Editing Observable

{{Short description|Any entity that can be measured}}
{{About|the use in physics|the use in statistics|Observable variable|the use in control theory|Observability|the use in software engineering|Observer pattern}}

{{More footnotes|date=May 2009}}

In [[physics]], an '''observable''' is a [[physical property]] or [[physical quantity]] that can be [[Measurement|measured]]. In [[classical mechanics]], an observable is a [[real number|real]]-valued "function" on the set of all possible system states, e.g., [[Position (vector)|position]] and [[momentum]]. In [[quantum mechanics]], an observable is an [[quantum operator|operator]], or [[gauge theory|gauge]], where the property of the [[quantum state]]  can be determined by some sequence of [[operational definition|operations]]. For example, these operations might involve submitting the system to various [[electromagnetic field]]s and eventually reading a value.

Physically meaningful observables must also satisfy [[linear map|transformation]] laws that relate observations performed by different [[observation|observer]]s in different [[frames of reference]]. These transformation laws are [[automorphism]]s of the [[state space (physics)|state space]], that is [[bijection|bijective]] [[transformation (mathematics)|transformation]]s that preserve certain mathematical properties of the space in question.

== Quantum mechanics ==

In [[quantum mechanics]], observables manifest as [[self-adjoint operators]] on a [[separable space|separable]] [[complex number|complex]] [[Hilbert space]] representing the [[quantum state space]].{{sfn | Teschl | 2014 | pp=65-66}} Observables assign values to outcomes of ''particular measurements'', corresponding to the [[eigenvalue]] of the operator. If these outcomes represent physically allowable states (i.e. those that belong to the Hilbert space) the eigenvalues are [[real number|real]]; however, the converse is not necessarily true.<ref>See page 20 of [http://bohr.physics.berkeley.edu/classes/221/1112/notes/hilbert.pdf Lecture notes 1 by Robert Littlejohn] {{Webarchive|url=https://web.archive.org/web/20230829114950/https://bohr.physics.berkeley.edu/classes/221/1112/notes/hilbert.pdf|date=2023-08-29}} for a mathematical discussion using the momentum operator as specific example.</ref>{{sfn|de la Madrid Modino|2001|pp=95-97}}<ref>{{cite book |last1=Ballentine |first1=Leslie |title=Quantum Mechanics: A Modern Development |date=2015 |publisher=World Scientific |isbn=978-9814578578 |page=49 |edition=2 |url=https://books.google.com/books?id=2JShngEACAAJ}}</ref> As a consequence, only certain measurements can determine the value of an observable for some state of a quantum system. In classical mechanics, ''any'' measurement can be made to determine the value of an observable.

The relation between the state of a quantum system and the value of an observable requires some [[linear algebra]] for its description.  In the [[mathematical formulation of quantum mechanics]], up to a [[Phase factor|phase constant]], [[pure state]]s are given by non-zero [[vector (geometry)|vector]]s in a [[Hilbert space]] ''V''. Two vectors '''v''' and '''w''' are considered to specify the same state if and only if <math>\mathbf{w} = c\mathbf{v}</math> for some non-zero <math>c \in \Complex</math>. Observables are given by [[self-adjoint operator]]s on ''V''. Not every self-adjoint operator corresponds to a physically meaningful observable.<ref>{{cite book |last1=Isham |first1=Christopher |author1link = Christopher Isham|title=Lectures On Quantum Theory: Mathematical And Structural Foundations |date=1995 |publisher=World Scientific |isbn=191129802X |pages=87–88 |url=https://books.google.com/books?id=vM02DwAAQBAJ}}</ref><ref>{{Citation | last1=Mackey | first1=George Whitelaw | author1-link=George Mackey | title=Mathematical Foundations of Quantum Mechanics | publisher=[[Dover Publications]] | location=New York | series=Dover Books on Mathematics | isbn=978-0-486-43517-6 | year=1963}}</ref><ref>{{Citation | last1=Emch | first1=Gerard G. | title=Algebraic methods in statistical mechanics and quantum field theory | publisher=[[Wiley-Interscience]] | isbn=978-0-471-23900-0 | year=1972}}</ref><ref>{{cite web |title=Not all self-adjoint operators are observables? |url=https://physics.stackexchange.com/questions/373357/not-all-self-adjoint-operators-are-observables |website=Physics Stack Exchange |access-date=11 February 2022}}</ref> Also, not all physical observables are associated with non-trivial self-adjoint operators. For example, in quantum theory, mass appears as a parameter in the Hamiltonian, not as a non-trivial operator.<ref>{{cite book |last1=Isham |first1=Christopher |title=Lectures On Quantum Theory: Mathematical And Structural Foundations |date=1995 |publisher=World Scientific |isbn=191129802X |pages=87–88 |url=https://books.google.com/books?id=vM02DwAAQBAJ}}</ref>

In the case of transformation laws in quantum mechanics, the requisite automorphisms are [[unitary operator|unitary]] (or [[antiunitary]]) [[linear transformation]]s of the Hilbert space ''V''.  Under [[Galilean relativity]] or [[special relativity]], the mathematics of frames of reference is particularly simple, considerably restricting the set of physically meaningful observables.

In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties.  Specifically, if a system is in a state described by a vector in a [[Hilbert space]], the measurement process affects the state in a non-deterministic but statistically predictable way. In particular, after a measurement is applied, the state description by a single vector may be destroyed, being replaced by a [[statistical ensemble]].  The [[reversible process (thermodynamics)|irreversible]] nature of measurement operations in quantum physics is sometimes referred to as the [[measurement problem]] and is described mathematically by [[quantum operation]]s. By the structure of quantum operations, this description is mathematically equivalent to that offered by the [[Many-worlds interpretation|relative state interpretation]] where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the [[partial trace]] of the state of the larger system.

In quantum mechanics, dynamical variables <math>A</math> such as position, translational (linear) [[momentum]], [[angular momentum operator|orbital angular momentum]], [[Spin (physics)|spin]], and [[total angular momentum]] are each associated with a [[self-adjoint operator]] <math>\hat{A}</math> that acts on the [[quantum state|state]] of the quantum system. The [[eigenvalues]] of operator <math>\hat{A}</math> correspond to the possible values that the dynamical variable can be observed as having. For example, suppose <math>|\psi_{a}\rangle</math> is an eigenket ([[eigenvector]]) of the observable <math>\hat{A}</math>, with eigenvalue <math>a</math>, and exists in a [[Hilbert space]]. Then
<math display="block">\hat{A}|\psi_a\rangle = a|\psi_a\rangle.</math>

This eigenket equation says that if a [[measurement]] of the observable <math>\hat{A}</math> is made while the system of interest is in the state <math>|\psi_a\rangle</math>, then the observed value of that particular measurement must return the eigenvalue <math>a</math> with certainty. However, if the system of interest is in the general state <math>|\phi\rangle \in \mathcal{H}</math> (and <math>|\phi\rangle</math> and <math>|\psi_a\rangle</math> are [[unit vector]]s, and the [[eigenspace]] of <math>a</math> is one-dimensional), then the eigenvalue <math>a</math> is returned with probability <math>|\langle \psi_a|\phi\rangle|^2</math>, by the [[Born rule]].

=== 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}}

==See also==
{{columns-list|colwidth=20em|
* [[Control theory]]
* [[Eigenvalue]]
* [[Expectation value]]
* [[Measure (mathematics)]]
* [[Measurement]]
* [[Observable universe]]
* [[Observer (quantum physics)]]
* [[Physical quantity]]
* [[Quantum state]]
* [[Statistical variable]]
* [[Operator (physics)#Table of QM operators|Table of QM operators]]
* [[Unobservable]]
}}

==References==
{{reflist}}

== Further reading ==
*{{cite book|last1=Auyang|first1=Sunny Y.|title=How is quantum field theory possible?|date=1995|publisher=Oxford University Press|location=New York, N.Y.|isbn=978-0195093452}}
*{{cite book | last=Cohen-Tannoudji | first=Claude | last2=Diu | first2=Bernard | last3=Laloë | first3=Franck | title=Quantum Mechanics, Volume 1 | publisher=John Wiley & Sons | publication-place=Weinheim | date=2019 | isbn=978-3-527-34553-3}}
*{{cite thesis |last=de la Madrid Modino |first= R. |date=2001 |title= Quantum mechanics in rigged Hilbert space language|url=https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en |degree=  PhD |publisher= Universidad de Valladolid}}
* {{cite book | last=Teschl | first=G. | title=Mathematical Methods in Quantum Mechanics | publisher=American Mathematical Soc. | publication-place=Providence (R.I) | date=2014 | isbn=978-1-4704-1704-8}}
*{{cite book|last1=von Neumann|first1=John|title=Mathematical foundations of quantum mechanics|date=1996|publisher=Princeton Univ. Press|location=Princeton, N.J.|isbn=978-0691028934|edition=12. print., 1. paperback print.|others=Translated by Robert T. Beyer}}
*{{cite book|last1=Varadarajan|first1=V.S.|title=Geometry of quantum theory|date=2007|publisher=Springer|location=New York|isbn=9780387493862|edition=2nd}}
*{{cite book|first1=Hermann |last1=Weyl |authorlink = Hermann Weyl|chapter=Appendix C: Quantum physics and causality |title=Philosophy of mathematics and natural science|date=2009|publisher=Princeton University Press|location=Princeton, N.J.|isbn=9780691141206|pages=253&ndash;265|others=Revised and augmented English edition based on a translation by Olaf Helmer}}
*{{cite book |last1=Moretti |first1=Valter |title=Spectral Theory and Quantum Mechanics: Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation |date=2017 |edition=2 |publisher=Springer |isbn=978-3319707068 |url=https://books.google.com/books?id=RNBJDwAAQBAJ}}
*{{cite book |last1=Moretti |first1=Valter |title=Fundamental Mathematical Structures of Quantum Theory: Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation |date=2019 |publisher=Springer |isbn=978-3030183462 |url=https://books.google.com/books?id=2UeeDwAAQBAJ}}

{{Quantum mechanics topics}}

[[Category:Control theory]]
[[Category:Mathematical physics]]
[[Category:Measurement]]
[[Category:Physical quantities]]
[[Category:Quantum mechanics]]
[[Category:Statistics]]