ਫਰਮਾ:ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ ਨਾਲ ਜਾਣ-ਪਛਾਣ ਲਈ ਸਰੋਤ ਵੇਖੋ

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[[File:Hydrogen Density Plots.png|thumb|286x286px|Solution to [[Schrödinger equation|Schrödinger's equation]] for the hydrogen atom at different energy levels. The brighter areas represent a higher probability of finding an [[electron]]]]
{{[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]}}
'''[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]''' ('''QM'''; also known as '''[[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]]''' or '''quantum theory''') including [[quantum field theory]], is a fundamental branch of [[physics]] concerned with processes involving, for example, [[atom]]s and [[photons]]. In such processes, said to be quantized, the [[Action (physics)|action]] has been observed to be only in integer multiples of the [[Planck constant]], a physical quantity that is exceedingly, indeed perhaps ultimately, small. This is utterly inexplicable in [[classical physics]].

[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] [[History of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]|gradually arose]] from [[Max Planck]]'s solution in 1900 to the [[black-body radiation]] problem (reported 1859) and [[Albert Einstein]]'s 1905 paper which offered a quantum-based theory to explain the [[photoelectric effect]] (reported 1887). [[old quantum theory|Early quantum theory]] was profoundly reconceived in the mid-1920s.

The reconceived theory is formulated in various [[mathematical formulations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]|specially developed mathematical formalisms]]. In one of them, a mathematical function, the [[wave function]], provides information about the [[probability amplitude]] of position, momentum, and other physical properties of a particle.

Important applications of quantum mechanical theory include [[superconducting magnet]]s, [[light-emitting diode]]s and the [[laser]], the [[transistor]] and [[semiconductor]]s such as the [[microprocessor]], [[medical imaging|medical and research imaging]] such as [[magnetic resonance imaging]] and [[electron microscope|electron microscopy]], and explanations for many biological and physical phenomena.

==History==
{{Modern physics}}
{{Main|History of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]}}
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.<ref name="Born & Wolf">[[Max Born]] & Emil Wolf, Principles of Optics, 1999, Cambridge University Press</ref> In 1803, [[Thomas Young (scientist)|Thomas Young]], an English [[polymath]], performed the famous [[Young's interference experiment|double-slit 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, [[Michael Faraday]] discovered [[cathode ray]]s. These studies were followed by 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 can be discrete, and the 1900 quantum hypothesis of [[Max Planck]].<ref>{{cite book |first=J. |last=Mehra |first2=H. |last2=Rechenberg |title=The historical development of quantum theory |location=New York |publisher=Springer-Verlag |year=1982 |isbn=0387906428 }}</ref> Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy elements) precisely matched the observed patterns of black-body radiation.

In 1896, [[Wilhelm Wien]] empirically determined a distribution law of black-body radiation,<ref>{{cite book |title=Quantum Generations: A History of Physics in the Twentieth Century |first1=Helge |last1=Kragh |publisher=Princeton University Press |year=2002 |isbn=0-691-09552-3 |page=58 |url=https://books.google.com/books?id=ELrFDIldlawC&pg=PA58}} [https://books.google.com/books?id=ELrFDIldlawC&pg=PA58 Extract of page 58]</ref> known as [[Wien approximation|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, Planck corrected this model using Boltzmann's statistical interpretation of thermodynamics and proposed what is now called [[Planck's law]], which led to the development of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]].

Following [[Max Planck]]'s solution in 1900 to the black-body radiation problem (reported 1859), [[Albert Einstein]] offered a quantum-based theory to explain the [[photoelectric effect]] (1905, reported 1887).  Around 1900-1910, the [[atomic theory]] and the [[corpuscular theory of light]]<ref>{{cite book |title=Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1 |first1=Ari  |last1=Ben-Menahem |publisher=Springer |year=2009 |isbn=3540688315 |page=3678 |url=https://books.google.com/books?id=9tUrarQYhKMC}} [https://books.google.com/books?id=9tUrarQYhKMC&pg=PA3678 Extract of page 3678]</ref> first came to be widely accepted as scientific fact; these latter theories can be viewed as quantum theories of [[matter]] and [[electromagnetic radiation]], respectively.

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 Andrews 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 orbit]]s, a concept also introduced by [[Arnold Sommerfeld]].<ref>{{cite journal|url=http://www.ias.ac.in/resonance/Volumes/15/12/1056-1059.pdf|journal=Resonance (journal)|volume=15|issue=12|year=2010|publisher=Indian Academy of Sciences|title=Peter Debye|author=E Arunan}}</ref> This phase is known as [[old quantum theory]].

According to Planck, each energy element (''E'')'' ''is proportional to its [[frequency]] (''ν''):
:<math> E = h \nu\ </math>
[[File:Max Planck (1858-1947).jpg|thumb|upright|[[Max Planck]] is considered the father of the quantum theory.]]
where ''h'' is [[Planck constant|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.<ref>{{cite book |authorlink=Thomas Samuel Kuhn |first=T. S. |last=Kuhn |title=Black-body theory and the quantum discontinuity 1894-1912 |publisher=Clarendon Press |location=Oxford |year=1978 |isbn=0195023838 }}</ref> In fact, he considered his [[quantum hypothesis]] a mathematical trick to get the right answer rather than a sizable discovery.<ref name="Kragh">{{citation | first = Helge | last = Kragh | url = http://physicsworld.com/cws/article/print/373 | title =  Max Planck: the reluctant revolutionary | publisher = PhysicsWorld.com | date = 1 December 2000}}</ref> However, in 1905 [[Albert Einstein]] interpreted Planck's quantum hypothesis [[local realism|realistically]] and used it to explain the [[photoelectric effect]], in which shining light on certain materials can eject electrons from the material. He won the 1921 Nobel Prize in Physics for this work.

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.<ref>{{cite journal |first=A. |last=Einstein |title={{lang|de|Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt}} |trans_title=On a heuristic point of view concerning the production and transformation of light |journal=[[Annalen der Physik]] |volume=17 |year=1905 |pages=132–148 |doi=10.1002/andp.19053220607 |bibcode = 1905AnP...322..132E |issue=6 }} Reprinted in ''The collected papers of Albert Einstein'', John Stachel, editor, Princeton University Press, 1989, Vol. 2, pp. 149-166, in German; see also ''Einstein's early work on the quantum hypothesis'', ibid. pp. 134-148.</ref>

[[File:Solvay conference 1927.jpg|left|thumb|260px|The 1927 [[Solvay Conference]] in [[Brussels]].]]
The foundations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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 [[:Category:Quantum physicists|others]]. The [[Copenhagen interpretation]] of [[Niels Bohr]] became widely accepted.
 
In the mid-1920s, developments in [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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 particle-like behavior in certain processes and measurements, light quanta came to be called [[photon]]s (1926). From Einstein's simple postulation was born a flurry of debating, theorizing, and testing. Thus, the entire field of [[[[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]]]] emerged, leading to its wider acceptance at the Fifth [[Solvay Conference]] in 1927.

It was found that [[subatomic particles]] and electromagnetic waves are neither simply particle nor wave but have certain properties of each. This originated the concept of [[wave–particle duality]].

By 1930, [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] had been further unified and formalized by the work of [[David Hilbert]], [[Paul Dirac]] and [[John von Neumann]]<ref>{{cite journal|last=van Hove|first=Leon|title=Von Neumann's contributions to [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]|journal=Bulletin of the American Mathematical Society|year=1958|volume=64|pages=Part2:95–99 |url=http://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10206-2/S0002-9904-1958-10206-2.pdf |format=PDF|doi=10.1090/s0002-9904-1958-10206-2}}</ref> with greater emphasis on [[measurement in [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]|measurement]], the statistical nature of our knowledge of reality, and [[Interpretations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]|philosophical speculation about the 'observer']]. It has since permeated many disciplines including [[quantum chemistry]], [[quantum electronics]], [[quantum optics]], and [[quantum information science]]. Its speculative modern developments include [[string theory]] and [[quantum gravity]] theories. It also provides a useful framework for many features of the modern [[periodic table|periodic table of elements]], and describes the behaviors of [[atoms]] during [[chemical bond]]ing and the flow of [[electron]]s in computer [[semiconductor]]s, and therefore plays a crucial role in many modern technologies.

While [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] was constructed to describe the world of the very small, it is also needed to explain some [[macroscopic]] phenomena such as [[superconductivity|superconductors]],<ref>[http://www.feynmanlectures.caltech.edu/III_21.html#Ch21-S5 ''[[The Feynman Lectures on Physics]]'' '''III''' 21-4 "...it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized that the phenomena of superconductivity presents us with just this situation.] accessdate=2015-11-24</ref>  and [[superfluid]]s.<ref>[http://research.physics.berkeley.edu/packard/publications/Articles/LT24_Berk_expts_on_macro_sup_effects.pdf Richard Packard (2006) "Berkeley Experiments on Superfluid Macroscopic Quantum Effects" ]  accessdate=2015-11-24</ref>

The word ''quantum'' derives from the [[Latin language|Latin]], meaning "how great" or "how much".<ref>{{cite web|url=http://www.merriam-webster.com/dictionary/quantum |title=Quantum - Definition and More from the Free Merriam-Webster Dictionary |publisher=Merriam-webster.com |date= |accessdate=2012-08-18}}</ref> In [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], it refers to a discrete unit assigned to certain [[Physical quantity|physical quantities]] such as the [[energy]] of an [[atom]] at rest (see Figure 1). The discovery that particles are discrete packets of energy with wave-like properties led to the branch of physics dealing with atomic and subatomic systems which is today called [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]. It underlies the [[mathematical]] framework of many fields of [[physics]] and [[chemistry]], including [[condensed matter physics]], [[solid-state physics]], [[atomic physics]], [[molecular physics]], [[computational physics]], [[computational chemistry]], [[quantum chemistry]], [[particle physics]], [[nuclear chemistry]], and [[nuclear physics]].<ref>http://mooni.fccj.org/~ethall/quantum/quant.htm</ref> Some fundamental aspects of the theory are still actively studied.<ref>{{cite web|url=http://ysfine.com/|title=ysfine.com|author=|date=|work=ysfine.com|accessdate=11 September 2015}}</ref>

[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] is essential to understanding the behavior of systems at [[atom]]ic 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, non-deterministic, ''smeared'', [[probability|probabilistic]] wave–particle [[Atomic orbital|orbital]] about the nucleus, defying the traditional assumptions of classical mechanics and [[electromagnetism]].<ref>{{Wayback|date=20091026095410 |url=http://geocities.com/mik_malm/quantmech.html |title=Oocities.com }}{{dead link|date=April 2014}}</ref>

[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] was initially developed to provide a better explanation and description of the atom, especially the differences in the [[spectrum|spectra]] of light emitted by different [[isotope]]s of the same [[chemical element]], as well as subatomic particles. In short, the quantum-mechanical atomic model has succeeded spectacularly in the realm where classical mechanics and electromagnetism falter.

Broadly speaking, [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] incorporates four classes of phenomena for which classical physics cannot account:

* [[quantization (physics)|quantization]] of [[Canonical conjugate variables|certain physical properties]]
* [[quantum entanglement]]
* [[uncertainty principle|principle of uncertainty]]
* [[wave–particle duality]]

==Mathematical formulations==
{{Main|Mathematical formulation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]}}
{{See also|Quantum logic}}

In the mathematically rigorous formulation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] developed by [[Paul Dirac]],<ref>P.A.M. Dirac, ''The Principles of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]'', Clarendon Press, Oxford, 1930.</ref> [[David Hilbert]],<ref>D. Hilbert ''Lectures on Quantum Theory'', 1915–1927</ref> [[John von Neumann]],<ref>J. von Neumann, ''Mathematische Grundlagen der Quantenmechanik'', Springer, Berlin, 1932 (English translation: ''Mathematical Foundations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]'', Princeton University Press, 1955).</ref> and [[Hermann Weyl]],<ref>H.Weyl "The Theory of Groups and [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]", 1931 (original title: "Gruppentheorie und Quantenmechanik").</ref> the possible states of a quantum mechanical system are symbolized<ref>[[Paul Adrien Maurice Dirac|Dirac, P.A.M.]] (1958). ''The Principles of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]'', 4th edition, Oxford University Press, Oxford UK, p.&nbsp;ix: "For this reason I have chosen the symbolic method, introducing the representatives later merely as an aid to practical calculation."</ref> as [[unit vector]]s (called ''state vectors''). Formally, these reside in a [[complex number|complex]] [[Separable space|separable]] [[Hilbert space]]—variously called the [[State space (physics)|''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 [[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 adjoint|Hermitian]] (precisely: by a [[self-adjoint operator|self-adjoint]]) linear [[Operator (physics)|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.

In the formalism of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], the state of a system at a given time is described by a [[complex number|complex]] [[wave function]], also referred to as state vector in a complex [[vector space]].<ref>{{cite book|title=[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] Symmetries, Second edition|first1=Walter|last1=Greiner|first2=Berndt|last2=Müller|publisher=Springer-Verlag|year=1994|isbn=3-540-58080-8|page=52|url=https://books.google.com/books?id=gCfvWx6vuzUC&pg=PA52}}, [https://books.google.com/books?id=gCfvWx6vuzUC&pg=PA52 Chapter 1, p. 52]</ref> This abstract mathematical object allows for the calculation of [[probability|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.<ref>{{cite web|url=http://www.aip.org/history/heisenberg/p08a.htm |title=Heisenberg - [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], 1925–1927: The Uncertainty Relations |publisher=Aip.org |date= |accessdate=2012-08-18}}</ref>

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 theorem|spectral decomposition]] of the corresponding operator. Heisenberg's [[uncertainty principle]] is represented by the statement that the operators corresponding to certain observables do not [[Commutator|commute]].

The [[probability|probabilistic]] nature of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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 experiment]]s. In the decades after the formulation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], the question of what constitutes a "measurement" has been extensively studied. Newer [[Interpretation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]|interpretations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]]] have been formulated that do away with the concept of "wave function collapse" (see, for example, the [[relative state interpretation]]). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become [[Quantum Entanglement|entangled]], so that the original quantum system ceases to exist as an independent entity. For details, see the article on [[measurement in [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]]].<ref name="google215">{{cite book|title=The Quantum Challenge: Modern Research on the Foundations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], Second edition|first1=George|last1=Greenstein|first2=Arthur|last2=Zajonc|publisher=Jones and Bartlett Publishers, Inc|year=2006|isbn=0-7637-2470-X|page=215|url=https://books.google.com/books?id=5t0tm0FB1CsC&pg=PA215}}, [https://books.google.com/books?id=5t0tm0FB1CsC&pg=PA215 Chapter 8, p. 215]
</ref>

Generally, [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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 [[Bohr model|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 [[probability amplitude|complex amplitude]], or quantum state nuclear attraction.<ref>{{cite web|url=http://www.actapress.com/PaperInfo.aspx?PaperID=25988&reason=500 |title=[Abstract&#93; Visualization of Uncertain Particle Movement |publisher=Actapress.com |date= |accessdate=2012-08-18}}</ref><ref>{{cite book
|title=The Dark Side of the Force: Economic Foundations of Conflict Theory|first1=Jack|last1=Hirshleifer|publisher=Campbridge University Press|year=2001|isbn=0-521-80412-4|page=265|url=https://books.google.com/books?id=W2J2IXgiZVgC&pg=PA265}}, [https://books.google.com/books?id=W2J2IXgiZVgC&pg=PA265 Chapter , p. ]</ref> 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 [[eigenstate]]s of the observable ("eigen" can be translated from [[German language|German]] as meaning "inherent" or "characteristic").<ref>{{cite web|url=http://www.dict.cc/german-english/eigen.html|title=dict.cc dictionary :: eigen :: German-English translation|author=|date=|work=dict.cc|accessdate=11 September 2015}}</ref>

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, [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] does not pinpoint the exact values of a particle's position and momentum (since they are [[Conjugate variables|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 ''[[Uncertainty principle|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 wave function will instantaneously be an eigenstate (or "generalized" eigenstate) of that observable. This process is known as [[wave function collapse]], a controversial and much-debated process<ref>{{cite web|url=http://www.phy.olemiss.edu/~luca/Topics/qm/collapse.html |title=Topics: Wave-Function Collapse |publisher=Phy.olemiss.edu |date=2012-07-27 |accessdate=2012-08-18}}</ref> 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 wave function collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wave function that is a [[wave packet]] centered around some mean position ''x''<sub>0</sub> (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.<ref name="google215"/> It is probable, but not certain, that it will be near ''x''<sub>0</sub>, 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''.<ref>{{cite web|url=http://farside.ph.utexas.edu/teaching/qmech/lectures/node28.html |title=Collapse of the wave-function |publisher=Farside.ph.utexas.edu |date= |accessdate=2012-08-18}}</ref>

The time evolution of a quantum state is described by the [[Schrödinger equation]], in which the [[Hamiltonian ([[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]])|Hamiltonian]] (the [[Operator (physics)|operator]] corresponding to the [[total energy]] of the system) generates the time evolution. The [[time evolution]] of wave functions is [[determinism|deterministic]] in the sense that - given a wave function at an ''initial'' time - it makes a definite prediction of what the wave function will be at any ''later'' time.<ref>{{cite web|url=http://www.reddit.com/r/philosophy/comments/8p2qv/determinism_and_naive_realism/ |title=Determinism and Naive Realism : philosophy |publisher=Reddit.com |date=2009-06-01 |accessdate=2012-08-18}}</ref>

During a [[quantum measurement|measurement]], on the other hand, the change of the initial wave function into another, later wave function is not deterministic, it is unpredictable (i.e., [[random]]). A time-evolution simulation can be seen here.<ref>{{cite web|author=Michael Trott |url=http://demonstrations.wolfram.com/TimeEvolutionOfAWavepacketInASquareWell/ |title=Time-Evolution of a Wavepacket in a Square Well&nbsp;— Wolfram Demonstrations Project |publisher=Demonstrations.wolfram.com |date= |accessdate=2010-10-15}}</ref><ref>{{cite web|author=Michael Trott |url=http://demonstrations.wolfram.com/TimeEvolutionOfAWavepacketInASquareWell/ |title=Time Evolution of a Wavepacket In a Square Well |publisher=Demonstrations.wolfram.com |date= |accessdate=2010-10-15}}</ref>

Wave functions change as time progresses. The [[Schrödinger equation]] describes how wave functions 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.<ref>{{cite book|title=A Textbook of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]|first1=Piravonu Mathews|last1=Mathews|first2=K.|last2=Venkatesan|publisher=Tata McGraw-Hill|year=1976|isbn=0-07-096510-2|page=36|url=https://books.google.com/books?id=_qzs1DD3TcsC&pg=PA36}}, [https://books.google.com/books?id=_qzs1DD3TcsC&pg=PA36 Chapter 2, p. 36]</ref>

[[Image:HAtomOrbitals.png|thumb|275px|Fig. 1: [[Probability density function|Probability densities]] corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: ''n'' = 1, 2, 3, ...) and angular momenta (increasing across from left to right: ''s'', ''p'', ''d'', ...). Brighter areas correspond to higher probability density in a position measurement. Such wave functions are directly comparable to [[Chladni's figures]] of [[acoustics|acoustic]] modes of vibration in [[classical physics]], and are modes of oscillation as well, possessing a sharp [[energy]] and, thus, a definite [[frequency]]. The [[angular momentum]] and energy are [[quantization (physics)|quantized]], and take '''only''' discrete values like those shown (as is the case for [[Resonant frequency|resonant frequencies]] in acoustics)]]

Some wave functions produce probability distributions that are constant, or independent of time—such as when in a [[Eigenstate#Schrödinger equation|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 [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] it is described by a static, [[spherical coordinate system|spherically symmetric]] wave function surrounding the nucleus ([[:Image:HAtomOrbitals.png|Fig. 1]]) (note, however, that only the lowest angular momentum states, labeled ''s'', are spherically symmetric).<ref>{{cite web|url=http://physics.ukzn.ac.za/~petruccione/Phys120/Wave%20Functions%20and%20the%20Schr%F6dinger%20Equation.pdf |title=Wave Functions and the Schrödinger Equation |format=PDF |date= |accessdate=2010-10-15}}{{dead link|date=August 2012}}</ref>

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 (waves)|phase]] encodes information about the [[Interference (wave propagation)|interference]] between quantum states. This gives rise to the "wave-like" behavior of quantum states. As it turns out, analytic solutions of the Schrödinger equation are available for only [[List of quantum-mechanical systems with analytical solutions|a very small number of relatively simple model Hamiltonians]], of which the [[quantum harmonic oscillator]], the [[particle in a box]], the [[dihydrogen cation]], 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.

There exist several techniques for generating approximate solutions, however. In the important method known as [[perturbation theory ([[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]])|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 "semi-classical equation of motion" approach, which applies to systems for which [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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]].

==Mathematically equivalent formulations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]==
There are numerous mathematically equivalent formulations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]. One of the oldest and most commonly used formulations is the "[[transformation theory ([[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]])|transformation theory]]" proposed by [[Paul Dirac]], which unifies and generalizes the two earliest formulations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] - [[matrix mechanics]] (invented by [[Werner Heisenberg]]) and [[Schrödinger equation|wave mechanics]] (invented by [[Erwin Schrödinger]]).<ref>[http://th-www.if.uj.edu.pl/acta/vol19/pdf/v19p0683.pdf]{{dead link|date=April 2014}}</ref>

Especially since [[Werner Heisenberg]] was awarded the [[Nobel Prize in Physics]] in 1932 for the creation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], 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 [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], and the use of probability amplitudes. Heisenberg himself acknowledges having learned matrices from Born, as published in a 1940 ''[[festschrift]]'' honoring [[Max Planck]].<ref>Nancy Thorndike Greenspan, "The End of the Certain World: The Life and Science of Max Born" (Basic Books, 2005), pp. 124-8 and 285-6.</ref>  In the matrix formulation, the [[quantum state|instantaneous state of a quantum system]] encodes the probabilities of its measurable properties, or "[[observable]]s". Examples of observables include [[energy]], [[position operator|position]], [[momentum operator|momentum]], and [[angular momentum]]. Observables can be either [[Continuous function|continuous]] (e.g., the position of a particle) or [[Discrete mathematics|discrete]] (e.g., the energy of an electron bound to a hydrogen atom).<ref>http://ocw.usu.edu/physics/classical-mechanics/pdf_lectures/06.pdf</ref> An alternative formulation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] is [[Feynman]]'s [[path integral formulation]], in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the [[action principle]] in classical mechanics.

==Interactions with other scientific theories==
The rules of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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 [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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 [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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.

{{unsolved|physics|In the [[correspondence limit]] of '''[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]''': Is there a preferred interpretation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]? How does the quantum description of reality, which includes elements such as the "[[superposition principle|superposition]] of states" and "[[wave function collapse]]", give rise to the reality we perceive?}}
When [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] was originally formulated, it was applied to models whose
correspondence limit was [[theory of ਰਿਲੇਟੀਵਿਟੀ|non-relativistic]] [[classical mechanics]]. For instance, the well-known model of the [[quantum harmonic oscillator]] uses an explicitly non-relativistic expression for the [[kinetic energy]] of the oscillator, and is thus a quantum version of the [[harmonic oscillator|classical harmonic oscillator]].

Early attempts to merge [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] with [[special ਰਿਲੇਟੀਵਿਟੀ]] 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 [[electromagnetism|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 [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], is to treat [[electric charge|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 <math>\scriptstyle -e^2/(4 \pi\ \epsilon_{_0}\ r)</math> [[Electric potential|Coulomb potential]]. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of [[photon]]s by [[charged particle]]s.

[[Field (physics)|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 [[quark]]s and [[gluon]]s. 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.<ref>{{cite web
|url=http://nobelprize.org/nobel_prizes/physics/laureates/1979/index.html
|title=The Nobel Prize in Physics 1979
|publisher=Nobel Foundation
|accessdate=2010-02-16}}</ref>

It has proven difficult to construct quantum models of [[gravity]], the remaining [[fundamental force]]. Semi-classical 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 ਰਿਲੇਟੀਵਿਟੀ]] (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.

Classical mechanics has also been extended into the [[complex domain]], with complex classical mechanics exhibiting behaviors similar to [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]].<ref>{{cite arXiv |title=Complex Elliptic Pendulum|author= Carl M. Bender, Daniel W. Hook, Karta Kooner |date=2009-12-31|eprint=1001.0131 |class=hep-th}}</ref>

===[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] and classical physics===
Predictions of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] have been verified experimentally to an extremely high degree of accuracy.<ref>See, for example, [[Precision tests of QED]]. The relativistic refinement of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] known as [[quantum electrodynamics]] (QED) has been shown to agree with experiment to within 1 part in 10<sup>8</sup> for some atomic properties.</ref> According to the [[correspondence principle]] between classical and [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], all objects obey the laws of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], and classical mechanics is just an approximation for large systems of objects (or a statistical [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] of a large collection of particles).<ref name="Tipler">{{cite book|last1=Tipler|first1=Paul|last2=Llewellyn|first2=Ralph|title=Modern Physics|edition=5|year=2008|publisher=[[W. H. Freeman and Company]]|isbn=978-0-7167-7550-8|pages=160–161}}</ref> The laws of classical mechanics thus follow from the laws of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] as a statistical average at the limit of large systems or large [[quantum number]]s.<ref>{{cite web|url=http://www.scribd.com/doc/5998949/Quantum-mechanics-course-iwhatisquantummechanics |title=[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] course iwhatisquantummechanics |publisher=Scribd.com |date=2008-09-14 |accessdate=2012-08-18}}</ref> However, [[Chaos theory|chaotic systems]] do not have good quantum numbers, and [[quantum chaos]] studies the relationship between classical and quantum descriptions in these systems.

[[Quantum coherence]] is an essential difference between classical and quantum theories as illustrated by the [[EPR paradox|Einstein–Podolsky–Rosen (EPR) paradox]] — an attack on a certain philosophical interpretation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] by an appeal to [[local realism]].<ref>A. Einstein, B. Podolsky, and N. Rosen, [http://www.drchinese.com/David/EPR.pdf ''Can quantum-mechanical description of physical reality be considered complete?''] Phys. Rev. '''47''' 777 (1935). [http://prola.aps.org/abstract/PR/v47/i10/p777_1]</ref> [[Quantum interference#Quantum interference|Quantum interference]] involves adding together ''[[probability amplitude]]s'', 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 long-range entanglement and other nonlocal phenomena characteristic of quantum systems.<ref>{{cite web|url=http://philsci-archive.pitt.edu/2328/1/handbook.pdf |title=Between classical and quantum� |format=PDF |date= |accessdate=2012-08-19}}</ref> 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.<ref>(see [[macroscopic quantum phenomena]], [[Bose–Einstein condensate]], and [[Quantum machine]])</ref> This is in accordance with the following observations:
* 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 [[molecule]]s 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 charge]]s under the rules of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]].<ref>{{cite web|url=http://academic.brooklyn.cuny.edu/physics/sobel/Nucphys/atomprop.html |title=Atomic Properties |publisher=Academic.brooklyn.cuny.edu |date= |accessdate=2012-08-18}}</ref>
* While the seemingly "exotic" behavior of matter posited by [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] and ਰਿਲੇਟੀਵਿਟੀ 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 "[[Isaac Newton|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 [[Speed of light|velocity of light]].<ref>http://assets.cambridge.org/97805218/29526/excerpt/9780521829526_excerpt.pdf</ref>

===Copenhagen interpretation of quantum versus classical kinematics===

A big difference between classical and [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] is that they use very different kinematic descriptions.<ref>[[Max Born|Born, M.]], [[Werner Heisenberg|Heisenberg, W.]], [[Pascual Jordan|Jordan, P.]] (1926). ''Z. Phys.'' '''35''': 557–615. Translated as 'On [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] {{math|II}}', pp. 321–385 in [[Bartel Leendert van der Waerden|Van der Waerden, B.L.]] (1967), ''Sources of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]'', North-Holland, Amsterdam, "The basic difference between the theory proposed here and that used hitherto ... lies in the characteristic kinematics ...", p. 385.</ref>

In [[Niels Bohr]]'s mature view, quantum mechanical phenomena are required to be experiments, with complete descriptions of all the devices for the system, preparative, intermediary, and finally measuring. The descriptions are in macroscopic terms, expressed in ordinary language, supplemented with the concepts of classical mechanics.<ref>[[Paul Dirac|Dirac, P.A.M.]] (1930/1958). ''The Principles of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]'', fourth edition, Oxford University Press, Oxford UK, p. 5: "A question about what will happen to a particular photon under certain conditions is not really very precise. To make it precise one must imagine some experiment performed having a bearing on the question, and enquire what will be the result of the experiment. Only questions about the results of experiments have a real significance and it is only such questions that theoretical physics has to consider."</ref><ref>[[Niels Bohr|Bohr, N.]] (1939). The Causality Problem in Atomic Physics, in ''New Theories in Physics, Conference organized in collaboration with the International Union of Physics and the Polish Intellectual Co-operation Committee, Warsaw, May 30th – June 3rd 1938'', International Institute of Intellectual Co-operation, Paris, 1939, pp. 11–30, reprinted in ''Niels Bohr, Collected Works'', volume 7 (1933 – 1958) edited by J. Kalckar, Elsevier, Amsterdam, ISBN 0-444-89892-1, pp. 303–322. "The essential lesson of the analysis of measurements in quantum theory is thus the emphasis on the necessity, in the account of the phenomena, of taking the whole experimental arrangement into consideration, in complete conformity with the fact that all unambiguous interpretation of the quantum mechanical formalism involves the fixation of the external conditions, defining the initial state of the atomic system and the character of the possible predictions as regards subsequent observable properties of that system. Any measurement in quantum theory can in fact only refer either to a fixation of the initial state or to the test of such predictions, and it is first the combination of both kinds which constitutes a well-defined phenomenon."</ref><ref>[[Niels Bohr|Bohr, N.]] (1948). On the notions of complementarity and causality, ''Dialectica'' '''2''': 312–319. "As a more appropriate way of expression, one may advocate limitation of the use of the word ''phenomenon'' to refer to observations obtained under specified circumstances, including an account of the whole experiment."</ref><ref>Ludwig, G. (1987). ''An Axiomatic Basis for [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]'', volume 2, ''[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] and Macrosystems'', translated by K. Just, Springer, Berlin, ISBN 978-3-642-71899-1, Chapter {{math|XIII}}, Special Structures in Preparation and Registration Devices, §1, Measurement chains, p. 132.</ref> The initial condition and the final condition of the system are respectively described by values in a configuration space, for example a position space, or some equivalent space such as a momentum space. [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] does not admit a completely precise description, in terms of both position and momentum, of an initial condition or "state" (in the classical sense of the word) that would support a precisely deterministic and causal prediction of a final condition.<ref name="Heisenberg 1927"/><ref>[[Herbert S. Green|Green, H.S.]] (1965). ''Matrix Mechanics'', with a foreword by [[Max Born]], P. Noordhoff Ltd, Groningen. "It is not possible, therefore, to provide 'initial conditions' for the prediction of the behaviour of atomic systems, in the way contemplated by classical physics. This is accepted by quantum theory, not merely as an experimental difficulty, but as a fundamental law of nature", p. 32.</ref> In this sense, advocated by Bohr in his mature writings, a quantum phenomenon is a process, a passage from initial to final condition, not an instantaneous "state" in the classical sense of that word.<ref>[[Léon Rosenfeld|Rosenfeld, L.]] (1957). Misunderstandings about the foundations of quantum theory, pp. 41–45 in ''Observation and Interpretation'', edited by S. Körner, Butterworths, London. "A phenomenon is therefore a process (endowed with the characteristic quantal wholeness) involving a definite type of interaction between the system and the apparatus."</ref><ref>[[Paul Dirac|Dirac, P.A.M.]] (1973). Development of the physicist's conception of nature, pp. 1–55 in ''The Physicist's Conception of Nature'', edited by J. Mehra, D. Reidel, Dordrecht, ISBN 90-277-0345-0, p. 5: "That led Heisenberg to his really masterful step forward, resulting in the new [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]. His idea was to build up a theory entirely in terms of quantities referring to two states."</ref> Thus there are two kinds of processes in [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]: stationary and transitional. For a stationary process, the initial and final condition are the same. For a transition, they are different. Obviously by definition, if only the initial condition is given, the process is not determined.<ref name="Heisenberg 1927">[[Werner Heisenberg|Heisenberg, W.]] (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, ''Z. Phys.'' '''43''': 172–198. Translation as 'The actual content of quantum theoretical kinematics and mechanics' [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19840008978.pdf here], "But in the rigorous formulation of the law of causality, — "If we know the present precisely,  we can calculate the future" — it is not the conclusion that is faulty, but the premise."</ref> Given its initial condition, prediction of its final condition is possible, causally but only probabilistically, because the Schrödinger equation is deterministic for wave function evolution, but the wave function describes the system only probabilistically.<ref>[[Max Born|Born, M.]] (1927). Physical aspects of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], ''Nature'' '''119''': 354–357, "These probabilities are thus dynamically determined. But what the system actually does is not determined ..."</ref><ref>[[Albert Messiah|Messiah, A.]] (1961). ''[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]'', volume 1, translated by G.M. Temmer from the French ''Mécanique Quantique'', North-Holland, Amsterdam, p. 157.</ref>

For many experiments, it is possible to think of the initial and final conditions of the system as being a particle. In some cases it appears that there are potentially several spatially distinct pathways or trajectories by which a particle might pass from initial to final condition. It is an important feature of the quantum kinematic description that it does not permit a unique definite statement of which of those pathways is actually followed. Only the initial and final conditions are definite, and, as stated in the foregoing paragraph, they are defined only as precisely as allowed by the configuration space description or its equivalent. In every case for which a quantum kinematic description is needed, there is always a compelling reason for this restriction of kinematic precision. An example of such a reason is that for a particle to be experimentally found in a definite position, it must be held motionless; for it to be experimentally found to have a definite momentum, it must have free motion; these two are logically incompatible.<ref>[[Niels Bohr|Bohr, N.]] (1928). The Quantum postulate and the recent development of atomic theory, ''Nature'' '''121''': 580–590.</ref><ref>[[Werner Heisenberg|Heisenberg, W.]] (1930). ''The Physical Principles of the Quantum Theory'', translated by C. Eckart and F.C. Hoyt, University of Chicago Press.</ref>

Classical kinematics does not primarily demand experimental description of its phenomena. It allows completely precise description of an instantaneous state by a value in phase space, the Cartesian product of configuration and momentum spaces. This description simply assumes or imagines a state as a physically existing entity without concern about its experimental measurability. Such a description of an initial condition, together with Newton's laws of motion, allows a precise deterministic and causal prediction of a final condition, with a definite trajectory of passage. [[Hamiltonian mechanics|Hamiltonian]] dynamics can be used for this. Classical kinematics also allows the description of a process analogous to the initial and final condition description used by [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]. [[Lagrangian mechanics]] applies to this.<ref>[[Herbert Goldstein|Goldstein, H.]] (1950). ''Classical Mechanics'', Addison-Wesley, ISBN 0-201-02510-8.</ref> For processes that need account to be taken of actions of a small number of [[Planck's constant|Planck constants]], classical kinematics is not adequate; [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] is needed.

===ਰਿਲੇਟੀਵਿਟੀ and [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]===
{{Main|Relativistic [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]}}

Even with the defining postulates of both Einstein's theory of general ਰਿਲੇਟੀਵਿਟੀ 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.<ref>"There is as yet no logically consistent and complete relativistic quantum field theory.", p. 4. &nbsp;— V. B. Berestetskii, [[Evgeny Lifshitz|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 0-08-016025-5</ref>

Einstein himself is well known for rejecting some of the claims of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]. While clearly contributing to the field, he did not accept many of the more "philosophical consequences and interpretations" of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], 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 [[Quantum entanglement|entanglement]], and used them to formulate the [[Einstein–Podolsky–Rosen paradox]] in the hope of showing that [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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 nonlocality|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.

According to the paper of J. Bell and the [[Copenhagen interpretation]]&mdash;the common interpretation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] by physicists since 1927 - and contrary to Einstein's ideas, [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] was ''not'', at the same time a "realistic" theory and  a "[[Principle of locality|local]]" theory.

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 high-security commercial applications in banking and government.

Gravity is negligible in many areas of particle physics, so that unification between general ਰਿਲੇਟੀਵਿਟੀ and [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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|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 interaction|fundamental forces of nature]] - the [[strong interaction|strong force]], [[electromagnetism]], the [[weak interaction|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).<ref>{{cite web|url=http://www.damtp.cam.ac.uk/strings02/dirac/hawking/|title=Stephen Hawking; Gödel and the end of physics|author=|date=|work=cam.ac.uk|accessdate=11 September 2015}}</ref>

===Attempts at a unified field theory===
{{Main|Grand unified theory}}
The quest to unify the [[fundamental force]]s through [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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 ਰਿਲੇਟੀਵਿਟੀ,<ref>{{cite web|url=https://books.google.com/books?id=6a-agBFWuyQC&pg=PA61&lpg=PA61&dq=most+accurately+tested+physical+theory+pulsar&source=bl&ots=rsrPvowECD&sig=S9MaY509U0pOgF86h2uvcpQQ_wU&hl=en&sa=X&ei=TwkTT6j9M8LXiAKSwNjTDQ&sqi=2&ved=0CE0Q6AEwBw#v=onepage&q=most%20accurately%20tested%20physical%20theory%20pulsar&f=false|title=The Nature of Space and Time|author=|date=|work=google.com|accessdate=11 September 2015}}</ref><ref>
{{Cite journal
 |author=Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, Makiko Nio
 |year=2012
 |title=Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant
 |journal=[[Physical Review Letters]]
 |volume=109
 |issue=11 |page=111807
 |doi=10.1103/PhysRevLett.109.111807
 |url=http://link.aps.org/doi/10.1103/PhysRevLett.109.111807
 |arxiv = 1205.5368v2 |bibcode = 2012PhRvL.109k1807A }}</ref> 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<sup>14</sup> GeV the three aforementioned forces are fused into a single unified field.<ref>{{cite book|last=Parker |first=B. |year=1993 |title=Overcoming some of the problems |pages=259–279 |publisher= |isbn=}}</ref> 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<sup>19</sup> GeV. However&nbsp;— and while special ਰਿਲੇਟੀਵਿਟੀ is parsimoniously incorporated into quantum electrodynamics&nbsp;— the expanded [[general ਰਿਲੇਟੀਵਿਟੀ]], 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 [[M-theory]], which is an attempt at describing the supersymmetrical based [[string theory]]. M-theory posits that our apparent 4-dimensional [[spacetime]] is, in reality, actually an 11-dimensional 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.

Another popular theory is [[Loop quantum gravity]] (LQG), a theory first proposed by [[Carlo Rovelli]] that describes the quantum properties of gravity. It is also a theory of [[quantum space]] and [[Chronon|quantum time]], because in general ਰਿਲੇਟੀਵਿਟੀ the geometry of spacetime is a manifestation of [[gravity]]. LQG is an attempt to merge and adapt standard [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] and standard [[general ਰਿਲੇਟੀਵਿਟੀ]]. 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<sup>−35</sup> 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.

==Philosophical implications==
{{Main|Interpretations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]}}

Since its inception, the many [[counter-intuitive]] aspects and results of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] have provoked strong [[philosophy|philosophical]] debates and many [[interpretations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]|interpretations]]. Even fundamental issues, such as [[Max Born]]'s basic [[Born rule|rules]] concerning [[probability amplitude]]s and [[probability distribution]]s, took decades to be appreciated by society and many leading scientists. [[Richard Feynman]] once said, "I think I can safely say that nobody understands [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]."<ref>The Character of Physical Law (1965) Ch. 6; also quoted in The New Quantum Universe (2003), by Tony Hey and Patrick Walters</ref>  According to [[Steven Weinberg]], "There is now in my opinion no entirely satisfactory interpretation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]."<ref>Weinberg, S. [http://arxiv.org/abs/1109.6462 "Collapse of the State Vector"], Phys. Rev. A 85, 062116 (2012).</ref>

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 [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] 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 well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the [[complementarity (physics)|conjugate]] nature of evidence obtained under different experimental situations.

[[Albert Einstein]], himself one of the founders of quantum theory, [[Bohr-Einstein debates|rejected]] the quantum theoretical doctrine that the state of a system depends on the experimental arrangement for its measurement. He held that underlying [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] there should be a theory that thoroughly and directly expresses the rule against [[action at a distance]]; in other words, he insisted on the [[principle of locality]]. He inferred that the present theory was incomplete, contrary to the Copenhagen doctrine that it is complete. He therefore produced a series of objections, the most famous of which has become known as the [[Einstein–Podolsky–Rosen paradox]].

[[John Stewart Bell|John Bell]] showed that this "EPR" paradox led to [[Bell's theorem|experimentally testable differences]] between [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] and theories that rely on added hidden variables. [[Bell test experiments|Experiments]] have been performed confirming the accuracy of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], thereby demonstrating that [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] cannot be improved upon by addition of hidden variables.<ref>{{cite web|url=http://plato.stanford.edu/entries/qm-action-distance/ |title=Action at a Distance in [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] (Stanford Encyclopedia of Philosophy) |publisher=Plato.stanford.edu |date=2007-01-26 |accessdate=2012-08-18}}</ref> The ''[[Bohr-Einstein debates]]'' provide a vibrant critique of the Copenhagen Interpretation from an [[epistemological]] point of view.

The [[Everett many-worlds interpretation]], formulated in 1956, holds that ''all'' the possibilities described by quantum theory ''simultaneously'' occur in a [[multiverse]] composed of mostly independent parallel universes.<ref>{{cite web|url=http://plato.stanford.edu/entries/qm-everett/ |title=Everett's Relative-State Formulation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] (Stanford Encyclopedia of Philosophy) |publisher=Plato.stanford.edu |date= |accessdate=2012-08-18}}</ref> This is not accomplished by introducing some "new axiom" to [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], 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 [[Quantum entanglement|entangled state]]. While the multiverse is deterministic, we perceive non-deterministic 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 Stewart Bell|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 [[Quantum entanglement|entangled]] with ''both'' the physicist who measured it ''and'' a huge number of other particles, some of which are [[photon]]s 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 test]]s, Cramer (1986) formulated his [[transactional interpretation]].<ref name="Cramer">[http://www.npl.washington.edu/npl/int_rep/tiqm/TI_toc.html The Transactional Interpretation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]] by John Cramer. ''Reviews of Modern Physics'' 58, 647-688, July (1986)</ref> [[Relational [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]]] appeared in the late 1990s as the modern derivative of the [[Copenhagen Interpretation]].

==Applications==
[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] has had enormous<ref>See, for example, [[the Feynman Lectures on Physics]] for some of the technological applications which use  [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], e.g., [[transistor]]s (vol '''III''', pp. 14-11 ff), [[integrated circuit]]s, which are follow-on technology in solid-state physics (vol '''II''', pp. 8-6), and [[laser]]s (vol '''III''', pp. 9-13).</ref> success in explaining many of the features of our universe. [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] is often the only tool available that can reveal the individual behaviors of the [[subatomic particle]]s that make up all forms of matter ([[electron]]s, [[proton]]s, [[neutron]]s, [[photon]]s, and others). [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] has strongly influenced [[string theory|string theories]], candidates for a [[theory of everything|Theory of Everything]] (see [[reductionism]]).

[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] is also critically important for understanding how individual atoms combine covalently to form [[molecules]]. The application of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] to [[chemistry]] is known as [[quantum chemistry]]. Relativistic [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] can, in principle, mathematically describe most of chemistry. [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] can also provide quantitative insight into [[Ionic bond|ionic]] and [[covalent bonding]] processes by explicitly showing which molecules are energetically favorable to which others and the magnitudes of the energies involved.<ref>{{cite book|url=https://books.google.com/books?id=vdXU6SD4_UYC |title=Introduction to [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] with Applications to Chemistry - Linus Pauling, E. Bright Wilson |publisher= |date= 1985-03-01|accessdate=2012-08-18|isbn=9780486648712}}</ref> Furthermore, most of the calculations performed in modern [[computational chemistry]] rely on [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]].

[[File:Rtd seq v3.gif|thumb|350px|right| A working mechanism of a resonant tunneling diode device, based on the phenomenon of quantum tunneling through potential barriers]]

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 [[Integrated circuit|microchip]]), the [[electron microscope]], and [[Magnetic Resonance Imaging|magnetic resonance imaging]] (MRI). The study of [[semiconductor]]s led to the invention of the [[diode]] and the [[transistor]], which are indispensable parts of modern [[electronics]] systems and devices.

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 computer]]s, which are expected to perform certain computational tasks exponentially faster than classical [[computer]]s. Instead of using classical bits, quantum computers use [[qubits]], which can be in [[quantum superposition|superpositions]] of states. Another active research topic is [[quantum teleportation]], which deals with techniques to transmit quantum information over arbitrary distances.

[[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 flash drive|USB drive]]s use quantum tunneling to erase their memory cells.

While [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] primarily applies to the smaller atomic regimes of matter and energy, some systems exhibit [[Mechanics#Classical versus quantum|quantum mechanical effects]] on a large scale. [[Superfluidity]], the frictionless flow of a liquid at temperatures near [[absolute zero]], is one well-known 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 theory also provides accurate descriptions for many previously unexplained phenomena, such as [[black-body radiation]] and the stability of the [[Atomic orbital|orbitals]] of electrons in atoms. It has also given insight into the workings of many different [[biological systems]], including [[smell receptors]] and [[protein structure]]s.<ref>{{cite web|last=Anderson |first=Mark |url=http://discovermagazine.com/2009/feb/13-is-quantum-mechanics-controlling-your-thoughts/article_view?b_start:int=1&-C |title=Is [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] Controlling Your Thoughts? &#124; Subatomic Particles |publisher=DISCOVER Magazine |date=2009-01-13 |accessdate=2012-08-18}}</ref> Recent work on [[photosynthesis]] has provided evidence that quantum correlations play an essential role in this fundamental process of plants and many other organisms.<ref>{{cite web|url=http://physicsworld.com/cws/article/news/41632 |title=[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] boosts photosynthesis |publisher=physicsworld.com |date= |accessdate=2010-10-23}}</ref> Even so, [[classical physics]] can often provide good approximations to results otherwise obtained by [[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]], typically in circumstances with large numbers of particles or large [[quantum number]]s. 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 [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] insignificant.
{{Clear}}

==Examples==

===Free particle===
For example, consider a [[free particle]]. In [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], 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 wave function 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 wave function, 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 ([[Distribution (mathematics)|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.<ref>{{Cite book
|title=[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], Second edition
|first1=P. C. W.
|last1=Davies
|first2=David S.
|last2=Betts
|publisher=Chapman and Hall
|year=1984
|isbn=0-7487-4446-0
|page=79
|url=https://books.google.com/books?id=XRyHCrGNstoC&pg=PA79}}, [https://books.google.com/books?id=XRyHCrGNstoC&pg=PA79 Chapter 6, p. 79]
</ref>
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]].<ref>{{cite book|url=https://books.google.com/books?id=tKm-Ekwke_UC |title=The Future of Complexity: Conceiving a Better Way to Understand Order and Chaos|first= Peter|last= Baofu|publisher= |date= 2007-12-31|accessdate=2012-08-18|isbn=9789812708991}}</ref>
[[File:QuantumDot wf.gif|thumb|600px|center|3D confined electron wave functions for each eigenstate in a Quantum Dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more ‘s-type’ and ‘p-type’. However, in a triangular dot, the wave functions are mixed due to confinement symmetry.]]

===Step potential===
{{Main|Solution of Schrödinger equation for a step potential}}
[[File:Qm step pot temp.png|thumb|right|400px|Scattering at a finite potential step of height ''V''<sub>0</sub>, shown in green. The amplitudes and direction of left- and right-moving waves are indicated. Yellow is the incident wave, blue are reflected and transmitted waves, red does not occur. ''E'' > ''V''<sub>0</sub> for this figure.]]

The potential in this case is given by:

:<math>V(x)= \begin{cases} 0, & x < 0, \\ V_0, & x \ge 0. \end{cases} </math>

The solutions are superpositions of left- and right-moving waves:

:<math>\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 </math>
:<math>\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</math>

where the [[wave vector]]s are related to the energy via

:<math>k_1=\sqrt{2m E/\hbar^2}</math>, and
:<math>k_2=\sqrt{2m (E-V_0)/\hbar^2}</math>

with coefficients A and B determined from the [[boundary condition]]s and by imposing a continuous [[derivative]] on the solution.

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.

===Rectangular potential barrier===
{{Main|Rectangular potential barrier}}
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 microscope|scanning tunneling microscopy]]. Quantum tunneling is central to physical phenomena involved in [[superlattice]]s.

===Particle in a box===
[[File:Infinite potential well.svg|thumb|1-dimensional potential energy box (or infinite potential well)]]
{{Main|Particle in a box}}
The particle in a one-dimensional 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 one-dimensional case in the <math>x</math> direction, the time-independent Schrödinger equation may be written<ref>Derivation of particle in a box, [http://chemistry.tidalswan.com/index.php?title=Quantum_Mechanics chemistry.tidalswan.com]</ref>

: <math> - \frac {\hbar ^2}{2m} \frac {d ^2 \psi}{dx^2} = E \psi.</math>

With the differential operator defined by

: <math> \hat{p}_x = -i\hbar\frac{d}{dx} </math>

the previous equation is evocative of the [[Kinetic energy#Kinetic energy of rigid bodies|classic kinetic energy analogue]],

: <math> \frac{1}{2m} \hat{p}_x^2 = E,</math>

with state <math>\psi</math> in this case having energy <math>E</math> coincident with the kinetic energy of the particle.

The general solutions of the Schrödinger equation for the particle in a box are

: <math> \psi(x) = A e^{ikx} + B e ^{-ikx} \qquad\qquad E =  \frac{\hbar^2 k^2}{2m}</math>

or, from [[Euler's formula]],

: <math> \psi(x) = C \sin kx + D \cos kx.\!</math>

The infinite potential walls of the box determine the values of ''C'', ''D'', and ''k'' at {{nowrap|''x'' {{=}} 0}} and {{nowrap|''x'' {{=}} ''L''}} where {{nowrap|''ψ''}} must be zero. Thus, at {{nowrap|''x'' {{=}} 0}},

:<math>\psi(0) = 0 = C\sin 0 + D\cos 0 = D\!</math>

and {{nowrap|''D'' {{=}} 0}}. At {{nowrap|''x'' {{=}} ''L''}},

:<math> \psi(L) = 0 = C\sin kL.\!</math>

in which ''C'' cannot be zero as this would conflict with the Born interpretation. Therefore, since {{nowrap|sin(''kL'') {{=}} 0}}, ''kL'' must be an integer multiple of [[Pi|π]],

:<math>k = \frac{n\pi}{L}\qquad\qquad n=1,2,3,\ldots.</math>

The quantization of energy levels follows from this constraint on ''k'', since

:<math>E = \frac{\hbar^2 \pi^2 n^2}{2mL^2} = \frac{n^2h^2}{8mL^2}.</math>

===Finite potential well===
{{Main|Finite potential well}}
A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth.

The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well.

===Harmonic oscillator===
{{Main|Quantum harmonic oscillator}}
[[File:QuantumHarmonicOscillatorAnimation.gif|thumb|300px|right|Some trajectories of a [[harmonic oscillator]] (i.e. a ball attached to a [[Hooke's law|spring]]) in [[classical mechanics]] (A-B) and [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] (C-H). In [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], the position of the ball is represented by a [[wave]] (called the [[wave function]]), with the [[real part]] shown in blue and the [[imaginary part]] shown in red. Some of the trajectories (such as C,D,E,and F) are [[standing wave]]s (or "[[stationary state]]s"). Each standing-wave frequency is proportional to a possible [[energy level]] of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have ''any'' energy.]]

As in the classical case, the potential for the quantum harmonic oscillator is given by

:<math>V(x)=\frac{1}{2}m\omega^2x^2</math>

This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The [[eigenstate]]s are given by

:<math>  \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. </math>

where ''H<sub>n</sub>'' are the [[Hermite polynomials]],

:<math>H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{-x^2}\right)</math>

and the corresponding energy levels are
:<math> E_n = \hbar \omega \left(n + {1\over 2}\right)</math>.

This is another example illustrating the quantization of energy for bound states.

==See also==

{{Portal|Physics}}
* [[Angular momentum diagrams ([[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]])]]
* [[EPR paradox]]
* [[Fractional [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]]]
* [[List of quantum-mechanical systems with analytical solutions]]
* [[Macroscopic quantum phenomena]]
* [[Phase space formulation]]
* [[Regularization (physics)]]
* [[Spherical basis]]

==Notes==
{{Reflist|colwidth=30em}}

==References==
{{Refbegin}}
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 [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]''. John Wiley. ISBN 0-486-42878-8
* {{cite book | first1= Brian | last1= Cox | first2= Jeff | last2= Forshaw | title= [[The Quantum Universe|The Quantum Universe: Everything That Can Happen Does Happen]]:  | publisher = Allen Lane | year = 2011 | isbn= 1-84614-432-9 }}
* [[Richard Feynman]], 1985. ''[[QED: The Strange Theory of Light and Matter]]'', [[Princeton University Press]]. ISBN 0-691-08388-6. Four elementary lectures on [[quantum electrodynamics]] and [[quantum field theory]], yet containing many insights for the expert.
* [[Giancarlo Ghirardi|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]]: 110-76.
* [[Victor Stenger]], 2000. ''Timeless Reality: Symmetry, Simplicity, and Multiple Universes''. Buffalo NY: Prometheus Books. Chpts. 5-8.  Includes [[cosmological]] and [[philosophical]] considerations.
More technical:
* [[Bryce DeWitt]], R. Neill Graham, eds., 1973. ''The Many-Worlds Interpretation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]'', Princeton Series in Physics, [[Princeton University Press]]. ISBN 0-691-08131-X
* {{cite book |last=Dirac |first=P. A. M. |authorlink=P. A. M. Dirac |year=1930 |title= [[Principles of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]|The Principles of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]]]|isbn=0-19-852011-5}} The beginning chapters make up a very clear and comprehensible introduction.
* [[Hugh Everett]], 1957, "Relative State Formulation of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]", ''Reviews of Modern Physics'' 29: 454-62.
* {{cite book |last1=Feynman |first1=Richard P. |authorlink1=Richard Feynman |last2=Leighton |first2=Robert B. |authorlink2=Robert B. Leighton |last3=Sands |first3=Matthew |year=1965 |title=[[The Feynman Lectures on Physics]] |volume=1-3 |publisher=[[Addison-Wesley]] |isbn=0-7382-0008-5}}
* {{cite book | author=Griffiths, David J.| title=Introduction to [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] (2nd ed.) | publisher=Prentice Hall |year=2004 |isbn=0-13-111892-7 | oclc=40251748}} A standard undergraduate text.
* [[Max Jammer]], 1966. ''The Conceptual Development of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]''. McGraw Hill.
* [[Hagen Kleinert]], 2004. ''Path Integrals in [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]], Statistics, Polymer Physics, and Financial Markets'', 3rd ed. Singapore: World Scientific. [http://www.physik.fu-berlin.de/~kleinert/b5 Draft of 4th edition.]
* Gunther Ludwig, 1968. ''Wave Mechanics''. London: Pergamon Press. ISBN 0-08-203204-1
* [[George Mackey]] (2004). ''The mathematical foundations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]''. Dover Publications. ISBN 0-486-43517-2.
* [[Albert Messiah]], 1966. ''[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]'' (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III.
* {{cite book | author=[[Roland Omnès|Omnès, Roland]] | title=Understanding [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] | publisher=Princeton University Press |year=1999 |isbn=0-691-00435-8 | oclc=39849482}}
* 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 [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]. ISBN 0-19-530573-6
* {{cite book | author=[[Transnational College of Lex]]| title=What is [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]? A Physics Adventure | publisher=Language Research Foundation, Boston |year=1996 |isbn=0-9643504-1-6 | oclc=34661512}}
* {{cite book |last=von Neumann |first=John |authorlink=John von Neumann |year=1955 |title=Mathematical Foundations of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] |publisher=Princeton University Press |isbn=0-691-02893-1}}
* [[Hermann Weyl]], 1950. ''The Theory of Groups and [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]'', Dover Publications.
* D. Greenberger, K. Hentschel, F. Weinert, eds., 2009. ''Compendium of [[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]], Concepts, experiments, history and philosophy'', Springer-Verlag, Berlin, Heidelberg.
{{Refend}}

==Further reading==
* {{cite book | author=Bernstein, Jeremy|title=Quantum Leaps| publisher=Belknap Press of Harvard University Press | location=Cambridge, Massachusetts|year=2009|isbn=978-0-674-03541-6|url=https://books.google.com/books?id=j0Me3brYOL0C&printsec=frontcover}}
* {{cite book | author=[[David Bohm|Bohm, David]] | title=Quantum Theory | publisher=[[Dover Publications]] | year=1989 | isbn=0-486-65969-0}}
* {{cite book | author=Eisberg, Robert; [[Robert Resnick|Resnick, Robert]] | title=[[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]] of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.) | publisher=Wiley | year=1985 | isbn=0-471-87373-X}}
* {{cite book | author=[[Liboff, Richard L.]] | title=Introductory [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] | publisher=Addison-Wesley | year=2002 | isbn=0-8053-8714-5}}
* {{cite book | author=Merzbacher, Eugen | title=[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] | publisher=Wiley, John & Sons, Inc | year=1998 | isbn=0-471-88702-1}}
* {{cite book | author=[[J. J. Sakurai|Sakurai, J. J.]] | title=Modern [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]| publisher=Addison Wesley | year=1994 | isbn=0-201-53929-2}}
* {{cite book | author=Shankar, R. | title=Principles of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] | publisher=Springer | year=1994| isbn=0-306-44790-8}}
* {{cite book | author=Stone, A. Douglas |title=Einstein and the Quantum | publisher=Princeton University Press | year=2013 | isbn=978-0-691-13968-5}}
* [[Martinus J. G. Veltman]], 2003  ''Facts and Mysteries in Elementary Particle Physics''.
* {{cite book | author=Shushi, Tomer|title=The Influence of Particle Interactions on the Existence of Quantum Particles Properties| publisher=Journal of Physical Science and Application | location=Haifa, Israel|year=2014|url=http://www.davidpublisher.org/Public/uploads/Contribute/55c84ab660c47.pdf}}

==External links==
{{Sister project links|s=[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]}}
* [http://www.toutestquantique.fr/ 3D animations, applications and research for basic quantum effects] (animations also available in [https://commons.wikimedia.org/wiki/Special:Contributions/Jubobroff commons.wikimedia.org] (Université paris Sud))
* [http://oyc.yale.edu/sites/default/files/notes_quantum_cookbook.pdf Quantum Cook Book] by R. Shankar, Open Yale PHYS 201 material (4pp)
* [http://www.lightandmatter.com/lm/ The Modern Revolution in Physics] - an online textbook.
* J. O'Connor and E. F. Robertson: [http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/The_Quantum_age_begins.html A history of [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]].]
* [http://www.quantiki.org/wiki/index.php/Introduction_to_Quantum_Theory Introduction to Quantum Theory at Quantiki.]
* [http://bethe.cornell.edu/ [[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]] Made Relatively Simple]: three video lectures by [[Hans Bethe]]
* [http://www.nonlocal.com/hbar/ H is for h-bar.]
* [http://www.freebookcentre.net/Physics/Quantum-Mechanics-Books.html [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] Books Collection]: Collection of free books
;Course material
* [http://qpdb.eu [[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]] Database - Fundamentals and Historical Background of Quantum Theory.]
* [http://arxiv.org/abs/quant-ph/0605180 Doron Cohen: Lecture notes in [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] (comprehensive, with advanced topics).]
* [[MIT OpenCourseWare]]: [http://ocw.mit.edu/OcwWeb/Chemistry/index.htm Chemistry].
* MIT OpenCourseWare: [http://ocw.mit.edu/OcwWeb/Physics/index.htm Physics]. See [http://ocw.mit.edu/OcwWeb/Physics/8-04Spring-2006/CourseHome/index.htm 8.04]
* [https://www.youtube.com/stanford#g/c/84C10A9CB1D13841 Stanford Continuing Education PHY 25: [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]] by [[Leonard Susskind]], see [http://continuingstudies.stanford.edu/courses/course.php?cid=20072_PHY%2025 course description] Fall 2007
* [http://www.physics.csbsju.edu/QM/ 5½ Examples in [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]]
* [http://www.imperial.ac.uk/quantuminformation/qi/tutorials Imperial College [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] Course.]
* [http://www.sparknotes.com/testprep/books/sat2/physics/chapter19section3.rhtml Spark Notes - [[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]].]
* [http://www.quantum-physics.polytechnique.fr/ [[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]] Online : interactive introduction to [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] (RS applets).]
* [http://www.didaktik.physik.uni-erlangen.de/quantumlab/english/index.html Experiments to the foundations of [[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]] with single photons.]
* [http://www.nanohub.org/topics/AQME AQME] : Advancing [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]] for Engineers&nbsp;— by T.Barzso, D.Vasileska and G.Klimeck online learning resource with simulation tools on [[nanohub]]
* [http://www.lsr.ph.ic.ac.uk/~plenio/lecture.pdf [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]] by Martin Plenio
* [http://farside.ph.utexas.edu/teaching/qm/389.pdf [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]] by Richard Fitzpatrick
* [http://nanohub.org/resources/2039 Online course on ''Quantum Transport'']

;FAQs
* [http://www.hedweb.com/manworld.htm Many-worlds or relative-state interpretation.]
* [http://www.mtnmath.com/faq/meas-qm.html Measurement in [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]].]

;Media
* [http://oyc.yale.edu/physics/phys-201#sessions PHYS 201: Fundamentals of Physics II] by Ramamurti Shankar, Open Yale Course
* [https://www.youtube.com/view_play_list?p=84C10A9CB1D13841 Lectures on [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]] by [[Leonard Susskind]]
* [http://www.newscientist.com/channel/fundamentals/quantum-world Everything you wanted to know about the quantum world]&nbsp;— archive of articles from ''[[New Scientist]]''.
* [http://www.sciencedaily.com/news/matter_energy/quantum_physics/ [[ਕੁਆਂਟਮ ਫਿਜ਼ਿਕਸ]] Research] from ''[[Science Daily]]''
* {{cite news|url=http://www.nytimes.com/2005/12/27/science/27eins.html?scp=1&sq=quantum%20trickery&st=cse|title=Quantum Trickery: Testing Einstein's Strangest Theory|date=December 27, 2005|work=The New York Times | first=Dennis | last=Overbye | accessdate=April 12, 2010}}
* [http://www.astronomycast.com/physics/ep-138-quantum-mechanics/ Audio: Astronomy Cast] [[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]&nbsp;— June 2009. [[Fraser Cain]] interviews [[Pamela L. Gay]].

;Philosophy
* {{sep|qm|"[[ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ]]"|Jenann Ismael}}
* {{sep|qt-measurement|"Measurement in Quantum Theory"|Henry Krips}}

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