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ve collapse quantum logic superdeterminism relational transactional advanced topics relativistic quantum mechanics quantum field theory quantum information science quantum computing quantum chaos decoherence epr paradox density matrix scattering theory quantum statistical mechanics quantum machine learning v t e in various interpretations of quantum mechanics wave function collapse also called reduction of the state vector 1 occurs when a wave function initially in a superposition of several eigenstates reduces to a single eigenstate due to interaction with the external world this interaction is called an observation and is the essence of a measurement in quantum mechanics which connects the wave function with classical observables such as position and momentum collapse is one of the two processes by which quantum systems evolve in time the other is the continuous evolution governed by the schrödinger equation 2 in the copenhagen interpretation wave function collapse connects quantum to classical models with a special role for the observer by contrast objective collapse proposes an origin in physical processes in the many worlds interpretation collapse does not exist all wave function outcomes occur while quantum decoherence accounts for the appearance of collapse historically werner heisenberg was the first to use the idea of wave function reduction to explain quantum measurement 3 4 mathematical description edit for an explanation of the notation used see bra ket notation for details on this formalism see quantum state in quantum mechanics each measurable physical quantity of a quantum system is called an observable which for example could be the position r displaystyle r and the momentum p displaystyle p but also energy e displaystyle e z displaystyle z components of spin s z displaystyle s_ z and so on the observable acts as a linear function on the states of the system its eigenvectors correspond to the quantum state i e eigenstate and the eigenvalues to the possible values of the observable the collection of eigenstates eigenvalue pairs represent all possible values of the observable writing ϕ i displaystyle phi _ i for an eigenstate and c i displaystyle c_ i for the corresponding observed value any arbitrary state of the quantum system can be expressed as a vector using bra ket notation ψ i c i ϕ i displaystyle psi rangle sum _ i c_ i phi _ i rangle the kets ϕ i displaystyle phi _ i rangle specify the different available quantum alternatives i e particular quantum states the wave function is a specific representation of a quantum state wave functions can therefore always be expressed as eigenstates of an observable though the converse is not necessarily true collapse edit to account for the experimental result that repeated measurements of a quantum system give the same results the theory postulates a collapse or reduction of the state vector upon observation 5 566 abruptly converting an arbitrary state into a single component eigenstate of the observable ψ i c i ϕ i ψ ϕ i displaystyle psi rangle sum _ i c_ i phi _ i rangle mapsto psi rangle phi _ i rangle where the arrow represents a measurement of the observable corresponding to the ϕ displaystyle phi basis 6 for any single event only one eigenvalue is measured chosen randomly from among the possible values meaning of the expansion coefficients edit the complex coefficients c i displaystyle c_ i in the expansion of a quantum state in terms of eigenstates ϕ i displaystyle phi _ i rangle ψ i c i ϕ i displaystyle psi rangle sum _ i c_ i phi _ i rangle can be written as an complex overlap of the corresponding eigenstate and the quantum state c i ϕ i ψ displaystyle c_ i langle phi _ i psi rangle they are called the probability amplitudes the square modulus c i 2 displaystyle c_ i 2 is the probability that a measurement of the observable yields the eigenstate ϕ i displaystyle phi _ i rangle the sum of the probability over all possible outcomes must be one 7 ψ ψ i c i 2 1 displaystyle langle psi psi rangle sum _ i c_ i 2 1 as examples individual counts in a double slit experiment with electrons appear at random locations on the detector after many counts are summed the distribution shows a wave interference pattern 8 in a stern gerlach experiment with silver atoms each particle appears in one of two areas unpredictably but the final conclusion has equal numbers of events in each area this statistical aspect of quantum measurements differs fundamentally from classical mechanics in quantum mechanics the only information we have about a system is its wave function and measurements of its wave function can only give statistical information 5 17 terminology edit the two terms reduction of the state vector or state reduction for short and wave function collapse are used to describe the same concept a quantum state is a mathematical description of a quantum system a quantum state vector uses hilbert space vectors for the description 9 159 reduction of the state vector replaces the full state vector with a single eigenstate of the observable the term wave function is typically used for a different mathematical representation of the quantum state one that uses spatial coordinates also called the position representation 9 324 when the wave function representation is used the reduction is called wave function collapse the measurement problem edit the schrödinger equation describes quantum systems but does not describe their measurement solutions to the equation include all possible observable values for measurements but measurements only result in one definite outcome this difference is called the measurement problem of quantum mechanics to predict measurement outcomes from quantum solutions the orthodox interpretation of quantum theory postulates wave function collapse and uses the born rule to compute the probable outcomes 10 despite the widespread quantitative success of these postulates scientists remain dissatisfied and have sought more detailed physical models rather than suspending the schrödinger equation during the process of measurement the measurement apparatus should be included and governed by the laws of quantum mechanics 11 127 physical approaches to collapse edit quantum theory offers no dynamical description of the collapse of the wave function viewed as a statistical theory no description is expected as fuchs and peres put it collapse is something that happens in our description of the system not to the system itself 12 various interpretations of quantum mechanics attempt to provide a physical model for collapse 13 816 three treatments of collapse can be found among the common interpretations the first group includes hidden variable theories like de broglie bohm theory here random outcomes only result from unknown values of hidden variables results from tests of bell s theorem shows that these variables would need to be non local the second group models measurement as quantum entanglement between the quantum state and the measurement apparatus this results in a simulation of classical statistics called quantum decoherence this group includes the many worlds interpretation and consistent histories models the third group postulates additional but as yet undetected physical basis for the randomness this group includes for example the objective collapse interpretations while models in all groups have contributed to better understanding of quantum theory no alternative explanation for individual events has emerged as more useful than collapse followed by statistical prediction with the born rule 13 819 the significance ascribed to the wave function varies from interpretation to interpretation and 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 citation needed quantum decoherence edit main article quantum decoherence quantum decoherence explains why a system interacting with an environment transitions from being a pure state exhibiting superpositions to a mixed state an incoherent combination of classical alternatives 14 this transition is fundamentally reversible as the combined state of system and environment is still pure but for all practical purposes irreversible in the same sense as in the second law of thermodynamics the environment is a very large and complex quantum system and it is not feasible to reverse their interaction decoherence is thus very important for explaining the classical limit of quantum mechanics but cannot explain wave function collapse as all classical alternatives are still present in the mixed state and wave function collapse selects only one of them 15 16 14 the form of decoherence known as environment induced superselection proposes that when a quantum system interacts with the environment the superpositions apparently reduce to mixtures of classical alternatives the combined wave function of the system and environment continue to obey the schrödinger equation throughout this apparent collapse 17 more importantly this is not enough to explain actual wave function collapse as decoherence does not reduce it to a single eigenstate 15 14 history edit the concept of wavefunction collapse was introduced by werner heisenberg in his 1927 paper on the uncertainty principle über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik and incorporated into the mathematical formulation of quantum mechanics by john von neumann in his 1932 treatise mathematische grundlagen der quantenmechanik 4 heisenberg did not try to specify exactly what the collapse of the wavefunction meant however he emphasized that it should not be understood as a physical process 18 niels bohr never mentions wave function collapse in his published work but he repeatedly cautioned that we must give up a pictorial representation despite the differences between bohr and heisenberg their views are often grouped together as the copenhagen interpretation of which wave function collapse is regarded as a key feature 19 john von neumann s influential 1932 work mathematical foundations of quantum mechanics took a more formal approach developing an ideal measurement scheme 20 21 1270 that postulated that there were two processes of wave function change the probabilistic non unitary non local discontinuous change brought about by observation and measurement state reduction or collapse the deterministic unitary continuous time evolution of an isolated system that obeys the schrödinger equation in 1957 hugh everett iii proposed a model of quantum mechanics that dropped von neumann s first postulate everett observed that the measurement apparatus was also a quantum system and its quantum interaction with the system under observation should determine the results he proposed that the discontinuous change is instead a splitting of a wave function representing the universe 21 1288 while everett s approach rekindled interest in foundational quantum mechanics it left core issues unresolved two key issues relate to origin of the observed classical results what causes quantum systems to appear classical and to resolve with the observed probabilities of the born rule 21 1290 20 5 beginning in 1970 h dieter zeh sought a detailed quantum decoherence model for the discontinuous change without postulating collapse further work by wojciech h zurek in 1980 lead eventually to a large number of papers on many aspects of the concept 22 decoherence assumes that every quantum system interacts quantum mechanically with its environment and such interaction is not separable from the system a concept called an open system 21 1273 decoherence has been shown to work very quickly and within a minimal environment but as yet it has not succeeded in providing a detailed model replacing the collapse postulate of orthodox quantum mechanics 21 1302 by explicitly dealing with the interaction of object and measuring instrument von neumann 2 described a quantum mechanical measurement scheme consistent with wave function collapse however he did not prove the necessity of such a collapse von neumann s projection postulate was conceived based on experimental evidence available during the 1930s in particular compton scattering later work refined the notion of measurements into the more easily discussed first kind that will give the same value when immediately repeated and the second kind that give different values when repeated 23 24 25 see also edit arrow of time interpretations of quantum mechanics quantum decoherence quantum interference quantum zeno effect schrödinger s cat stern gerlach experiment universal wave function wave function collapse algorithm references edit penrose roger may 1996 on gravity s role in quantum state reduction general relativity and gravitation 28 5 581 600 bibcode 1996gregr 28 581p doi 10 1007 bf02105068 issn 0001 7701 1 2 j von neumann 1932 mathematische grundlagen der quantenmechanik in german berlin springer j von neumann 1955 mathematical foundations of quantum mechanics princeton university press 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 1 2 kiefer claus 2003 on the interpretation of quantum theory from copenhagen to the present day in castell lutz ischebeck otfried eds time quantum and information berlin heidelberg springer berlin heidelberg pp 291 299 arxiv quant ph 0210152 doi 10 1007 978 3 662 10557 3_19 isbn 978 3 642 07892 7 1 2 griffiths david j schroeter darrell f 2018 introduction to quantum mechanics 3 ed cambridge new york ny cambridge university press isbn 978 1 107 18963 8 hall brian c 2013 quantum theory for mathematicians graduate texts in mathematics new york springer p 68 isbn 978 1 4614 7115 8 griffiths david j 2005 introduction to quantum mechanics 2e upper saddle river new jersey pearson prentice hall p 107 isbn 0 13 111892 7 bach roger pope damian liou sy hwang batelaan herman 2013 03 13 controlled double slit electron diffraction new journal of physics 15 3 033018 iop publishing arxiv 1210 6243 bibcode 2013njph 15c3018b doi 10 1088 1367 2630 15 3 033018 issn 1367 2630 s2cid 832961 1 2 messiah albert 1966 quantum mechanics north holland john wiley sons isbn 0 486 40924 4 zurek wojciech hubert 2003 05 22 decoherence einselection and the quantum origins of the classical reviews of modern physics 75 3 715 775 arxiv quant ph 0105127 bibcode 2003rvmp 75 715z doi 10 1103 revmodphys 75 715 issn 0034 6861 susskind leonard friedma...