reactions involving the atom and of the bonding interactions it has with other atoms. ... For other uses, see Electron (disambiguation). ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Experimental physics is the part of physics that deals with experiments and observations pertaining to natural/physical phenomena, as opposed to theoretical physics. ... Theoretical physics employs mathematical models and abstractions of physics in an attempt to explain experimental data taken of the natural world. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Look up quantum in Wiktionary, the free dictionary. ... For other uses, see Mechanic (disambiguation). ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... “Accuracy” redirects here. ... Physical Phenomena are observable events which are explained by physics or raise some question about matter, light, or spacetime. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ... An electron orbital may refer to: An atomic orbital A molecular orbital Electron configuration Category: ... Macroscopic is commonly used to describe physical objects that are measurable and observable by the naked eye. ... For other uses, see Atom (disambiguation). ... Helium atom (schematic) Showing two protons (red), two neutrons (green) and two electrons (yellow). ...

Overview

The word “quantum” came from the Latin word which means "what quantity". In quantum mechanics, it refers to a discrete(separate) unit that quantum theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1, at right). The discovery that waves have discrete energy packets (called quanta) that behave in a manner similar to particles led to the branch of physics that deals with atomic and subatomic systems which we today call quantum mechanics. It is the underlying mathematical framework of many fields of physics and chemistry, including condensed matter physics, solid-state physics, atomic physics, molecular physics, computational chemistry, quantum chemistry, particle physics, and nuclear physics. The foundations of quantum mechanics were established during the first half of the twentieth century by Werner Heisenberg, Max Planck, Louis de Broglie, Albert Einstein, Niels Bohr, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli and others. Some fundamental aspects of the theory are still actively studied. For other uses, see Atom (disambiguation). ... Surface waves in water This article is about waves in the most general scientific sense. ... In physics quanta is the plural of quantum. ... Helium atom (schematic) Showing two protons (red), two neutrons (green) and two electrons (yellow). ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses, see Chemistry (disambiguation). ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... Atomic physics (or atom physics) is the field of physics that studies atoms as isolated systems comprised of electrons and an atomic nucleus. ... Molecular physics is the study of the physical properties of molecules and of the chemical bonds between atoms that bind them into molecules. ... Computational chemistry is a branch of chemistry that uses the results of theoretical chemistry incorporated into efficient computer programs to calculate the structures and properties of molecules and solids, applying these programs to complement the information obtained by actual chemical experiments, predict hitherto unobserved chemical phenomena, and solve related problems. ... Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics and quantum field theory to address issues and problems in chemistry. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Nuclear physics is the branch of physics concerned with the nucleus of the atom. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... “Planck” redirects here. ... Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892–March 19, 1987), was a French physicist and Nobel Prize laureate. ... “Einstein” redirects here. ... Niels Henrik David Bohr (October 7, 1885 – November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. ... Schrödinger in 1933, when he was awarded the Nobel Prize in Physics Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... Max Born (December 11, 1882 – January 5, 1970) was a German physicist and mathematician. ... A separate article covers Saint John Neumann, the American priest. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... This article is about the Austrian-Swiss physicist. ... Below is a list of famous physicists. ...

It is currently necessary to use quantum mechanics to understand the behavior of systems at atomic length scales and smaller. For example, if Newtonian mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus. However, in the natural world the electrons normally remain in an unknown orbital path around the nucleus, defying classical electromagnetism. For other uses, see Atom (disambiguation). ... It has been suggested that this article or section be merged with Classical mechanics. ... For other uses, see Electron (disambiguation). ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ...

Quantum mechanics was initially developed to provide a better explanation of the atom, especially the spectra of light emitted by different atomic species. The quantum theory of the atom developed as an explanation for the electron's staying in its orbital, which could not be explained by Newton's laws of motion and by Maxwell's laws of classical electromagnetism. In most modern usages of the word spectrum, there is a unifying theme of between extremes at either end. ... For other uses, see Light (disambiguation). ... For other uses, see Isotope (disambiguation). ... In chemistry, an atomic orbital is the region in which an electron may be found around a single atom. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell (written by Oliver Heaviside), that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...

In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function (sometimes referred to as orbitals in the case of atomic electrons), and more generally, elements of a complex vector space. This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, with arbitrary accuracy. For instance, electrons may be considered to be located somewhere within a region of space, but with their exact positions being unknown. Contours of constant probability, often referred to as “clouds” may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. It should be stressed that the electron itself is not spread out over such cloud regions. It is either in a particular region of space, or it is not [citation needed]. Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle. In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Probability is the likelihood that something is the case or will happen. ... In physics, especially in quantum mechanics, conjugate variables are pairs of variables that share an uncertainty relation. ... In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ...

The other exemplar that led to quantum mechanics was the study of electromagnetic waves such as light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta, Albert Einstein exploited this idea to show that an electromagnetic wave such as light could be described by a particle called the photon with a discrete energy dependent on its frequency. This led to a theory of unity between subatomic particles and electromagnetic waves called wave–particle duality in which particles and waves were neither one nor the other, but had certain properties of both. While quantum mechanics describes the world of the very small, it also is needed to explain certain “macroscopic quantum systems” such as superconductors and superfluids. Exemplar, in the sense developed by philosopher of science Thomas Kuhn, is a well known usage of a scientific theory. ... Electromagnetic radiation is a propagating wave in space with electric and magnetic components. ... “Einstein” redirects here. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. ... In physics and chemistry, wave-particle duality is a conceptualization that all objects in our universe exhibit properties of both waves and of particles. ... Macroscopic is commonly used to describe physical objects that are measurable and observable by the naked eye. ... A magnet levitating above a high-temperature superconductor, cooled with liquid nitrogen. ... Helium II will creep along surfaces in order to find its own level - after a short while, the levels in the two containers will equalize. ...

Broadly speaking, quantum mechanics incorporates four classes of phenomena that classical physics cannot account for: (i) the quantization (discretization) of certain physical quantities, (ii) wave-particle duality, (iii) the uncertainty principle, and (iv) quantum entanglement. Each of these phenomena is described in detail in subsequent sections. In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... Canonical conjugate variables in physics are pairs of variables that share an uncertainty relation. ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ... In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ... It has been suggested that Quantum coherence be merged into this article or section. ...

History

Main article: History of quantum mechanics

The history of quantum mechanics began essentially with the 1838 discovery of cathode rays by Michael Faraday, the 1859 statement of the black body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete, and the 1900 quantum hypothesis by Max Planck that any energy is radiated and absorbed in quantities divisible by discrete ‘energy elements’ ε such that each of these energy elements is proportional to the frequency ν with which they each individually radiate energy, as defined by the following formula: Niels Bohr’s 1913 quantum model of the atom, which incorporated an explanation of Johannes Rydbergs 1888 formula, Max Planck’s 1900 quantum hypothesis, i. ... Alternative meanings: There is also an Electric-type Pok mon named Electrode. ... Michael Faraday, FRS (September 22, 1791 – August 25, 1867) was an English chemist and physicist (or natural philosopher, in the terminology of that time) who contributed to the fields of electromagnetism and electrochemistry. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ... Gustav Robert Kirchhof (March 12, 1824 – October 17, 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. ... Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 – Duino near Trieste, September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. ... “Planck” redirects here. ... For other uses, see Frequency (disambiguation). ...

where h is Planck's Action Constant. Although Planck insisted that this was simply an aspect of the absorption and radiation of energy and had nothing to do with the physical reality of the energy itself, in 1905, to explain the photoelectric effect (1839), i.e. that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, as based on Planck’s quantum hypothesis, that light itself consists of individual quanta, which later came to be called photons (1926). From Einstein's simple postulation was borne a flurry of debating, theorizing and testing, and thus, the entire field of quantum physics. A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... A diagram illustrating the emission of electrons from a metal plate, requiring energy gained from an incoming photon to be more than the work function of the material. ... “Einstein” redirects here. ... For other uses, see Light (disambiguation). ... In physics, the photon (from Greek φως, phōs, meaning light) is the quantum of the electromagnetic field; for instance, light. ... Fig. ...

Relativity and quantum mechanics

The modern world of physics is notably founded on two tested and demonstrably sound theories of general relativity and quantum mechanics —theories which appear to contradict one another. The defining postulates of both Einstein's theory of relativity and quantum theory are indisputably supported by rigorous and repeated empirical evidence. However, while they do not directly contradict each other theoretically (at least with regard to primary claims), they are resistant to being incorporated within one cohesive model.

Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly inventive in this field, he did not accept the more philosophic consequences and interpretations of quantum mechanics, such as the lack of deterministic causality and the assertion that a single subatomic particle can occupy numerous areas of space at one time. He also was the first to notice some of the apparently exotic consequences of entanglement and used them to formulate the Einstein-Podolsky-Rosen paradox, in the hope of showing that quantum mechanics has unacceptable implications. This was 1935, but in 1964 it was shown by John Bell (see Bell inequality) that Einstein's assumption that quantum mechanics is correct, but has to be completed by hidden variables, was based on wrong philosophical assumptions: according to the paper of J. Bell and the Copenhagen interpretation (the common interpretation of quantum mechanics by physicists for decades), and contrary to Einstein's ideas, quantum mechanics is It has been suggested that Quantum coherence be merged into this article or section. ... In quantum mechanics, the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. ... In quantum mechanics, the EPR paradox is a thought experiment which demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. ... Early twentieth century studies of the physics of very small-scale phenomena led to the Copenhagen interpretation. ...

• neither a "realistic" theory (since quantum measurements do not state pre-existing properties, but rather they prepare properties)

• nor a local theory (essentially not, because the state vector determines simultaneously the probability amplitudes at all sites, ).

The Einstein-Podolsky-Rosen paradox shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner, although the two particles can be an arbitrary distance apart; however, this effect does not violate causality, since no transfer of information happens. These experiments are the basis of some of the most topical applications of the theory, quantum cryptography, which works well, although at small distances of typically 100 km, being on the market since 2004. This article is about the principle of locality in physics. ... Causality or causation denotes the relationship between one event (called cause) and another event (called effect) which is the consequence (result) of the first. ... Quantum cryptography, or quantum key distribution (QKD), uses quantum mechanics to guarantee secure communication. ...

There do exist quantum theories which incorporate special relativity—for example, quantum electrodynamics (QED), which is currently the most accurately tested physical theory ^[1]—and these lie at the very heart of modern particle physics. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those applications. However, the lack of a correct theory of quantum gravity is an important issue in cosmology. Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... QED can mean several different things: Q.E.D. Latin Quod erat demonstrandum, used at the end of mathematical proofs The QED project intended to construct a formalized database of all mathematical knowledge The QED text editor program Quantum electrodynamics, a field of physics Quantum Effect Devices, a maker of... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... Cosmology, from the Greek: κοσμολογία (cosmologia, κόσμος (cosmos) order + λογια (logia) discourse) is the study of the Universe in its totality, and by extension, humanitys place in it. ...

Attempts at a unified theory

Main article: Quantum gravity

Inconsistencies arise when one tries to join the quantum laws with general relativity, a more elaborate description of spacetime which incorporates gravitation. Resolving these inconsistencies has been a major goal of twentieth- and twenty-first-century physics. Many prominent physicists, including Stephen Hawking, have labored in the attempt to discover a "Grand Unification Theory" that combines not only different models of subatomic physics, but also derives the universe's four forces—the strong force, electromagnetism, weak force, and gravity— from a single force or phenomenon. Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... For other uses of this term, see Spacetime (disambiguation). ... Gravity redirects here. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999 in the... 20XX redirects here. ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ... For the album, see Grand Unification (album). ... The strong interaction or strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four fundamental interactions of nature. ... Gravity is a force of attraction that acts between bodies that have mass. ...

Quantum mechanics and classical physics

Predictions of quantum mechanics have been verified experimentally to a very high degree of accuracy. Thus, the current logic of correspondence principle between classical and quantum mechanics is that all objects obey laws of quantum mechanics, and classical mechanics is just a quantum mechanics of large systems (or a statistical quantum mechanics of a large collection of particles). Laws of classical mechanics thus follow from laws of quantum mechanics at the limit of large systems or large quantum numbers. In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. ...

Main differences between classical and quantum theories have already been mentioned above in the remarks on the Einstein-Podolsky-Rosen paradox. Essentially the difference boils down to the statement that quantum mechanics is coherent (addition of amplitudes), whereas classical theories are incoherent (addition of intensities). Thus, such quantities as coherence lengths and coherence times come into play. For microscopic bodies the extension of the system is certainly much smaller than the coherence length; for macroscopic bodies one expects that it should be the other way round. In quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory. ... Quantum coherence refers to the condition of a quantum system whose constituents are in-phase. ... Coherence is from Latin cohaerere = stick together, to be connected with, logically consistent. ...

This is in accordance with the following observations:

Many “macroscopic” properties of “classic” systems are direct consequences of quantum behavior of its parts. For example, stability of bulk matter (which consists of atoms and molecules which would quickly collapse under electric forces alone), rigidity of this matter, mechanical, thermal, chemical, optical and magnetic properties of this matter—they are all results of interaction of electric charges under the rules of quantum mechanics. 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ...

Because seemingly exotic behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with extremely fast-moving or extremely tiny particles, the laws of classical “Newtonian” physics still remain accurate in predicting the behavior of surrounding (“large”) objects—of the order of the size of large molecules and bigger.

Despite the proposal of many novel ideas, the unification of quantum mechanics—which reigns in the domain of the very small—and general relativity—a superb description of the very large—remains, tantalizingly, a future possibility. (See quantum gravity, string theory.) Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory This box:      String theory is a model of fundamental physics, whose building blocks are one-dimensional extended objects called strings, rather than the zero...

Theory

There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the transformation theory proposed by Cambridge theoretical physicist Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg)^[2] and wave mechanics (invented by Erwin Schrödinger). The term transformation theory refers to a procedure used by P. A. M. Dirac in his early formulation of quantum theory, from around 1927. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves. ... Schrödinger in 1933, when he was awarded the Nobel Prize in Physics Bust of Schrödinger, in the courtyard arcade of the main building, University of Vienna, Austria. ...

In this formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom). Probability densities for the electron at different quantum numbers (l) In quantum mechanics, the quantum state of a system is a set of numbers that fully describe a quantum system. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... Look up position in Wiktionary, the free dictionary. ... This article is about momentum in physics. ... This gyroscope remains upright while spinning due to its angular momentum. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ...

Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is, the probability of obtaining each of the possible outcomes from measuring an observable. Naturally, these probabilities will depend on the quantum state at the instant of the measurement. There are, however, certain states that are associated with a definite value of a particular observable. These are known as "eigenstates" of the observable ("eigen" can be roughly translated from German as inherent or as a characteristic). In the everyday world, it is natural and intuitive to think of everything being in an eigenstate of every observable. Everything appears to have a definite position, a definite momentum, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values for the position or momentum of a certain particle in a given space in a finite time; rather, it only provides a range of probabilities of where that particle might be. Therefore, it became necessary to use different words for (a) the state of something having an uncertainty relation and (b) a state that has a definite value. The latter is called the "eigenstate" of the property being measured. A probability distribution describes the values and probabilities that a random event can take place. ...

For example, consider a free particle. In quantum mechanics, there is wave-particle duality so the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave, of arbitrary shape and extending over all of space, called a wave function. The position and momentum of the particle are observables. The Uncertainty Principle of quantum mechanics states that both the position and the momentum cannot simultaneously be known with infinite precision at the same time. However, one can measure just the position alone of a moving free particle creating an eigenstate of position with a wavefunction that is very large at a particular position x, and almost zero everywhere else. If one performs a position measurement on such a wavefunction, the result x will be obtained with almost 100% probability. In other words, the position of the free particle will almost be known. This is called an eigenstate of position (mathematically more precise: a generalized eigenstate (eigendistribution) ). If the particle is in an eigenstate of position then its momentum is completely unknown. An eigenstate of momentum, on the other hand, has the form of a plane wave. It can be shown that the wavelength is equal to h/p, where h is Planck's constant and p is the momentum of the eigenstate. If the particle is in an eigenstate of momentum then its position is completely blurred out. In physics a free particle is a particle that is never under the influence of an external force Classical Free Particle The classical free particle is characterized simply by a fixed velocity. ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ... Probability densities for the electron at different quantum numbers (l) In quantum mechanics, the quantum state of a system is a set of numbers that fully describe a quantum system. ... Surface waves in water This article is about waves in the most general scientific sense. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ... The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction. ... For other uses, see Wavelength (disambiguation). ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

Usually, a system will not be in an eigenstate of whatever observable we are interested in. However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or generalized eigenstate) of that observable. This process is known as wavefunction collapse. It involves expanding the system under study to include the measurement device, so that a detailed quantum calculation would no longer be feasible and a classical description must be used. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x₀, neither an eigenstate of position nor of momentum. When one measures the position of the particle, it is impossible to predict with certainty the result that we will obtain. It is probable, but not certain, that it will be near x₀, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x. In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ... The wave packet is one of the most widely misunderstood and misused concepts in physics. ...

Wave functions can change as time progresses. An equation known as the Schrödinger equation describes how wave functions change in time, a role similar to Newton's second law in classical mechanics. The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. This also has the effect of turning position eigenstates (which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position eigenstates. For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ...

Some wave functions produce probability distributions that are constant in time. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus ( Fig. 1). (Note that only the lowest angular momentum states, labeled s, are spherically symmetric). For other uses, see Electron (disambiguation). ... For other uses, see Atom (disambiguation). ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ... A point plotted using the spherical coordinate system In mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis, and the azimuth angle... Image File history File links HAtomOrbitals. ...

The time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time. During a measurement, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i.e., random. This article is about the general notion of determinism in philosophy. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... Random redirects here. ...

The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr-Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Interpretations of quantum mechanics have been formulated to do away with the concept of "wavefunction collapse"; see, for example, the relative state interpretation. The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics. Probability is the likelihood that something is the case or will happen. ... Niels Bohr with Albert Einstein at Paul Ehrenfests home in Leiden (December 1925) The Bohr-Einstein debates is a popular name given to what was actually a series of epistemological challenges presented by Albert Einstein against what has come to be called the standard or Copenhagen interpretation of quantum... It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ... The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics, based on Hugh Everetts relative-state formulation. ... Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ...

Mathematical formulation

Main article: Mathematical formulation of quantum mechanics

See also: Quantum logic

In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally-Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues. The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... In mathematical physics and quantum mechanics, quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... For other persons named John Neumann, see John Neumann (disambiguation). ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... This article does not cite its references or sources. ... In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution. For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ...

The inner product between two state vectors is a complex number known as a probability amplitude. During a measurement, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator - which explains the choice of Hermitian operators, for which all the eigenvalues are real. We can find the probability distribution of an observable in a given state by computing the spectral decomposition of the corresponding operator. Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute. In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more uncertain we might say that that characteristic is for the system. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ...

The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the wave-like behavior of quantum states. This article is about a portion of a periodic process. ... For other uses, see Interference (disambiguation). ...

It turns out that analytic solutions of Schrödinger's equation are only available for a small number of model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen-molecular ion and the hydrogen atom are the most important representatives. Even the helium atom, which contains just one more electron than hydrogen, defies all attempts at a fully analytic treatment. There exist several techniques for generating approximate solutions. For instance, in the method known as perturbation theory one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak potential energy. Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior. The deviations can be calculated based on the classical motion. This approach is important for the field of quantum chaos. The field of quantum mechanics is based on solutions to the Schrödinger equation, which is often represented (in its non-relativistic form) as Where is the wavefunction of the system, H is the Hamiltonian operator, and T and V are the operators for the kinetic energy and potential energy... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with... Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. ... General Name, symbol, number