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|language=de}}<br>
:{{cite book
|author=J. von Neumann.
|year=1955
|title=Mathematical Foundations of Quantum Mechanics
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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>
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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.<ref name="Bach Pope Liou Batelaan 2013 p=033018">{{cite journal | last1=Bach | first1=Roger | last2=Pope | first2=Damian | last3=Liou | first3=Sy-Hwang | last4=Batelaan | first4=Herman | title=Controlled double-slit electron diffraction | journal=New Journal of Physics | publisher=IOP Publishing | volume=15 | issue=3 | date=2013-03-13 | issn=1367-2630 | doi=10.1088/1367-2630/15/3/033018 | page=033018 | arxiv=1210.6243 | bibcode=2013NJPh...15c3018B | s2cid=832961 | url=https://iopscience.iop.org/article/10.1088/1367-2630/15/3/033018}}</ref> 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.<ref name=GriffithsSchroeter3rd/>{{rp|17}}
==Terminology==
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