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Quantum physics
$Delta x , Delta p ge frac{hbar}{2}$
Quantum mechanics
Introduction to... Mathematical formulation of... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Quantum mechanics (QM, or quantum theory) is a physical science dealing with the behaviour of matter and energy on the scale of atoms and subatomic particles / waves. ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ...
Fundamental concepts
Decoherence · Interference Uncertainty · Exclusion Transformation theory Ehrenfest theorem · Measurement Superposition · Entanglement 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. ... For other uses, see Interference (disambiguation). ... 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. ... The term transformation theory refers to a procedure used by P. A. M. Dirac in his early formulation of quantum theory, from around 1927. ... 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. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... 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. ...
Experiments
Double-slit experiment Davisson-Germer experiment Stern–Gerlach experiment Bell's inequality experiment Popper's experiment Schrödinger's cat Double-slit diffraction and interference pattern The double-slit experiment consists of letting light diffract through two slits, which produces fringes or wave-like interference patterns on a screen. ... In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow moving electrons at a crystalline Nickel target. ... In quantum mechanics, the Sternâ€“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. ... SchrÃ¶dingers Cat: When the nucleus (bottom left) decays, the Geiger counter (bottom centre) may sense it and trigger the release of the gas. ...
Equations
Schrödinger equation Pauli equation Klein-Gordon equation Dirac equation For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The Pauli equation is a SchrÃ¶dinger equation which handles spin. ... The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the SchrÃ¶dinger 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-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ...
Advanced theories
Quantum field theory Wightman axioms Quantum electrodynamics Quantum chromodynamics Quantum gravity Feynman diagram Quantum field theory (QFT) is the quantum theory of fields. ... In physics the Wightman axioms are an attempt of mathematically stringent, axiomatic formulation of quantum field theory. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... 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. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ...
Interpretations
Copenhagen · Ensemble Hidden variables · Transactional Many-worlds · Consistent histories Quantum logic Consciousness causes collapse It has been suggested that Quantum mechanics, philosophy and controversy be merged into this article or section. ... The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ... The Ensemble Interpretation, or Statistical Interpretation of Quantum Mechanics, is an interpretation that can be viewed as a minimalist interpretation. ... In physics, a hidden variable theory is urged by a minority of physicists who argue that the statistical nature of quantum mechanics implies that quantum mechanics is incomplete; it is really applicable only to ensembles of particles; new physical phenomena beyond quantum mechanics are needed to explain an individual event. ... 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. ... Consciousness causes collapse is the theory that observation by a conscious observer is responsible for the wavefunction collapse in quantum mechanics. ...
Scientists
Planck · Schrödinger Heisenberg · Bohr · Pauli Dirac · Bohm · Born de Broglie · von Neumann Einstein · Feynman Everett · Penrose · Others â€œPlanckâ€ redirects here. ... SchrÃ¶dinger in 1933, when he was awarded the Nobel Prize in Physics Bust of SchrÃ¶dinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... Werner Karl Heisenberg (December 5, 1901 â€“ 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 â€“ 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 1922. ... This article is about the Austrian-Swiss physicist. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... David Bohm. ... Max Born (December 11, 1882 in Breslau â€“ January 5, 1970 in GÃ¶ttingen) was a mathematician and physicist. ... Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892â€“March 19, 1987), was a French physicist and Nobel Prize laureate. ... For other persons named John Neumann, see John Neumann (disambiguation). ... â€œEinsteinâ€ redirects here. ... This article is about the physicist. ... Hugh Everett III (November 11, 1930 â€“ 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. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Below is a list of famous physicists. ...
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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.^[1] A barrier, in terms of quantum tunnelling, may be a form of energy state analogous to a "hill" or incline in classical mechanics, which would suggest that passage through or over such a barrier would be impossible without sufficient energy. However, because of differences in terms of scale and interaction between quantum and classical mechanics, practical applications of the latter would be inaccurate. For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... It has been suggested that nanopowder be merged into this article or section. ... 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. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ... An energy level is a quantified stable energy, which a physical system can have; the term is most commonly used in reference to the electron configuration of electrons, in atoms or molecules. ...

On the quantum scale, objects exhibit wave-like behavior; in quantum theory, quanta moving against a potential energy "hill" can be described by their wave-function, which represents the probability of finding that particle in a certain location at either side of the "hill". If this function describes the particle as being on the other side of the "hill", then there is the probability that it has moved through, rather than over it, and has thus "tunnelled". In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ... Look up quantum in Wiktionary, the free dictionary. ... A potential well is the region surrounding a local minimum of potential energy. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

1 History
2 Semi-classical calculation
3 In fiction
4 See also
5 References
- 5.1 Notes
- 5.2 Books

History

By 1928, George Gamow had solved the theory of the alpha decay of a nucleus via tunneling. Classically, the particle is confined to the nucleus because of the high energy requirement to escape the very strong potential. Under this system, it takes an enormous amount of energy to pull apart the nucleus. In quantum mechanics, however, there is a probability the particle can tunnel through the potential and escape. Gamow solved a model potential for the nucleus and derived a relationship between the half-life of the particle and the energy of the emission. Year 1928 (MCMXXVIII) was a leap year starting on Sunday (link will display full calendar) of the Gregorian calendar. ... George Gamow (pronounced GAM-off) (March 4, 1904 â€“ August 19, 1968) , born Georgiy Antonovich Gamov (Ð“ÐµÐ¾Ñ€Ð³Ð¸Ð¹ ÐÐ½Ñ‚Ð¾Ð½Ð¾Ð²Ð¸Ñ‡ Ð“Ð°Ð¼Ð¾Ð²) was a Ukrainian born physicist and cosmologist. ... Alpha decay Alpha decay is a type of radioactive decay in which an atom emits an alpha particle (two protons and two neutrons bound together into a particle identical to a helium nucleus) and transforms (or decays) into an atom with a mass number 4 less and atomic number 2... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ... In physics, a potential may refer to the scalar potential or to the vector potential. ...

Alpha decay via tunneling was also solved concurrently by Ronald Gurney and Edward Condon. Shortly thereafter, both groups considered whether particles could also tunnel into the nucleus. Edward Uhler Condon (March 2, 1902 â€“ March 26, 1974) was a distinguished nuclear physicist, a pioneer in quantum mechanics, a participant in the development of radar and nuclear weapons in World War II, research director of Corning Glass, director of the National Bureau of Standards, and president of the American...

After attending a seminar by Gamow, Max Born recognized the generality of quantum-mechanical tunneling. He realized that the tunneling phenomenon was not restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems. Today the theory of tunneling is even applied to the early cosmology of the universe. A seminar is, generally, a form of academic instruction, either at a university or offered by a commercial or professional organization. ... Max Born (December 11, 1882 in Breslau â€“ January 5, 1970 in GÃ¶ttingen) was a mathematician and physicist. ... Nuclear physics is the branch of physics concerned with the nucleus of the atom. ... This article is about the physics subject. ... For other uses, see Universe (disambiguation). ...

Quantum tunneling was later applied to other situations, such as the cold emission of electrons, and perhaps most importantly semiconductor and superconductor physics. Phenomena such as field emission, important to flash memory, are explained by quantum tunneling. Tunneling is a source of major current leakage in Very-large-scale integration (VLSI) electronics, and results in the substantial power drain and heating effects that plague high-speed and mobile technology. Properties The electron (also called negatron, commonly represented as e−) is a subatomic particle. ... A semiconductor is a solid whose electrical conductivity is in between that of a conductor and that of an insulator, and can be controlled over a wide range, either permanently or dynamically. ... Superconductivity is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of electrical resistance and the damping of the interior magnetic field (the Meissner effect. ... Field emission, also known as Fowler-Nordheim tunneling, is a form of quantum tunneling in which electrons pass through a barrier in the presence of a high electric field. ... A USB flash drive. ... It has been suggested that VHSIC be merged into this article or section. ...

Another major application is in electron-tunneling microscopes (see scanning tunneling microscope) which can resolve objects that are too small to see using conventional microscopes. Electron tunneling microscopes overcome the limiting effects of conventional microscopes (optical aberrations, wavelength limitations) by scanning the surface of an object with tunneling electrons. Image of reconstruction on a clean Au(100) surface. ... A microscope (Greek: micron = small and scopos = aim) is an instrument for viewing objects that are too small to be seen by the naked or unaided eye. ... Aberration in optical systems (lenses, prisms, mirrors or series of them intended to produce a sharp image) generally leads to blurring of the image. ... For other uses, see Wavelength (disambiguation). ... For other uses, see Electron (disambiguation). ...

It has been found that quantum tunneling may be the mechanism used by enzymes to speed up reactions in lifeforms to millions of times their normal speed^[2]. Ribbon diagram of the enzyme TIM, surrounded by the space-filling model of the protein. ...

Semi-classical calculation

Let us consider the time-independent Schrödinger equation for one particle, in one dimension, under the influence of a hill potential $V (x)$ . For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... 2-dimensional renderings (ie. ...

$-frac{hbar^2}{2m} frac{d^2}{dx^2} Psi(x) + V(x) Psi(x) = E Psi(x)$

$frac{d^2}{dx^2} Psi(x) = frac{2m}{hbar^2} left( V(x) - E right) Psi(x)$

Now let us recast the wave function $Ψ(x)$ as the exponential of a function.

Ψ(x) = e Φ(x)

$Phi''(x) + Phi'(x)^2 = frac{2m}{hbar^2} left( V(x) - E right)$

Now let us separate $Φ'(x)$ into real and imaginary parts using real valued functions A and B.

Φ'(x) = A (x) + i B (x)

$A'(x) + A(x)^2 - B(x)^2 = frac{2m}{hbar^2} left( V(x) - E right)$ ,

because the pure imaginary part needs to vanish due to the real-valued right-hand side:

$ileft(B'(x) - 2 A(x) B(x)right) = 0$

Next we want to take the semi-classical approximation to solve this. That means we expand each function as a power series in $hbar$ . From the equations we can already see that the power series must start with at least an order of $hbar^{-1}$ to satisfy the real part of the equation. But as we want a good classical limit, we also want to start with as high a power of Planck's constant as possible. In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

$A(x) = frac{1}{hbar} sum_{k=0}^infty hbar^k A_k(x)$

$B(x) = frac{1}{hbar} sum_{k=0}^infty hbar^k B_k(x)$

The constraints on the lowest order terms are as follows.

$A_0(x)^2 - B_0(x)^2 = 2m left( V(x) - E right)$

A 0 (x) B 0 (x) = 0

If the amplitude varies slowly as compared to the phase, we set $A 0 (x) = 0$ and get

$B_0(x) = pm sqrt{ 2m left( E - V(x) right) }$

Which is obviously only valid when you have more energy than potential - classical motion. After the same procedure on the next order of the expansion we get

$Psi(x) approx C frac{ e^{i int dx sqrt{frac{2m}{hbar^2} left( E - V(x) right)} + theta} }{sqrt[4]{frac{2m}{hbar^2} left( E - V(x) right)}}$

On the other hand, if the phase varies slowly as compared to the amplitude, we set $B 0 (x) = 0$ and get

$A_0(x) = pm sqrt{ 2m left( V(x) - E right) }$

Which is obviously only valid when you have more potential than energy - tunnelling motion. Grinding out the next order of the expansion yields

$Psi(x) approx frac{ C_{+} e^{+int dx sqrt{frac{2m}{hbar^2} left( V(x) - E right)}} + C_{-} e^{-int dx sqrt{frac{2m}{hbar^2} left( V(x) - E right)}}}{sqrt[4]{frac{2m}{hbar^2} left( V(x) - E right)}}$

It is apparent from the denominator, that both these approximate solutions are bad near the classical turning point $E = V (x)$ . What we have are the approximate solutions away from the potential hill and beneath the potential hill. Away from the potential hill, the particle acts similarly to a free wave - the phase is oscillating. Beneath the potential hill, the particle undergoes exponential changes in amplitude.

In a specific tunneling problem, we might already suspect that the transition amplitude be proportional to $e^{-int dx sqrt{frac{2m}{hbar^2} left( V(x) - E right)}}$ and thus the tunneling be exponentially dampened by large deviations from classically allowable motion.

But to be complete we must find the approximate solutions everywhere and match coefficients to make a global approximate solution. We have yet to approximate the solution near the classical turning points $E = V (x)$ .

Let us label a classical turning point $x 1$ . Now because we are near $E = V (x 1)$ , we can easily expand $frac{2m}{hbar^2}left(V(x)-Eright)$ in a power series.

$frac{2m}{hbar^2}left(V(x)-Eright) = v_1 (x - x_1) + v_2 (x - x_1)^2 + cdots$

Let us only approximate to linear order $frac{2m}{hbar^2}left(V(x)-Eright) = v_1 (x - x_1)$

$frac{d^2}{dx^2} Psi(x) = v_1 (x - x_1) Psi(x)$

This differential equation looks deceptively simple. Its solutions are Airy functions. In mathematics, the Airy function Ai(x) is a special function, i. ...

$Psi(x) = C_A Aileft( sqrt[3]{v_1} (x - x_1) right) + C_B Bileft( sqrt[3]{v_1} (x - x_1) right)$

Hopefully this solution should connect the far away and beneath solutions. Given the 2 coefficients on one side of the classical turning point, we should be able to determine the 2 coefficients on the other side of the classical turning point by using this local solution to connect them. We should be able to find a relationship between $C,θ$ and $C +, C -$ .

Fortunately the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationship can be found as follows.

$C_{+} = frac{1}{2} C cos{left(theta - frac{pi}{4}right)}$

$C_{-} = - C sin{left(theta - frac{pi}{4}right)}$

Now we can easily construct global solutions and solve tunneling problems.

The transmission coefficient, $left| frac{C_{mbox{outgoing}}}{C_{mbox{incoming}}} right|^2$ , for a particle tunneling through a single potential barrier is found to be The transmission coefficient, , for a particle tunneling through a single barrier potential is found to be: Where are the 2 classical turning points for the potential barrier. ...

$T = frac{e^{-2int_{x_1}^{x_2} dx sqrt{frac{2m}{hbar^2} left( V(x) - E right)}}}{ left( 1 + frac{1}{4} e^{-2int_{x_1}^{x_2} dx sqrt{frac{2m}{hbar^2} left( V(x) - E right)}} right)^2}$

Where $x 1, x 2$ are the 2 classical turning points for the potential barrier. If we take the classical limit of all other physical parameters much larger than Planck's constant, abbreviated as $hbar rightarrow 0$ , we see that the transmission coefficient correctly goes to zero. This classical limit would have failed in the unphysical, but much simpler to solve, situation of a square potential. In Quantum Mechanics, a square potential is a one dimensional problem which can be used to demonstrate the phenomenon of Quantum tunneling. ...

In fiction

Quantum tunnelling is a plot device in the 2007 video game Supreme Commander, where it is used as both a tool by the main characters and the player, and as an aim in the plot of the game. In the fictional world of the 27th century, Quantum Tunnelling is the main method of space travel, and the only thing linking together the various exoplanetary human settlements. At the start of every game, the player's armoured command unit is quantum tunnelled to the battlefield. Using a "Quantum gateway", the player is able to use quantum tunnelling to call in additional support commanders. Computer and video games redirects here. ... Supreme Commander - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... Edward White on a spacewalk during the Gemini 4 mission. ... An extrasolar planet, or exoplanet, is a planet beyond the Solar System. ...

In the single player campaign, shutting down the whole quantum gate network is the main objective of the Cybran faction, a relatively young nation of cyborgs.^[3] The factions of Supreme Commander are comprised of three fictional factions from the game Supreme Commander, a real-time strategy game developed by Gas Powered Games and was first released on February 16, 2007. ... For other uses, see Cyborg (disambiguation). ...

References

Notes

^ Razavy, Mohsen. (2003)., p1
^ http://www.seedmagazine.com/news/2006/04/the_quantum_shortcut.php
^ Cam Shea (2007-01-22). The World of Supreme Commander: A tale of three factions. IGN Australia. Retrieved on 2007-04-26.

For other uses, see IGN (disambiguation). ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 116th day of the year (117th in leap years) in the Gregorian calendar. ...

Books

Razavy, Mohsen (2003). Quantum Theory of Tunneling. World Scientific. ISBN 981-238-019-1.
Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.
Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.

Categories: Physics | Particle physics | Quantum mechanics Richard L. Liboff is a U.S. physicist who has authored five books and nearly 150 other publications in variety of fields, including plasma physics, planetary physics, cosmology, quantum chaos, and quantum billiards. ...

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