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{{Quantum book backlink|Measurement and information}}

In [[quantum mechanics]], '''wave function collapse''', also called '''reduction of the state vector''', is the process by which a [[wave function]]—initially in a [[quantum superposition]] of several [[eigenstate]]s—reduces to a single eigenstate when a measurement yields a definite outcome.<ref name="Grundlagen">{{cite book
|author=J. von Neumann
|year=1955
|title=Mathematical Foundations of Quantum Mechanics
|publisher=Princeton University Press
}}</ref> Collapse is one of the two ways quantum systems are commonly described as evolving in time; the other is the continuous deterministic evolution governed by the [[Schrödinger equation]].<ref name="Grundlagen" />

[[File:Wave-Collapse.png|thumb|400px|right|Visualization of quantum measurement regimes: POVM allows non-orthogonal outcomes, weak measurement minimally perturbs the state, and projective measurement collapses the state onto an eigenbasis.]]

== Mathematical description ==

A quantum state may be expanded in a basis of eigenstates <math>\{|\phi_i\rangle\}</math> of an observable:

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

Here the coefficients <math>c_i</math> are [[probability amplitude]]s, given by

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

When the observable is measured, the state is postulated to collapse to one of the eigenstates:

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

The probability that the outcome corresponding to <math>|\phi_i\rangle</math> is obtained is

<math display="block">
P(i)=|c_i|^2,
</math>

with normalization

<math display="block">
\sum_i |c_i|^2 = 1.
</math>

This collapse postulate is introduced to account for the fact that an immediately repeated measurement gives the same result.<ref name="GriffithsSchroeter">{{Cite book |last1=Griffiths |first1=David J. |last2=Schroeter |first2=Darrell F. |title=Introduction to Quantum Mechanics |edition=3 |publisher=Cambridge University Press |year=2018 |isbn=978-1-107-18963-8}}</ref>

== Physical meaning ==

Wave function collapse connects the probabilistic quantum description with definite measurement outcomes such as position, momentum, or spin.<ref name="Hall">{{Cite book |last=Hall |first=Brian C. |title=Quantum Theory for Mathematicians |publisher=Springer |year=2013 |isbn=978-1-4614-7115-8 |page=68}}</ref> For an individual event, only one outcome is observed, even though the pre-measurement state may be a superposition of many possibilities.

Examples include the [[double-slit experiment]], where each particle is detected at a definite location although many events build up an interference pattern, and the [[Stern–Gerlach experiment]], where each atom is observed in one of the discrete spin channels.<ref name="Bach2013">{{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 | volume=15 | issue=3 | year=2013 | article-number=033018 | doi=10.1088/1367-2630/15/3/033018 }}</ref>

== The measurement problem ==

The Schrödinger equation predicts a continuous evolution containing all possible outcomes in superposition, but an actual measurement yields only one definite result. This tension is known as the [[measurement problem]] of quantum mechanics.<ref name="Zurek2003">{{Cite journal |last=Zurek |first=Wojciech Hubert |title=Decoherence, einselection, and the quantum origins of the classical |journal=Reviews of Modern Physics |volume=75 |issue=3 |pages=715–775 |year=2003 |doi=10.1103/RevModPhys.75.715}}</ref>

To make predictions, orthodox quantum mechanics combines unitary evolution with the [[Born rule]] and the collapse postulate. Although this framework is extremely successful experimentally, the physical status of collapse remains debated.<ref name="Susskind">{{Cite book |last1=Susskind |first1=Leonard |last2=Friedman |first2=Art |title=Quantum Mechanics: The Theoretical Minimum |publisher=Basic Books |year=2014 |isbn=978-0-465-06290-4}}</ref>

== Interpretations ==

Different interpretations of quantum mechanics treat collapse in different ways.

* In the [[Copenhagen interpretation]], collapse is taken as a fundamental part of measurement theory.
* In the [[many-worlds interpretation]], collapse does not occur; instead, all outcomes persist in different branches.
* In [[objective-collapse theory]], collapse is treated as a real physical process.
* In approaches based on [[quantum decoherence]], interaction with the environment explains why classical alternatives appear, though decoherence by itself does not select a single outcome.<ref name="Schlosshauer">{{cite journal |last=Schlosshauer |first=Maximilian |title=Decoherence, the measurement problem, and interpretations of quantum mechanics |journal=Reviews of Modern Physics |volume=76 |issue=4 |pages=1267–1305 |year=2005 |doi=10.1103/RevModPhys.76.1267}}</ref>

== History ==

The idea of wave function reduction appeared early in the development of quantum mechanics. [[Werner Heisenberg]] used it in 1927 in discussing quantum measurement, and [[John von Neumann]] gave it a systematic mathematical formulation in 1932.<ref name="Kiefer">{{Cite book |last=Kiefer |first=Claus |chapter=On the Interpretation of Quantum Theory — from Copenhagen to the Present Day |title=Time, Quantum and Information |publisher=Springer |year=2003 |pages=291–299 |doi=10.1007/978-3-662-10557-3_19}}</ref>

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Measurement collapse|1}}

Physics:Quantum Measurement collapse - Revision history

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