Editing Wave function collapse (section)

==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 state#Pure 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===
The [[complex number|complex]] coefficients <math>\{c_{i}\}</math> in the expansion of a quantum state in terms of eigenstates <math>\{| \phi_i \rangle\}</math>, 
<math display=block> | \psi \rangle = \sum_i c_i | \phi_i \rangle.</math>
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 value#Complex 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>

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}}