Wave function collapse: Difference between revisions

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Calculations of [[quantum decoherence]] show that when a quantum system interacts with the environment, the superpositions ''apparently'' reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation throughout this ''apparent'' collapse.<ref name=Zurek>{{cite journal |last=Zurek |first=Wojciech Hubert |title=Quantum Darwinism |journal=Nature Physics |year=2009 |volume=5 |issue=3 |pages=181–188 |doi=10.1038/nphys1202 |arxiv = 0903.5082 |bibcode = 2009NatPh...5..181Z |s2cid=119205282}}</ref> More importantly, this is not enough to explain ''actual'' wave function collapse, as decoherence does not reduce it to a single eigenstate.<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 |arxiv = quant-ph/0312059 |bibcode = 2004RvMP...76.1267S |s2cid=7295619}}</ref><ref name="Stanford1">{{cite encyclopedia

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==Mathematical description==

{{About||an explanation of the notation used|Bra–ket notation|details on this formalism|Quantum state}}

In quantum mechanics each measurable physical quantity of a quantum system is called an [[observable]] which, for example, could be the position <math>r</math> and the momentum <math>p</math> but also energy <math>E</math>, <math>z</math> components of spin (<math>s_{z}</math>), and so on. The observable acts as a [[linear mapping|linear function]] on the states of the system; its eigenvectors correspond to the quantum state (i.e. [[Quantum_stateQuantum state#Pure_states_of_wave_functionsPure states of wave functions|eigenstate]]) and the [[eigenvalue]]s to the possible values of the observable. The collection of eigenstates/eigenvalue pairs represent all possible values of the observable. Writing <math>\phi_i</math> for an eigenstate and <math>c_i</math> for the corresponding observed value, any arbitrary state of the quantum system can be expressed as a vector using [[bra–ket notation]]:

<math display=block> | \psi \rangle = \sum_i c_i | \phi_i \rangle.</math>

The kets <math>\{| \phi_i \rangle\}</math> 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===

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,<ref name=GriffithsSchroeter3rd>{{Cite book |last=Griffiths |first=David J. |title=Introduction to quantum mechanics |last2=Schroeter |first2=Darrell F. |date=2018 |publisher=Cambridge University Press |isbn=978-1-107-18963-8 |edition=3 |___location=Cambridge ; New York, NY}}</ref>{{rp|566|q=to account for the fact that an immediately repeated measurement yields the same result, we are forced to assume that the act of measurement collapses the wave function,}} abruptly converting an arbitrary state into a single component eigenstate of the observable:

:<math> | \psi \rangle = \sum_i c_i | \phi_i \rangle \rightarrow |\psi'\rangle = |\phi_i\rangle.</math>

where the arrow represents a measurement of the observable corresponding to the <math>\phi</math> basis.<ref>{{Cite book |last=Hall |first=Brian C. |title=Quantum theory for mathematicians |date=2013 |publisher=Springer |isbn=978-1-4614-7115-8 |series=Graduate texts in mathematics |___location=New York |page=68}}</ref>

For any single event, only one eigenvalue is measured, chosen randomly from among the possible values.

===Meaning of the expansion coefficients===

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can be written as an (complex) overlap of the corresponding eigenstate and the quantum state:

<math display=block> c_i = \langle \phi_i | \psi \rangle .</math>

They are called the [[probability amplitude]]s. The [[Absolute_valueAbsolute value#Complex_numbersComplex numbers|square modulus]] <math>|c_{i}|^{2}</math> is the probability that a measurement of the observable yields the eigenstate <math>| \phi_i \rangle</math>. The sum of the probability over all possible outcomes must be one:<ref>{{cite book|last=Griffiths|first=David J.|title=Introduction to Quantum Mechanics, 2e|year=2005|publisher=Pearson Prentice Hall|___location=Upper Saddle River, New Jersey|isbn=0131118927|pages=107}}</ref>

:<math>\langle \psi|\psi \rangle = \sum_i |c_i|^2 = 1.</math>

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==Terminology==

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.<ref name=messiah>{{Cite book|last=Messiah|first=Albert|title=Quantum Mechanics|date=1966|publisher=North Holland, John Wiley & Sons|isbn=0486409244|language=en}}</ref>{{rp|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".<ref name=messiah/>{{rp|324}} When the wave function representation is used, the "reduction" is called "wave function collapse".

== The measurement problem ==

The [[Schrödinger equation]] describes quantum systems but does not describe their measurement. Solution to the equations 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.<ref>{{Cite journal |last=Zurek |first=Wojciech Hubert |date=2003-05-22 |title=Decoherence, einselection, and the quantum origins of the classical |url=https://link.aps.org/doi/10.1103/RevModPhys.75.715 |journal=Reviews of Modern Physics |language=en |volume=75 |issue=3 |pages=715–775 |doi=10.1103/RevModPhys.75.715 |issn=0034-6861|arxiv=quant-ph/0105127 }}</ref> Despite the widespread quantitative success of these postulates scientists remain dissatisfied and have sought more detailed physical models. Rather than suspending the SchrodingerSchrödinger equation during the process of measurement, the measurement apparatus should be included and governed by the laws of quantum mechanics.<ref>{{Cite book |last=Susskind |first=Leonard |title=Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics] |last2=Friedman |first2=Art |last3=Susskind |first3=Leonard |date=2014 |publisher=Basic Books |isbn=978-0-465-06290-4 |series=The theoretical minimum / Leonard Susskind and George Hrabovsky |___location=New York, NY}}</ref>{{rp|127}}

==Physical approaches to collapse==

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".<ref name=FuchsPeresNo>{{Cite journal |last=Fuchs |first=Christopher A. |last2=Peres |first2=Asher |date=2000-03-01 |title=Quantum Theory Needs No ‘Interpretation’ |url=https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=7596444653d614458ee7aea0422dabfc95ace3e6 |journal=Physics Today |language=en |volume=53 |issue=3 |pages=70–71 |doi=10.1063/1.883004 |issn=0031-9228}}</ref>

Various [[interpretations of quantum mechanics]] attempt to provide a physical model for collapse.<ref name=Stamatescu>{{Cite book |last=Stamatescu |first=Ion-Olimpiu |url=https://link.springer.com/10.1007/978-3-540-70626-7_230 |title=Wave Function Collapse |date=2009 |publisher=Springer Berlin Heidelberg |isbn=978-3-540-70622-9 |editor-last=Greenberger |editor-first=Daniel |___location=Berlin, Heidelberg |pages=813–822 |language=en |doi=10.1007/978-3-540-70626-7_230 |editor-last2=Hentschel |editor-first2=Klaus |editor-last3=Weinert |editor-first3=Friedel}}</ref>{{rp|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 [[Bell test|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 interpretation]]s. 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.<ref name=Stamatescu/>{{rp|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.{{cncitation needed| reason=the ontological wave function literature should be represented, the paragraph based on Stamatescu is too compact now.|date=March 2024}}

===Quantum decoherence===

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==History==

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''.<ref>{{cite arXiv |author=C. Kiefer |year=2002 |title=On the interpretation of quantum theory—from Copenhagen to the present day |eprint=quant-ph/0210152 }}</ref> 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.<ref>{{cite journal |author=G. Jaeger |year=2017 |title="Wave-Packet Reduction" and the Quantum Character of the Actualization of Potentia |journal=Entropy |volume=19 |issue=10 |pages=13

|doi=10.3390/e19100513|bibcode=2017Entrp..19..513J |doi-access=free }}</ref> 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.<ref>{{cite journal|title=Niels Bohr on the wave function and the classical/quantum divide |author=Henrik Zinkernagel |year=2016 |doi=10.1016/j.shpsb.2015.11.001 |journal=Studies in History and Philosophy of Modern Physics |volume=53 |pages=9–19 |arxiv = 1603.00353|bibcode=2016SHPMP..53....9Z |s2cid=18890207 |quote=Among Bohr scholars it is common to assert that Bohr never mentions the wave function collapse (see e.g. Howard, 2004 and Faye, 2008). It is true that in Bohr’s published writings, he does not discuss the status or existence of this standard component in the popular image of the Copenhagen interpretation. }}</ref>

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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.<ref name=SchlosshauerReview/>{{rp|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]].<ref name=SchlosshauerReview/>{{rp|1290}}<ref name=HartleQMCosmology/>{{rp|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.<ref>{{Cite journal |last=Camilleri |first=Kristian |date=2009-12-01 |title=A history of entanglement: Decoherence and the interpretation problem |url=https://www.sciencedirect.com/science/article/pii/S1355219809000562 |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |series=On The History Of The Quantum |volume=40 |issue=4 |pages=290–302 |doi=10.1016/j.shpsb.2009.09.003 |issn=1355-2198}}</ref> 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".<ref name=SchlosshauerReview/>{{rp|1273}} Decoherence has been shown to work very quickly and within a minimal environment, but as yet it has not succeeded in a providing a detailed model replacing the collapse postulate of orthodox quantum mechanics.<ref name=SchlosshauerReview/>{{rp|1302}}

By explicitly dealing with the interaction of object and measuring instrument, von Neumann<ref name="Grundlagen"/> 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.<ref>