Editing Observable (section)

=== Compatible and incompatible observables in quantum mechanics ===
A crucial difference between classical quantities and quantum mechanical observables is that some pairs of quantum observables may not be simultaneously measurable, a property referred to as [[complementarity (physics)|complementarity]]. This is mathematically expressed by non-[[commutativity]] of their corresponding operators, to the effect that the [[commutator (physics)|commutator]]
<math display="block">\left[\hat{A}, \hat{B}\right] := \hat{A}\hat{B} - \hat{B}\hat{A} \neq \hat{0}.</math>

This inequality expresses a dependence of measurement results on the order in which measurements of observables <math>\hat{A}</math> and <math>\hat{B}</math> are performed. A measurement of <math>\hat{A}</math> alters the quantum state in a way that is incompatible with the subsequent measurement of <math>\hat{B}</math> and vice versa.

Observables corresponding to commuting operators are called ''compatible observables''. For example, momentum along say the <math>x</math> and <math>y</math> axes are compatible.  Observables corresponding to non-commuting operators are called ''incompatible observables'' or ''complementary variables''. For example, the position and momentum along the same axis are incompatible.<ref name=messiah>{{Cite book|last=Messiah|first=Albert|authorlink = Albert Messiah|title=Quantum Mechanics|date=1966|publisher=North Holland, John Wiley & Sons|isbn=0486409244|language=en}}</ref>{{rp|155}} 

Incompatible observables cannot have a complete set of common [[eigenfunction]]s. Note that there can be some simultaneous eigenvectors of <math>\hat{A}</math> and <math>\hat{B}</math>, but not enough in number to constitute a complete [[basis (vector space)|basis]].<ref>{{Cite book|last=Griffiths|first=David J.|authorlink = David J. Griffiths|url=https://books.google.com/books?id=0h-nDAAAQBAJ|title=Introduction to Quantum Mechanics|date=2017|publisher=Cambridge University Press|isbn=978-1-107-17986-8|pages=111|language=en}}</ref>{{sfn | Cohen-Tannoudji | Diu | Laloë | 2019 | p=232}}