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| {{quantum mechanics}}
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| In [[quantum mechanics]], '''wave function collapse''' is the phenomenon in which a [[wave function]]—initially in a [[quantum superposition|superposition]] of several [[eigenstates]]—appears to reduce to a single eigenstate after interaction with a measuring apparatus.<ref name=Griffiths>{{cite book|last=Griffiths|first=David J.|title=Introduction to Quantum Mechanics, 2e|year=2005|publisher=Pearson Prentice Hall|location=Upper Saddle River, NJ|isbn=0131118927|pages=106–109}}</ref> It is the essence of [[measurement in quantum mechanics]], and connects the wave function with classical [[observable]]s like [[position]] and [[momentum]]. Collapse is one of two processes by which [[quantum system]]s evolve in time; the other is continuous evolution via the [[Schrödinger equation]].<ref name="Grundlagen">
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| {{cite book
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| |author=J. von Neumann
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| |year=1932
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| |title=Mathematische Grundlagen der Quantenmechanik
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| |publisher=[[Springer (publisher)|Springer]]
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| |location=Berlin
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| }} {{De icon}}<br>
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| :{{cite book
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| |author=J. von Neumann
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| |year=1955
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| |title=Mathematical Foundations of Quantum Mechanics
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| |publisher=[[Princeton University Press]]
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| }} {{En icon}}</ref> However in this role, collapse is merely a [[black box]] for [[Reversible process (thermodynamics)|thermodynamically irreversible]] interaction with a classical environment.<ref name=Schlosshauer>{{cite journal|last=Schlosshauer|first=Maximilian|title=Decoherence, the measurement problem, and interpretations of quantum mechanics|journal=Rev. Mod. Phys.|year=2005|volume=76|issue=4|pages=1267–1305|doi=10.1103/RevModPhys.76.1267|url=http://rmp.aps.org/abstract/RMP/v76/i4/p1267_1|accessdate=28 February 2013|arxiv = quant-ph/0312059 |bibcode = 2004RvMP...76.1267S }}</ref> Calculations of [[quantum decoherence]] predict ''apparent'' wave function collapse when a superposition forms between the quantum system's states and the environment's states. Significantly, the combined wave function of the system and environment continue to obey the [[Schrödinger equation]].<ref name=Zurek>{{cite journal|last=Zurek|first=Wojciech Hubert|title=Quantum Darwinism|journal=Nature Physics|year=2009|volume=5|pages=181–188|doi=10.1038/nphys1202|url=http://www.nature.com/nphys/journal/v5/n3/full/nphys1202.html|accessdate=28 February 2013|arxiv = 0903.5082 |bibcode = 2009NatPh...5..181Z }}</ref>
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| When the [[Copenhagen interpretation]] was first expressed, [[Niels Bohr]] postulated wave function collapse to cut the [[quantum mechanics|quantum world]] from the [[classical mechanics|classical]].<ref>{{cite journal|last=Bohr|first=N|title=The quantum postulate and the recent development of atomic theory.|journal=Nature|year=1928|volume=121|pages=580–590|accessdate=28 February 2013|bibcode = 1928Natur.121..580B |doi = 10.1038/121580a0 }}</ref> This tactical move allowed quantum theory to develop without distractions from interpretational worries. Nevertheless it was debated, for if collapse were a fundamental physical phenomenon, rather than just the [[epiphenomenon]] of some other process, it would mean nature were fundamentally [[stochastic]], i.e. [[nondeterministic]], an undesirable property for a theory.<ref name=Schlosshauer/><ref>{{cite web
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| |author=L. Bombelli
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| |date=
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| |url=http://www.phy.olemiss.edu/~luca/Topics/qm/collapse.html
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| |title=Wave-Function Collapse in Quantum Mechanics
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| |work=Topics in Theoretical Physics
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| |accessdate=2010-10-13
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| }}</ref> This issue remained until quantum decoherence entered mainstream opinion after its reformulation in the 1980s.<ref name=Schlosshauer/><ref name="Zurek"/><ref>{{cite web
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| |author=M. Pusey, J. Barrett, T. Rudolph
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| |year=2012
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| |url=http://arxiv.org/pdf/1111.3328v2.pdf
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| |title=On the reality of the quantum state
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| }}</ref> Decoherence explains the perception of wave function collapse in terms of interacting large- and small-scale quantum systems, and is commonly taught at the graduate level (e.g. the [[Claude Cohen-Tannoudji|Cohen-Tannoudji]] textbook).<ref>
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| {{cite book
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| |author=C. Cohen-Tannoudji
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| |year=1973; revised 2006
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| |title=Quantum Mechanics (2 volumes)
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| |publisher=Wiley
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| |pages=22
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| |location=New York
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| }}{{en icon}}</ref> The [[quantum filtering]] approach<ref name=Belavkin79>{{cite techreport
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| | author = V. P. Belavkin
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| | title = Optimal Measurement and Control in Quantum Dynamical Systems
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| | institute = Preprint Instytut Fizyki
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| | publisher = Copernicus University, Torun
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| | number = 411
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| | pages = 3-38
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| | year = 1979
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| | arxiv = quant-ph/0208108}}</ref><ref name=Belavkin92>{{cite journal
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| | author = V. P. Belavkin
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| | title = Quantum stochastic calculus and quantum nonlinear filtering
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| | journal = Journal of Multivariate Analysis
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| | volume = 42
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| | number = 2
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| | year = 1992
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| | pages = 171–201
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| | doi = 10.1016/0047-259X(92)90042-E
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| | arxiv = math/0512362}}</ref><ref name=Belavkin99>{{cite journal
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| | author = V. P. Belavkin
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| | title = Measurement, filtering and control in quantum open dynamical systems
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| | journal = Reports on Mathematical Physics
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| | volume = 43
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| | number = 3
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| | pages = A405–A425
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| | year = 1999
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| | doi = 10.1016/S0034-4877(00)86386-7
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| | arxiv = quant-ph/0208108|bibcode = 1999RpMP...43..405B }}</ref> and the introduction of quantum causality non-demolition principle<ref name=Belavkin94>{{cite journal
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| | author = V. P. Belavkin
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| | title= Nondemolition principle of quantum measurement theory
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| | journal = Foundations of Physics
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| | volume = 24
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| | number = 5
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| | year = 1994
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| | pages = 685–714
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| | doi = 10.1007/BF02054669
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| | arxiv = quant-ph/0512188 |bibcode = 1994FoPh...24..685B }}</ref> allows for a classical-environment derivation of wave function collapse from the stochastic [[Schrödinger equation]].
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| ==Mathematical description==
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| Before collapse, the [[wave function]] may be any [[square-integrable]] function. This function is expressible as a linear combination of the [[eigenstate]]s of any [[observable]]. Observables represent [[classical mechanics|classical]] dynamical variables, and when one is measured by a [[Observer (quantum mechanics)|classical observer]], the wave function is [[vector projection|projected]] onto a random eigenstate of that observable. The observer simultaneously measures the classical value of that observable to be the [[eigenvalue]] of the final state.<ref name="Griffiths"/>
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| ===Mathematical background===
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| {{about||an explanation of the notation used|Bra–ket notation|details on this formalism|quantum state}}
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| The [[quantum state]] of a physical system is described by a wave function (in turn – an element of a [[projective space|projective]] [[Hilbert space]]). This can be expressed in Dirac or [[bra-ket notation]] as a vector:
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| :<math> | \psi \rangle = \sum_i c_i | \phi_i \rangle .</math>
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| The kets <math>\scriptstyle { | \phi_1 \rangle, | \phi_2 \rangle, | \phi_3 \rangle \cdots } </math>, specify the different quantum "alternatives" available - a particular quantum state. They form an [[orthonormal]] [[eigenvector]] [[basis (linear algebra)|basis]], formally
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| :<math>\langle \phi_i | \phi_j \rangle = \delta_{ij}.</math> | |
| Where <math>\delta_{ij}</math> represents the [[Kronecker delta]].
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| An observable (i.e. measurable parameter of the system) is associated with each eigenbasis, with each quantum alternative having a specific value or [[eigenvalue]], ''e''<sub>i</sub>, of the observable. A "measurable parameter of the system" could be the usual position '''r''' and the momentum '''p''' of (say) a particle, but also its energy ''E'', z-components of spin (''s<sub>z</sub>''), orbital (''L<sub>z</sub>'') and total angular (''J<sub>z</sub>'') momenta etc. In the basis representation these are respectively <math>\scriptstyle { | \mathbf{r},t \rangle = | x,t \rangle + | y,t \rangle + | z,t \rangle, | \mathbf{p},t \rangle = | p_x,t \rangle + | p_y,t \rangle + | p_z,t \rangle, | E \rangle, | s_z \rangle, | L_z \rangle, | J_z \rangle, \cdots } </math>.
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| The coefficients ''c''<sub>1</sub>, ''c''<sub>2</sub>, ''c''<sub>3</sub>... are the [[probability amplitude]]s corresponding to each basis <math>\scriptstyle { | \phi_1 \rangle, | \phi_2 \rangle, | \phi_3 \rangle \cdots } </math>. These are [[complex numbers]]. The [[Absolute value#Definition and properties|moduli square]] of ''c<sub>i</sub>'', that is |''c<sub>i</sub>''|<sup>2</sup> = ''c<sub>i</sub>''*''c<sub>i</sub>'' (* denotes [[complex conjugate]]), is the probability of measuring the system to be in the state <math>\scriptstyle | \phi_i \rangle </math>.
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| For simplicity in the following, all wave functions are assumed to be [[Normalizable wave function|normalized]]; the total probability of measuring all possible states is unity:
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| :<math>\langle \psi|\psi \rangle = \sum_i |c_i|^2 = 1.</math>
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| === The process of collapse ===
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| With these definitions it is easy to describe the process of collapse. For any observable, the wave function is initially some [[linear combination]] of the eigenbasis <math>\{ |\phi_i\rangle \}</math> of that observable. When an external agency (an observer, experimenter) measures the observable associated with the eigenbasis <math>\{| \phi_i \rangle\}</math>, the wave function ''collapses'' from the full <math>| \psi \rangle</math> to just ''one'' of the basis eigenstates, <math>| \phi_i \rangle</math>, that is:
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| : <math>|\psi\rangle \rightarrow |\phi_i\rangle.</math>
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| The probability of collapsing to a given eigenstate <math>| \phi_k \rangle</math> is the [[Born probability]], <math>P_k=| c_k |^2 </math>. Post-measurement, other elements of the wave function vector, <math>c_{i \neq k}</math>, have "collapsed" to zero, and <math>c_k=1</math>.
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| More generally, collapse is defined for an operator <math>\hat{Q}</math> with eigenbasis <math>\{|\phi_i\rang\}</math>. If the system is in state <math>|\psi\rang</math>, and <math>\hat{Q}</math> is measured, the probability of collapsing the system to state <math>|\phi_i\rang</math> (and measuring <math>| \phi_i \rang</math>) would be <math>|\lang\psi|\phi_i\rang|^2</math>. Note that this is ''not'' the probability that the particle is in state <math>| \phi_i \rangle</math>; it is in state <math>|\psi\rang</math> until cast to an eigenstate of <math>\hat{Q}</math>.
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| However, we never observe collapse to a single eigenstate of a continuous-spectrum operator (e.g. [[position operator|position]], [[momentum operator|momentum]], or a [[free particle|scattering]] [[Hamiltonian operator|Hamiltonian]]), because such eigenfunctions are non-normalizable. In these cases, the wave function will partially collapse to a linear combination of "close" eigenstates (necessarily involving a spread in eigenvalues) that embodies the imprecision of the measurement apparatus. The more precise the measurement, the tighter the range. Calculation of probability proceeds identically, except with an integral over the expansion coefficient <math>c (q, t) dq</math>.<ref name=GriffithsEigenfunctions>{{cite book|last=Griffiths|first=David J.|title=Introduction to Quantum Mechanics, 2e|year=2005|publisher=Pearson Prentice Hall|location=Upper Saddle River, NJ|isbn=0131118927|pages=100–105}}</ref> This phenomenon is unrelated to the [[uncertainty principle]], although increasingly precise measurements of one operator (e.g. position) will naturally homogenize the expansion coefficient of wave function with respect to another, [[Observable#Incompatibility_of_observables_in_quantum_mechanics|incompatible]] operator (e.g. momentum), lowering the probability of measuring any particular value of the latter.
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| ===The determination of preferred-basis===
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| {{fringe-section|date=December 2013}}
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| The complete set of [[orthogonal functions]] which a [[wave function]] will collapse to is also called preferred-basis.<ref name=Schlosshauer/> There lacks theoretical foundation for the preferred-basis to be the [[eigenstate]]s of [[observable]]s such as [[position operator|position]], [[momentum operator|momentum]], etc. In fact the [[eigenstate]]s of [[position operator|position]] are not even physical due to the infinite energy associated with them. A better way to obtain the preferred-basis is to derive them from a basic principle that [[wave function]] evolves continuously. Since [[Schrödinger equation]] is supposed to govern the evolution of [[wave function]] once a collapse process completes, the collapse equation needs to end at [[Schrödinger equation]]. It is proved that only appropriate basis functions are able to make the collapse equation to end at [[Schrödinger equation]].<ref>{{cite web
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| |author=S. Mei
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| |year=2013
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| |url=http://arxiv.org/pdf/1311.4405v1.pdf
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| |title=on the origin of preferred-basis and evolution pattern of wave function
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| }}</ref> Those functions are, e.g., energy [[eigenfunction]]s for isolated sub-systems or quasi-position [[eigenfunction]]s for sub-systems that at the end of the collapse interact with other objects by approximate [[algebraic function]]s of distance in the system [[Hamiltonian operator|Hamiltonian]].
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| In [[quantum decoherence]], an important [[Einselection|einselected]] [[eigen basis|basis]] is the set of [[eigenstate]]s of [[position operator|position]]. Quasi-position eigenstates such as those for a collapse process are not considered as valid [[Einselection|einselected]] [[eigen basis|basis]]. If this claim is correct, then there must be principles that validate [[position operator|position]] [[eigenstate]]s but invalidate quasi-position eigenstates. Unfortunately such principles are yet to be discovered. On the other side, wave function collapse may be fundamental, and its preferred-basis is also the [[Einselection|einselected]] [[eigen basis|basis]].
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| ===Quantum decoherence===
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| {{Main|Quantum decoherence#Mathematical details}}
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| Wave function collapse is not fundamental from the perspective of [[quantum decoherence]].<ref name="zurek03">[[Wojciech H. Zurek]], Decoherence, [[einselection]], and the quantum origins of the classical,''Reviews of Modern Physics'' 2003, 75, 715 or http://arxiv.org/abs/quant-ph/0105127</ref> There are several equivalent approaches to deriving collapse, like the [[quantum decoherence#Density matrix approach|density matrix approach]], but each has the same effect: decoherence irreversibly converts the "averaged" or "environmentally traced over" density matrix from a pure state to a reduced mixture, giving the appearance of wave function collapse.
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| ==History and context== | |
| The concept of wavefunction collapse was introduced by [[Werner Heisenberg]] in his 1927 paper on the [[uncertainty principle]], "Über den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik", and incorporated into the [[mathematical formulation of quantum mechanics]] by [[John von Neumann]], in his 1932 treatise ''Mathematische Grundlagen der Quantenmechanik''.<ref>{{cite arxiv |author=C. Kiefer |year=2002 |title=On the interpretation of quantum theory – from Copenhagen to the present day |class=quant-ph |eprint=quant-ph/0210152 }}</ref> Consistent with Heisenberg, von Neumann postulated that there were two processes of wave function change:
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| # The [[probability|probabilistic]], non-[[unitary transformation|unitary]], [[local realism|non-local]], discontinuous change brought about by observation and [[quantum measurement|measurement]], as outlined above.
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| # The [[deterministic]], unitary, continuous [[time evolution]] of an isolated system that obeys the [[Schrödinger equation]] (or a relativistic equivalent, i.e. the [[Dirac equation]]).
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| In general, quantum systems exist in [[quantum superposition|superpositions]] of those basis states that most closely correspond to classical descriptions, and, in the absence of measurement, evolve according to the Schrödinger equation. However, when a measurement is made, the wave function collapses—from an observer's perspective—to just one of the basis states, and the property being measured uniquely acquires the eigenvalue of that particular state, <math>\lambda_i</math>. After the collapse, the system again evolves according to the Schrödinger equation.
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| By explicitly dealing with the [[Measurement in quantum mechanics#Wavefunction collapse|interaction of object and measuring instrument]], von Neumann<ref name="Grundlagen"/> has attempted to create consistency of the two processes of wave function change.
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| He was able to prove the ''possibility'' of a quantum mechanical measurement scheme consistent with wave function collapse. However, he did not prove the ''necessity'' of such a collapse. Although von Neumann's projection postulate is often presented as a normative description of quantum measurement, it was conceived by taking into account experimental evidence available during the 1930s (in particular the [[Mathematical formulation of quantum mechanics#The problem of measurement|Compton-Simon experiment]] has been paradigmatic), and many important [[Measurement in quantum mechanics#Wavefunction collapse|present-day measurement procedures]] do not satisfy it (so-called measurements of the second kind).<ref>
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| {{cite book
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| |author=W. Pauli
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| |year=1958
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| |chapter=Die allgemeinen Prinzipien der Wellenmechanik
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| |editor=S. Flügge
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| |title=Handbuch der Physik
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| |volume=V |page=73
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| |publisher=[[Springer-Verlag]]
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| |location=[[Berlin]]
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| }} {{De icon}}</ref><ref>
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| {{cite journal
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| |author=L. Landau and R. Peierls
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| |year=1931
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| |title=Erweiterung des Unbestimmtheitsprinzips für die relativistische Quantentheorie|journal=[[Zeitschrift fur Physik]]
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| |volume=69
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| |issue=1-2 |pages=56
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| |doi=10.1007/BF01391513
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| |bibcode = 1931ZPhy...69...56L }} {{De icon}})</ref><ref>Discussions of measurements of the second kind can be found in most treatments on the foundations of quantum mechanics, for instance, {{cite book
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| |author=J. M. Jauch
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| |year=1968
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| |title=Foundations of Quantum Mechanics
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| |publisher=[[Addison-Wesley]]
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| |page=165
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| |isbn=
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| }}; {{cite book
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| |author=B. d'Espagnat
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| |year=1976
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| |title=Conceptual Foundations of Quantum Mechanics
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| |publisher=[[W. A. Benjamin]]
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| |pages=18, 159
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| |isbn=
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| }}; and {{cite book
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| |author=W. M. de Muynck
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| |year=2002
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| |title=Foundations of Quantum Mechanics: An Empiricist Approach
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| |page=section 3.2.4 |nopp=yes
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| |publisher=[[Kluwer Academic Publishers]]
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| |isbn=
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| }}.</ref>
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| The existence of the wave function collapse is required in
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| * the [[Copenhagen interpretation]]
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| * the [[objective collapse interpretation]]s
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| * the [[transactional interpretation]]
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| * the [[Interpretation of quantum mechanics#von Neumann/Wigner interpretation: consciousness causes the collapse|von Neumann interpretation]] in which [[consciousness causes collapse]].
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| On the other hand, the collapse is considered a redundant or optional approximation in
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| * the [[Consistent histories]] approach, self-dubbed "Copenhagen done right"
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| * the [[Bohm interpretation]]
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| * the [[Many-worlds interpretation]]
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| * the [[Ensemble Interpretation]]
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| The cluster of phenomena described by the expression ''wave function collapse'' is a fundamental problem in the interpretation of quantum mechanics, and is known as the [[measurement problem]]. The problem is deflected by the Copenhagen Interpretation, which postulates that this is a special characteristic of the "measurement" process. [[Hugh Everett III|Everett]]'s [[many-worlds interpretation]] deals with it by discarding the collapse-process, thus reformulating the relation between measurement apparatus and system in such a way that the linear laws of quantum mechanics are universally valid; that is, the only process according to which a quantum system evolves is governed by the Schrödinger equation or some [[theory of relativity|relativistic]] equivalent.
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| Originating from Everett's theory, but no longer tied to it, is the physical process of [[decoherence]], which causes an ''apparent'' collapse. Decoherence is also important for the [[consistent histories]] interpretation. A general description of the evolution of quantum mechanical systems is possible by using [[density matrix|density operators]] and [[quantum operation]]s. In this formalism (which is closely related to the [[C*-algebra]]ic formalism) the collapse of the wave function corresponds to a non-unitary quantum operation.
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| The significance ascribed to the wave function varies from interpretation to interpretation, and varies even within an interpretation (such as the Copenhagen Interpretation). If the wave function merely encodes an observer's knowledge of the universe then the wave function collapse corresponds to the receipt of new information. This is somewhat analogous to the situation in classical physics, except that the classical "wave function" does not necessarily obey a wave equation. If the wave function is physically real, in some sense and to some extent, then the collapse of the wave function is also seen as a real process, to the same extent.
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| ==See also==
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| *[[Arrow of time]]
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| *[[Interpretation of quantum mechanics]]
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| *[[Quantum decoherence]]
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| *[[Quantum interference]]
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| *[[Schrödinger's cat]]
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| *[[Zeno effect]]
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| ==References==
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| {{reflist}}
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| [[Category:Quantum measurement]]
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| [[Category:Concepts in physics]]
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