Quantum mechanics
Quantum mechanics 

\Delta x\cdot\Delta p_x \geqslant \frac{\hbar}{2} 
Equations

Scientists

Quantum mechanics (QM; also known as quantum physics, or quantum theory) is a fundamental branch of physics which deals with physical phenomena at nanoscopic scales where the action is on the order of the Planck constant. It departs from classical mechanics primarily at the quantum realm of atomic and subatomic length scales. Quantum mechanics provides a mathematical description of much of the dual particlelike and wavelike behavior and interactions of energy and matter. Quantum mechanics provides a substantially useful framework for many features of the modern periodic table of elements including the behavior of atoms during chemical bonding and has played a significant role in the development of many modern technologies.
In advanced topics of quantum mechanics, some of these behaviors are macroscopic (see macroscopic quantum phenomena) and emerge at only extreme (i.e., very low or very high) energies or temperatures (such as in the use of superconducting magnets). In the context of quantum mechanics, the wave–particle duality of energy and matter and the uncertainty principle provide a unified view of the behavior of photons, electrons, and other atomicscale objects.
The mathematical formulations of quantum mechanics are abstract. A mathematical function, the wavefunction, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. Mathematical manipulations of the wavefunction usually involve bra–ket notation which requires an understanding of complex numbers and linear functionals. The wavefunction formulation treats the particle as a quantum harmonic oscillator, and the mathematics is akin to that describing acoustic resonance. Many of the results of quantum mechanics are not easily visualized in terms of classical mechanics. For instance, in a quantum mechanical model the lowest energy state of a system, the ground state, is nonzero as opposed to a more "traditional" ground state with zero kinetic energy (all particles at rest). Instead of a traditional static, unchanging zero energy state, quantum mechanics allows for far more dynamic, chaotic possibilities, according to John Wheeler.
The earliest versions of quantum mechanics were formulated in the first decade of the 20th century. About this time, the atomic theory and the corpuscular theory of light (as updated by Einstein)^{[1]} first came to be widely accepted as scientific fact; these latter theories can be viewed as quantum theories of matter and electromagnetic radiation, respectively. Early quantum theory was significantly reformulated in the mid1920s by Werner Heisenberg, Max Born and Pascual Jordan, (matrix mechanics); Louis de Broglie and Erwin Schrödinger (wave mechanics); and Wolfgang Pauli and Satyendra Nath Bose (statistics of subatomic particles). Moreover, the Copenhagen interpretation of Niels Bohr became widely accepted. By 1930, quantum mechanics had been further unified and formalized by the work of David Hilbert, Paul Dirac and John von Neumann^{[2]} with a greater emphasis placed on measurement in quantum mechanics, the statistical nature of our knowledge of reality, and philosophical speculation about the role of the observer. Quantum mechanics has since permeated throughout many aspects of 20thcentury physics and other disciplines including quantum chemistry, quantum electronics, quantum optics, and quantum information science. Much 19thcentury physics has been reevaluated as the "classical limit" of quantum mechanics and its more advanced developments in terms of quantum field theory, string theory, and speculative quantum gravity theories.
The name quantum mechanics derives from the observation that some physical quantities can change only in discrete amounts (Latin quanta), and not in a continuous (cf. analog) way.
Contents
 History 1
 Mathematical formulations 2
 Mathematically equivalent formulations of quantum mechanics 3

Interactions with other scientific theories 4
 Quantum mechanics and classical physics 4.1
 Relativity and quantum mechanics 4.2
 Attempts at a unified field theory 4.3
 Philosophical implications 5
 Applications 6

Examples 7
 7.1 Free particle
 Step potential 7.2
 Rectangular potential barrier 7.3
 7.4 Particle in a box
 Finite potential well 7.5
 Harmonic oscillator 7.6
 See also 8
 Notes 9
 References 10
 Further reading 11
 External links 12
History
Modern physics 

{ i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) = \hat H \Psi(\mathbf{r},\,t)} 
History of modern physics 
Founders

Concepts

Branches
Philosophy of Science · Philosophy of physics
Mathematical logic · Mathematical physics Grand Unified Theory · Standard model Quantum mechanics · Quantum field theory Electroweak interaction · Antimatter Weak interaction · Quantum electrodynamics Strong interaction · Quantum chromodynamics Particle physics · Nuclear physics Atomic, molecular, and optical physics Condensed matter physics Quantum statistical mechanics Quantum information · Quantum Computation Nonlinear dynamics · Photonics · Biophysics Neurophysics · Quantum mind Plasma physics · Neutrino astronomy Special relativity · General relativity Dark matter · Dark energy Astrophysics · Cosmology Supersymmetry · String theory Quantum gravity · Multiverse Quantum chaos · Complex systems 
Scientific inquiry into the wave nature of light began in the 17th and 18th centuries when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations.^{[3]} In 1803, Thomas Young, an English polymath, performed the famous doubleslit experiment that he later described in a paper entitled "On the nature of light and colours". This experiment played a major role in the general acceptance of the wave theory of light.
In 1838, with the discovery of cathode rays by Michael Faraday, these studies were followed by the 1859 statement of the blackbody radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, and the 1900 quantum hypothesis of Max Planck.^{[4]} Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" (or "energy elements") precisely matched the observed patterns of blackbody radiation.
In 1896, Wilhelm Wien empirically determined a distribution law of blackbody radiation,^{[5]} known as Wien's law in his honor. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it was valid only at high frequencies, and underestimated the radiance at low frequencies. Later, Max Planck corrected this model using Boltzmann statistical interpretation of thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics.
Among the first to study quantum phenomena in nature were Arthur Compton, C.V. Raman, and Pieter Zeeman, each of whom has a quantum effect named after him. Robert A. Millikan studied the Photoelectric effect experimentally and Albert Einstein developed a theory for it. At the same time Niels Bohr developed his theory of the atomic structure which was later confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept also introduced by Arnold Sommerfeld.^{[6]} This phase is known as Old quantum theory.
According to Planck, each energy element, E, is proportional to its frequency, ν:
 E = h \nu\
where h is Planck's constant. Planck (cautiously) insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.^{[7]} In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.^{[8]} However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect in which shining light on certain materials can eject electrons from the material.
The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wilhelm Wien, Satyendra Nath Bose, Arnold Sommerfeld and others. In the mid1920s, developments in quantum mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the "Old Quantum Theory". Out of deference to their particlelike behavior in certain processes and measurements, light quanta came to be called photons (1926). From Einstein's simple postulation was born a flurry of debating, theorizing, and testing. Thus the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927.
The other exemplar that led to quantum mechanics was the study of electromagnetic waves, such as visible and ultraviolet 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 further developed this idea to show that an electromagnetic wave such as light could also be described as a particle (later called the photon) with a discrete quantum of energy that was dependent on its frequency.^{[9]} Einstein was able to use the photon theory of light to explain the photoelectric effect for which he won the 1921 Nobel Prize in Physics. This led to a theory of unity between subatomic particles and electromagnetic waves in which particles and waves are neither simply particle nor wave but have certain properties of each. This originated the concept of wave–particle duality.
While quantum mechanics traditionally described the world of the very small, it is also needed to explain certain recently investigated
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 Quantum mechanics


 "Quantum Mechanics" entry by Jenann Ismael in the Stanford Encyclopedia of Philosophy
 "Measurement in Quantum Theory" entry by Henry Krips in the Stanford Encyclopedia of Philosophy
 Philosophy
 PHYS 201: Fundamentals of Physics II by Ramamurti Shankar, Open Yale Course
 Lectures on Quantum Mechanics by Leonard Susskind
 Everything you wanted to know about the quantum world — archive of articles from New Scientist.
 Quantum Physics Research from Science Daily
 Overbye, Dennis (December 27, 2005). "Quantum Trickery: Testing Einstein's Strangest Theory". The New York Times. Retrieved April 12, 2010.
 Audio: Astronomy Cast Quantum Mechanics — June 2009. Fraser Cain interviews Pamela L. Gay.
 Media
 Manyworlds or relativestate interpretation.
 Measurement in Quantum mechanics.
 FAQs
 Quantum Physics Database  Fundamentals and Historical Background of Quantum Theory.
 Doron Cohen: Lecture notes in Quantum Mechanics (comprehensive, with advanced topics).
 MIT OpenCourseWare: Chemistry.
 MIT OpenCourseWare: Physics. See 8.04
 Stanford Continuing Education PHY 25: Quantum Mechanics by Leonard Susskind, see course description Fall 2007
 5½ Examples in Quantum Mechanics
 Imperial College Quantum Mechanics Course.
 Spark Notes  Quantum Physics.
 Quantum Physics Online : interactive introduction to quantum mechanics (RS applets).
 Experiments to the foundations of quantum physics with single photons.
 AQME : Advancing Quantum Mechanics for Engineers — by T.Barzso, D.Vasileska and G.Klimeck online learning resource with simulation tools on nanohub
 Quantum Mechanics by Martin Plenio
 Quantum Mechanics by Richard Fitzpatrick
 Quantum TransportOnline course on
 Course material
 3D animations, applications and research for basic quantum effects (animations also available in (Université paris Sud))
 Quantum Cook Book by R. Shankar, Open Yale PHYS 201 material (4pp)
 The Modern Revolution in Physics  an online textbook.
 J. O'Connor and E. F. Robertson: A history of quantum mechanics.
 Introduction to Quantum Theory at Quantiki.
 Quantum Physics Made Relatively Simple: three video lectures by Hans Bethe
 H is for hbar.
 Quantum Mechanics Books Collection: Collection of free books
External links
 Bernstein, Jeremy (2009). Quantum Leaps. Cambridge, Massachusetts: Belknap Press of Harvard University Press.
 Eisberg, Robert;
 Merzbacher, Eugen (1998). Quantum Mechanics. Wiley, John & Sons, Inc.
 Shankar, R. (1994). Principles of Quantum Mechanics. Springer.
 Stone, A. Douglas (2013). Einstein and the Quantum. Princeton University Press.
Further reading
The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus.
 Chester, Marvin (1987) Primer of Quantum Mechanics. John Wiley. ISBN 0486428788
 Cox, Brian; Forshaw, Jeff (2011).
 Richard Feynman, 1985. QED: The Strange Theory of Light and Matter, Princeton University Press. ISBN 0691083886. Four elementary lectures on quantum electrodynamics and quantum field theory, yet containing many insights for the expert.
 Ghirardi, GianCarlo, 2004. Sneaking a Look at God's Cards, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using algebra, trigonometry, and bra–ket notation can be passed over on a first reading.
 N. David Mermin, 1990, "Spooky actions at a distance: mysteries of the QT" in his Boojums all the way through. Cambridge University Press: 11076.
 Victor Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Chpts. 58. Includes cosmological and philosophical considerations.
More technical:
 Bryce DeWitt, R. Neill Graham, eds., 1973. The ManyWorlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press. ISBN 069108131X
 The beginning chapters make up a very clear and comprehensible introduction.
 Hugh Everett, 1957, "Relative State Formulation of Quantum Mechanics", Reviews of Modern Physics 29: 45462.
 Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. A standard undergraduate text.
 Max Jammer, 1966. The Conceptual Development of Quantum Mechanics. McGraw Hill.
 Hagen Kleinert, 2004. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed. Singapore: World Scientific. Draft of 4th edition.
 Gunther Ludwig, 1968. Wave Mechanics. London: Pergamon Press. ISBN 0082032041
 George Mackey (2004). The mathematical foundations of quantum mechanics. Dover Publications. ISBN 0486435172.
 Albert Messiah, 1966. Quantum Mechanics (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III.
 Scerri, Eric R., 2006. The Periodic Table: Its Story and Its Significance. Oxford University Press. Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics. ISBN 0195305736
 Hermann Weyl, 1950. The Theory of Groups and Quantum Mechanics, Dover Publications.
 D. Greenberger, K. Hentschel, F. Weinert, eds., 2009. Compendium of quantum physics, Concepts, experiments, history and philosophy, SpringerVerlag, Berlin, Heidelberg.
References
 ^ BenMenahem, Ari (2009). Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1. Springer. p. 3678. , Extract of page 3678
 ^ van Hove, Leon (1958). "Von Neumann's contributions to quantum mechanics" (PDF). Bulletin of the American Mathematical Society 64: Part2:95–99.
 ^ Max Born & Emil Wolf, Principles of Optics, 1999, Cambridge University Press
 ^ Mehra, J.; Rechenberg, H. (1982). The historical development of quantum theory. New York: SpringerVerlag.
 ^ Kragh, Helge (2002). Quantum Generations: A History of Physics in the Twentieth Century. Princeton University Press. p. 58. , Extract of page 58
 ^ E Arunan (2010). "Peter Debye". Resonance (journal) (Indian Academy of Sciences) 15 (12).
 ^
 ^ Kragh, Helge (1 December 2000), Max Planck: the reluctant revolutionary, PhysicsWorld.com
 ^ Einstein, A. (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt" [On a heuristic point of view concerning the production and transformation of light]. Reprinted in The collected papers of Albert Einstein, John Stachel, editor, Princeton University Press, 1989, Vol. 2, pp. 149166, in German; see also Einstein's early work on the quantum hypothesis, ibid. pp. 134148.
 ^ "Quantum interference of large organic molecules". Nature.com. Retrieved April 20, 2013.
 ^ "Quantum  Definition and More from the Free MerriamWebster Dictionary". Merriamwebster.com. Retrieved 20120818.
 ^ http://mooni.fccj.org/~ethall/quantum/quant.htm
 ^ Compare the list of conferences presented here
 ^ Oocities.com at the Wayback Machine (archived October 26, 2009)
 ^ P.A.M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 1930.
 ^ D. Hilbert Lectures on Quantum Theory, 1915–1927
 ^ J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932 (English translation: Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955).
 ^ H.Weyl "The Theory of Groups and Quantum Mechanics", 1931 (original title: "Gruppentheorie und Quantenmechanik").
 ^ Greiner, Walter; Müller, Berndt (1994). Quantum Mechanics Symmetries, Second edition. SpringerVerlag. p. 52. , Chapter 1, p. 52
 ^ "Heisenberg  Quantum Mechanics, 1925–1927: The Uncertainty Relations". Aip.org. Retrieved 20120818.
 ^ ^{a} ^{b} Greenstein, George; Zajonc, Arthur (2006). The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics, Second edition. Jones and Bartlett Publishers, Inc. p. 215. , Chapter 8, p. 215
 ^ "[Abstract] Visualization of Uncertain Particle Movement". Actapress.com. Retrieved 20120818.
 ^ Hirshleifer, Jack (2001). The Dark Side of the Force: Economic Foundations of Conflict Theory. Campbridge University Press. p. 265. , Chapter , p.

^ Dict.cc
De.pons.eu  ^ "Topics: WaveFunction Collapse". Phy.olemiss.edu. 20120727. Retrieved 20120818.
 ^ "Collapse of the wavefunction". Farside.ph.utexas.edu. Retrieved 20120818.
 ^ "Determinism and Naive Realism : philosophy". Reddit.com. 20090601. Retrieved 20120818.
 ^ Michael Trott. "TimeEvolution of a Wavepacket in a Square Well — Wolfram Demonstrations Project". Demonstrations.wolfram.com. Retrieved 20101015.
 ^ Michael Trott. "Time Evolution of a Wavepacket In a Square Well". Demonstrations.wolfram.com. Retrieved 20101015.
 ^ Mathews, Piravonu Mathews; Venkatesan, K. (1976). A Textbook of Quantum Mechanics. Tata McGrawHill. p. 36. , Chapter 2, p. 36
 ^ "Wave Functions and the Schrödinger Equation" (PDF). Retrieved 20101015.
 ^ [1]
 ^ Nancy Thorndike Greenspan, "The End of the Certain World: The Life and Science of Max Born" (Basic Books, 2005), pp. 1248 and 2856.
 ^ http://ocw.usu.edu/physics/classicalmechanics/pdf_lectures/06.pdf
 ^ "The Nobel Prize in Physics 1979". Nobel Foundation. Retrieved 20100216.
 ^ Carl M. Bender, Daniel W. Hook, Karta Kooner (20091231). "Complex Elliptic Pendulum". arXiv:1001.0131 [hepth].
 ^ See, for example, Precision tests of QED. The relativistic refinement of quantum mechanics known as quantum electrodynamics (QED) has been shown to agree with experiment to within 1 part in 10^{8} for some atomic properties.
 ^ Tipler, Paul; Llewellyn, Ralph (2008). Modern Physics (5 ed.).
 ^ "Quantum mechanics course iwhatisquantummechanics". Scribd.com. 20080914. Retrieved 20120818.
 ^ A. Einstein, B. Podolsky, and N. Rosen, Can quantummechanical description of physical reality be considered complete? Phys. Rev. 47 777 (1935). [2]
 ^ "Between classical and quantum�" (PDF). Retrieved 20120819.
 ^ (see macroscopic quantum phenomena, Bose–Einstein condensate, and Quantum machine)
 ^ "Atomic Properties". Academic.brooklyn.cuny.edu. Retrieved 20120818.
 ^ http://assets.cambridge.org/97805218/29526/excerpt/9780521829526_excerpt.pdf
 ^ "There is as yet no logically consistent and complete relativistic quantum field theory.", p. 4. — V. B. Berestetskii, E. M. Lifshitz, L P Pitaevskii (1971). J. B. Sykes, J. S. Bell (translators). Relativistic Quantum Theory 4, part I. Course of Theoretical Physics (Landau and Lifshitz) ISBN 0080160255
 ^ Stephen Hawking; Gödel and the end of physics
 ^ Excerpt from an article by Roger Penrose
 ^ "Life on the lattice: The most accurate theory we have". Latticeqcd.blogspot.com. 20050603. Retrieved 20101015.
 ^ Parker, B. (1993). Overcoming some of the problems. pp. 259–279.
 ^ The Character of Physical Law (1965) Ch. 6; also quoted in The New Quantum Universe (2003), by Tony Hey and Patrick Walters
 ^ Weinberg, S. "Collapse of the State Vector", Phys. Rev. A 85, 062116 (2012).
 ^ "Action at a Distance in Quantum Mechanics (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. 20070126. Retrieved 20120818.
 ^ "Everett's RelativeState Formulation of Quantum Mechanics (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved 20120818.
 ^ The Transactional Interpretation of Quantum Mechanics by John Cramer. Reviews of Modern Physics 58, 647688, July (1986)
 ^ See, for example, the Feynman Lectures on Physics for some of the technological applications which use quantum mechanics, e.g., transistors (vol III, pp. 1411 ff), integrated circuits, which are followon technology in solidstate physics (vol II, pp. 86), and lasers (vol III, pp. 913).
 ^ Introduction to Quantum Mechanics with Applications to Chemistry  Linus Pauling, E. Bright Wilson. 19850301.
 ^ Anderson, Mark (20090113). "Is Quantum Mechanics Controlling Your Thoughts?  Subatomic Particles". DISCOVER Magazine. Retrieved 20120818.
 ^ "Quantum mechanics boosts photosynthesis". physicsworld.com. Retrieved 20101023.
 ^ Davies, P. C. W.; Betts, David S. (1984). Quantum Mechanics, Second edition. Chapman and Hall. p. 79. , Chapter 6, p. 79
 ^ Baofu, Peter (20071231). The Future of Complexity: Conceiving a Better Way to Understand Order and Chaos.
 ^ Derivation of particle in a box, chemistry.tidalswan.com
Notes
 Angular momentum diagrams (quantum mechanics)
 EPR paradox
 Fractional quantum mechanics
 List of quantummechanical systems with analytical solutions
 Macroscopic quantum phenomena
 Phase space formulation
 Spherical basis
See also
This is another example illustrating the quantization of energy for bound states.
 E_n = \hbar \omega \left(n + {1\over 2}\right).
and the corresponding energy levels are
 H_n(x)=(1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{x^2}\right)
where H_{n} are the Hermite polynomials,
 \psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{  \frac{m\omega x^2}{2 \hbar}} \cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), \qquad n = 0,1,2,\ldots.
This problem can either be treated by directly solving the Schrödinger, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The eigenstates are given by
 V(x)=\frac{1}{2}m\omega^2x^2
As in the classical case, the potential for the quantum harmonic oscillator is given by
Harmonic oscillator
The finite potential well problem is mathematically more complicated than the infinite particleinabox problem as the wavefunction is not pinned to zero at the walls of the well. Instead, the wavefunction must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well.
A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth.
Finite potential well
 E = \frac{\hbar^2 \pi^2 n^2}{2mL^2} = \frac{n^2h^2}{8mL^2}.
The quantization of energy levels follows from this constraint on k, since
 k = \frac{n\pi}{L}\qquad\qquad n=1,2,3,\ldots.
in which C cannot be zero as this would conflict with the Born interpretation. Therefore, since sin(kL) = 0, kL must be an integer multiple of π,
 \psi(L) = 0 = C\sin kL.\!
and D = 0. At x = L,
 \psi(0) = 0 = C\sin 0 + D\cos 0 = D\!
The infinite potential walls of the box determine the values of C, D, and k at x = 0 and x = L where ψ must be zero. Thus, at x = 0,
 \psi(x) = C \sin kx + D \cos kx.\!
or, from Euler's formula,
 \psi(x) = A e^{ikx} + B e ^{ikx} \qquad\qquad E = \frac{\hbar^2 k^2}{2m}
The general solutions of the Schrödinger equation for the particle in a box are
with state \psi in this case having energy E coincident with the kinetic energy of the particle.
 \frac{1}{2m} \hat{p}_x^2 = E,
the previous equation is evocative of the classic kinetic energy analogue,
 \hat{p}_x = i\hbar\frac{d}{dx}
With the differential operator defined by
  \frac {\hbar ^2}{2m} \frac {d ^2 \psi}{dx^2} = E \psi.
The particle in a onedimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and infinite potential energy everywhere outside that region. For the onedimensional case in the x direction, the timeindependent Schrödinger equation may be written^{[61]}
Particle in a box
This is a model for the quantum tunneling effect which plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy. Quantum tunneling is central to physical phenomena involved in superlattices.
Rectangular potential barrier
Each term of the solution can be interpreted as an incident, reflected, or transmitted component of the wave, allowing the calculation of transmission and reflection coefficients. Notably, in contrast to classical mechanics, incident particles with energies greater than the potential step are partially reflected.
with coefficients A and B determined from the boundary conditions and by imposing a continuous derivative on the solution.
 k_1=\sqrt{2m E/\hbar^2}, and
 k_2=\sqrt{2m (EV_0)/\hbar^2}
where the wave vectors are related to the energy via
 \psi_1(x)= \frac{1}{\sqrt{k_1}} \left(A_\rightarrow e^{i k_1 x} + A_\leftarrow e^{ik_1x}\right)\quad x<0
 \psi_2(x)= \frac{1}{\sqrt{k_2}} \left(B_\rightarrow e^{i k_2 x} + B_\leftarrow e^{ik_2x}\right)\quad x>0
The solutions are superpositions of left and rightmoving waves:
 V(x)= \begin{cases} 0, & x < 0, \\ V_0, & x \ge 0. \end{cases}
The potential in this case is given by:
Step potential
For example, consider a free particle. In quantum mechanics, there is wave–particle duality, so the properties of the particle can be described as the properties of a wave. Therefore, its quantum state can be represented as a wave of arbitrary shape and extending over space as a wave function. The position and momentum of the particle are observables. The Uncertainty Principle states that both the position and the momentum cannot simultaneously be measured with complete precision. However, one can measure the position (alone) of a moving free particle, creating an eigenstate of position with a wavefunction that is very large (a Dirac delta) at a particular position x, and zero everywhere else. If one performs a position measurement on such a wavefunction, the resultant x will be obtained with 100% probability (i.e., with full certainty, or complete precision). This is called an eigenstate of position—or, stated in mathematical terms, a generalized position eigenstate (eigendistribution). If the particle is in an eigenstate of position, then its momentum is completely unknown. On the other hand, if the particle is in an eigenstate of momentum, then its position is completely unknown.^{[59]} In an eigenstate of momentum having a plane wave form, 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.^{[60]}
Free particle
Examples
. Since classical formulas are much simpler and easier to compute than quantum formulas, classical approximations are used and preferred when the system is large enough to render the effects of quantum mechanics insignificant.quantum numbers can often provide good approximations to results otherwise obtained by quantum physics, typically in circumstances with large numbers of particles or large classical physics Even so, [58]
Quantum theory also provides accurate descriptions for many previously unexplained phenomena, such as
While quantum mechanics primarily applies to the smaller atomic regimes of matter and energy, some systems exhibit quantum mechanical effects on a large scale. Superfluidity, the frictionless flow of a liquid at temperatures near absolute zero, is one wellknown example. So is the closely related phenomenon of superconductivity, the frictionless flow of an electron gas in a conducting material (an electric current) at sufficiently low temperatures.
Quantum tunneling is vital to the operation of many devices. Even in the simple light switch, the electrons in the electric current could not penetrate the potential barrier made up of a layer of oxide without quantum tunneling. Flash memory chips found in USB drives use quantum tunneling to erase their memory cells.
Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to more fully develop quantum cryptography, which will theoretically allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than classical computers. Instead of using classical bits, quantum computers use qubits, which can be in superpositions of states. Another active research topic is quantum teleportation, which deals with techniques to transmit quantum information over arbitrary distances.
A great deal of modern technological inventions operate at a scale where quantum effects are significant. Examples include the laser, the transistor (and thus the microchip), the electron microscope, and magnetic resonance imaging (MRI). The study of semiconductors led to the invention of the diode and the transistor, which are indispensable parts of modern electronics systems and devices.
Quantum mechanics is also critically important for understanding how individual atoms combine covalently to form molecules. The application of quantum mechanics to chemistry is known as quantum chemistry. Relativistic quantum mechanics can, in principle, mathematically describe most of chemistry. Quantum mechanics can also provide quantitative insight into ionic and covalent bonding processes by explicitly showing which molecules are energetically favorable to which others and the magnitudes of the energies involved.^{[56]} Furthermore, most of the calculations performed in modern computational chemistry rely on quantum mechanics.
Quantum mechanics has had enormous^{[55]} success in explaining many of the features of our universe. Quantum mechanics is often the only tool available that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, protons, neutrons, photons, and others). Quantum mechanics has strongly influenced string theories, candidates for a Theory of Everything (see reductionism).
Applications
The Everett manyworlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a multiverse composed of mostly independent parallel universes.^{[53]} This is not accomplished by introducing some "new axiom" to quantum mechanics, but on the contrary, by removing the axiom of the collapse of the wave packet. All of the possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a real physical  not just formally mathematical, as in other interpretations  quantum superposition. Such a superposition of consistent state combinations of different systems is called an entangled state. While the multiverse is deterministic, we perceive nondeterministic behavior governed by probabilities, because we can only observe the universe (i.e., the consistent state contribution to the aforementioned superposition) that we, as observers, inhabit. Everett's interpretation is perfectly consistent with John Bell's experiments and makes them intuitively understandable. However, according to the theory of quantum decoherence, these "parallel universes" will never be accessible to us. The inaccessibility can be understood as follows: once a measurement is done, the measured system becomes entangled with both the physicist who measured it and a huge number of other particles, some of which are photons flying away at the speed of light towards the other end of the universe. In order to prove that the wave function did not collapse, one would have to bring all these particles back and measure them again, together with the system that was originally measured. Not only is this completely impractical, but even if one could theoretically do this, it would have to destroy any evidence that the original measurement took place (including the physicist's memory). In light of these Bell tests, Cramer (1986) formulated his transactional interpretation.^{[54]} Relational quantum mechanics appeared in the late 1990s as the modern derivative of the Copenhagen Interpretation.
Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement. Einstein held that there should be a local hidden variable theory underlying quantum mechanics and, consequently, that the present theory was incomplete. He produced a series of objections to quantum theory, the most famous of which has become known as the Einstein–Podolsky–Rosen paradox. John Bell showed that this "EPR" paradox led to experimentally testable differences between quantum mechanics and local realistic theories. Experiments have been performed confirming the accuracy of quantum mechanics, thereby demonstrating that the physical world cannot be described by any local realistic theory.^{[52]} The BohrEinstein debates provide a vibrant critique of the Copenhagen Interpretation from an epistemological point of view.
The Copenhagen interpretation  due largely to the Danish theoretical physicist Niels Bohr  remains the quantum mechanical formalism that is currently most widely accepted amongst physicists, some 75 years after its enunciation. According to this interpretation, the probabilistic nature of quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but instead must be considered a final renunciation of the classical idea of "causality." It is also believed therein that any welldefined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the conjugate nature of evidence obtained under different experimental situations.
Since its inception, the many counterintuitive aspects and results of quantum mechanics have provoked strong philosophical debates and many interpretations. Even fundamental issues, such as Max Born's basic rules concerning probability amplitudes and probability distributions, took decades to be appreciated by society and many leading scientists. Richard Feynman once said, "I think I can safely say that nobody understands quantum mechanics."^{[50]} According to Steven Weinberg, "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics."^{[51]}
Philosophical implications
Another popular theory is Loop quantum gravity (LQG), a theory that describes the quantum properties of gravity. It is also a theory of quantum space and quantum time, because in general relativity the geometry of spacetime is a manifestation of gravity. LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. The main output of the theory is a physical picture of space where space is granular. The granularity is a direct consequence of the quantization. It has the same nature of the granularity of the photons in the quantum theory of electromagnetism or the discrete levels of the energy of the atoms. But here it is space itself which is discrete. More precisely, space can be viewed as an extremely fine fabric or network "woven" of finite loops. These networks of loops are called spin networks. The evolution of a spin network over time, is called a spin foam. The predicted size of this structure is the Planck length, which is approximately 1.616×10^{−35} m. According to theory, there is no meaning to length shorter than this (cf. Planck scale energy). Therefore LQG predicts that not just matter, but also space itself, has an atomic structure. Loop quantum Gravity was first proposed by Carlo Rovelli.
The quest to unify the fundamental forces through quantum mechanics is still ongoing. Quantum electrodynamics (or "quantum electromagnetism"), which is currently (in the perturbative regime at least) the most accurately tested physical theory in competition with general relativity,^{[47]}^{[48]} ^{(blog)} has been successfully merged with the weak nuclear force into the electroweak force and work is currently being done to merge the electroweak and strong force into the electrostrong force. Current predictions state that at around 10^{14} GeV the three aforementioned forces are fused into a single unified field,^{[49]} Beyond this "grand unification", it is speculated that it may be possible to merge gravity with the other three gauge symmetries, expected to occur at roughly 10^{19} GeV. However — and while special relativity is parsimoniously incorporated into quantum electrodynamics — the expanded general relativity, currently the best theory describing the gravitation force, has not been fully incorporated into quantum theory. One of those searching for a coherent TOE is Edward Witten, a theoretical physicist who formulated the Mtheory, which is an attempt at describing the supersymmetrical based string theory. Mtheory posits that our apparent 4dimensional spacetime is, in reality, actually an 11dimensional spacetime containing 10 spatial dimensions and 1 time dimension, although 7 of the spatial dimensions are  at lower energies  completely "compactified" (or infinitely curved) and not readily amenable to measurement or probing.
Attempts at a unified field theory
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 particular applications. However, the lack of a correct theory of quantum gravity is an important issue in cosmology and the search by physicists for an elegant "Theory of Everything" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th and 21st century physics. Many prominent physicists, including Stephen Hawking, have labored for many years in the attempt to discover a theory underlying everything. This TOE would combine not only the different models of subatomic physics, but also derive the four fundamental forces of nature  the strong force, electromagnetism, the weak force, and gravity  from a single force or phenomenon. While Stephen Hawking was initially a believer in the Theory of Everything, after considering Gödel's Incompleteness Theorem, he has concluded that one is not obtainable, and has stated so publicly in his lecture "Gödel and the End of Physics" (2002).^{[46]}
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. Quantum entanglement forms the basis of quantum cryptography, which is used in highsecurity commercial applications in banking and government.
According to the paper of J. Bell and the Copenhagen interpretation—the common interpretation of quantum mechanics by physicists since 1927  and contrary to Einstein's ideas, quantum mechanics was not, at the same time a "realistic" theory and a "local" theory.
Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly contributing to the field, he did not accept many of the more "philosophical consequences and interpretations" of quantum mechanics, such as the lack of deterministic causality. He is famously quoted as saying, in response to this aspect, "My God does not play with dice". He also had difficulty with the assertion that a single subatomic particle can occupy numerous areas of space at one time. However, he was also 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 had unacceptable implications if taken as a complete description of physical reality. This was 1935, but in 1964 it was shown by John Bell (see Bell inequality) that  although Einstein was correct in identifying seemingly paradoxical implications of quantum mechanical nonlocality  these implications could be experimentally tested. Alain Aspect's initial experiments in 1982, and many subsequent experiments since, have definitively verified quantum entanglement.
Even with the defining postulates of both Einstein's theory of general relativity and quantum theory being indisputably supported by rigorous and repeated empirical evidence, and while they do not directly contradict each other theoretically (at least with regard to their primary claims), they have proven extremely difficult to incorporate into one consistent, cohesive model.^{[45]}
Relativity and quantum mechanics
 Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.^{[43]}
 While the seemingly "exotic" behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with particles of extremely small size or velocities approaching the speed of light, the laws of classical, often considered "Newtonian", physics remain accurate in predicting the behavior of the vast majority of "large" objects (on the order of the size of large molecules or bigger) at velocities much smaller than the velocity of light.^{[44]}
Quantum coherence is an essential difference between classical and quantum theories as illustrated by the Einstein–Podolsky–Rosen (EPR) paradox — an attempt to disprove quantum mechanics by an appeal to local realism.^{[40]} Quantum interference involves adding together probability amplitudes, whereas classical "waves" infer that there is an adding together of intensities. For microscopic bodies, the extension of the system is much smaller than the coherence length, which gives rise to longrange entanglement and other nonlocal phenomena characteristic of quantum systems.^{[41]} Quantum coherence is not typically evident at macroscopic scales, though an exception to this rule may occur at extremely low temperatures (i.e. approaching absolute zero) at which quantum behavior may manifest itself macroscopically.^{[42]} This is in accordance with the following observations:
Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy.^{[37]} According to the correspondence principle between classical and quantum mechanics, all objects obey the laws of quantum mechanics, and classical mechanics is just an approximation for large systems of objects (or a statistical quantum mechanics of a large collection of particles).^{[38]} The laws of classical mechanics thus follow from the laws of quantum mechanics as a statistical average at the limit of large systems or large quantum numbers.^{[39]} However, chaotic systems do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems.
Quantum mechanics and classical physics
Classical mechanics has also been extended into the complex domain, with complex classical mechanics exhibiting behaviors similar to quantum mechanics.^{[36]}
It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semiclassical approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity (the most accurate theory of gravity currently known) and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research, and theories such as string theory are among the possible candidates for a future theory of quantum gravity.
Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of subnuclear particles such as quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as electroweak theory), by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg. These three men shared the Nobel Prize in Physics in 1979 for this work.^{[35]}
Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction. The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical \scriptstyle e^2/(4 \pi\ \epsilon_{_0}\ r) Coulomb potential. This "semiclassical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.
When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was nonrelativistic classical mechanics. For instance, the wellknown model of the quantum harmonic oscillator uses an explicitly nonrelativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.
In the correspondence limit of quantum mechanics: Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the "superposition of states" and "wavefunction collapse", give rise to the reality we perceive? 
The rules of quantum mechanics are fundamental. They assert that the state space of a system is a Hilbert space and that observables of that system are Hermitian operators acting on that space—although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical mechanics when a system moves to higher energies or—equivalently—larger quantum numbers, i.e. whereas a single particle exhibits a degree of randomness, in systems incorporating millions of particles averaging takes over and, at the high energy limit, the statistical probability of random behaviour approaches zero. In other words, classical mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the classical or correspondence limit. One can even start from an established classical model of a particular system, then attempt to guess the underlying quantum model that would give rise to the classical model in the correspondence limit.
Interactions with other scientific theories
Especially since Werner Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, the role of Max Born in the development of QM was overlooked until the 1954 Nobel award. The role is noted in a 2005 biography of Born, which recounts his role in the matrix formulation of quantum mechanics, and the use of probability amplitudes. Heisenberg himself acknowledges having learned matrices from Born, as published in a 1940 festschrift honoring Max Planck.^{[33]} In the matrix 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).^{[34]} An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantummechanical amplitude is considered as a sum over all possible classical and nonclassical paths between the initial and final states. This is the quantummechanical counterpart of the action principle in classical mechanics.
There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the "transformation theory" proposed by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics  matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).^{[32]}
Mathematically equivalent formulations of quantum mechanics
There exist several techniques for generating approximate solutions, however. In the important method known as perturbation theory, one uses the analytic result for a simple quantum mechanical model to generate a result for a more complicated model that is related to the simpler model by (for one example) the addition of a weak potential energy. Another method is the "semiclassical equation of motion" approach, which applies to systems for which quantum mechanics produces only weak (small) deviations from classical behavior. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of quantum chaos.
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 "wavelike" behavior of quantum states. As it turns out, analytic solutions of the Schrödinger equation are available for only a very small number of relatively simple 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 does the hydrogen atom—has defied all attempts at a fully analytic treatment.
Some wave functions produce probability distributions that are constant, or independent of time—such as when in a stationary state of constant energy, time vanishes in the absolute square of the wave function. 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, however, that only the lowest angular momentum states, labeled s, are spherically symmetric).^{[31]}
Wave functions change as time progresses. The Schrödinger equation describes how wavefunctions change in time, playing 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 with time. This also has the effect of turning a position eigenstate (which can be thought of as an infinitely sharp wave packet) into a broadened wave packet that no longer represents a (definite, certain) position eigenstate.^{[30]}
During a measurement, on the other hand, the change of the initial wavefunction into another, later wavefunction is not deterministic, it is unpredictable (i.e., random). A timeevolution simulation can be seen here.^{[28]}^{[29]}
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 the time evolution. 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.^{[27]}
In the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate. Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum (since they are conjugate pairs) or its energy and time (since they too are conjugate pairs); rather, it provides only a range of probabilities in which that particle might be given its momentum and momentum probability. Therefore, it is helpful to use different words to describe states having uncertain values and states having definite values (eigenstates). Usually, a system will not be in an eigenstate of the observable (particle) 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, a controversial and muchdebated process^{[25]} that involves expanding the system under study to include the measurement device. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of the wavefunction 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 x_{0} (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.^{[21]} It is probable, but not certain, that it will be near x_{0}, 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.^{[26]}
Generally, quantum mechanics does not assign definite values. Instead, it makes a prediction using a probability distribution; that is, it describes the probability of obtaining the possible outcomes from measuring an observable. Often these results are skewed by many causes, such as dense probability clouds. Probability clouds are approximate, but better than the Bohr model, whereby electron location is given by a probability function, the wave function eigenvalue, such that the probability is the squared modulus of the complex amplitude, or quantum state nuclear attraction.^{[22]}^{[23]} Naturally, these probabilities will depend on the quantum state at the "instant" of the measurement. Hence, uncertainty is involved in the value. 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 translated from German as meaning "inherent" or "characteristic").^{[24]}
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 BohrEinstein 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. Newer interpretations of quantum mechanics have been formulated that 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.^{[21]}
According to one interpretation, as the result of a measurement the wave function containing the probability information for a system collapses from a given initial state to a particular eigenstate. The possible results of a measurement are the eigenvalues of the operator representing the observable—which explains the choice of Hermitian operators, for which all the eigenvalues are real. The probability distribution of an observable in a given state can be found 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 the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function, also referred to as state vector in a complex vector space.^{[19]} 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 accuracy. For instance, electrons may be considered (to a certain probability) to be located somewhere within a given region of space, but with their exact positions 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. Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate momentum.^{[20]}
In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac,^{[15]} David Hilbert,^{[16]} John von Neumann,^{[17]} and Hermann Weyl^{[18]} the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors"). Formally, these reside in a complex separable Hilbert space—variously called the "state space" or the "associated Hilbert space" of the system—that is well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective 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 squareintegrable 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 selfadjoint) 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 attain only those discrete eigenvalues.
Mathematical formulations
 quantization of certain physical properties
 wave–particle duality
 principle of uncertainty
 quantum entanglement.
Broadly speaking, quantum mechanics incorporates four classes of phenomena for which classical physics cannot account:
Quantum mechanics was initially developed to provide a better explanation and description of the atom, especially the differences in the spectra of light emitted by different isotopes of the same element, as well as subatomic particles. In short, the quantummechanical atomic model has succeeded spectacularly in the realm where classical mechanics and electromagnetism falter.
Quantum mechanics is essential to understanding the behavior of systems at atomic length scales and smaller. If the physical nature of an atom was solely described by classical mechanics electrons would not "orbit" the nucleus since orbiting electrons emit radiation (due to circular motion) and would eventually collide with the nucleus due to this loss of energy. This framework was unable to explain the stability of atoms. Instead, electrons remain in an uncertain, nondeterministic, "smeared", probabilistic, wave–particle orbital about the nucleus, defying the traditional assumptions of classical mechanics and electromagnetism.^{[14]}
The word quantum derives from the Latin, meaning "how great" or "how much".^{[11]} In quantum mechanics, it refers to a discrete unit that quantum theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1). The discovery that particles are discrete packets of energy with wavelike properties led to the branch of physics dealing with atomic and subatomic systems which is today called quantum mechanics. It underlies the mathematical framework of many fields of physics and chemistry, including condensed matter physics, solidstate physics, atomic physics, molecular physics, computational physics, computational chemistry, quantum chemistry, particle physics, nuclear chemistry, and nuclear physics.^{[12]} Some fundamental aspects of the theory are still actively studied.^{[13]}
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