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{{About|observables in physics|the use of the term "observable" in [[control theory]]|Observability}}
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In [[physics]], particularly in [[quantum physics]], a system '''observable''' is a measurable operator, or gauge, where the property of the [[Quantum state|system state]]  can be determined by some sequence of physical [[operational definition|operations]]. For example, these operations might involve submitting the system to various [[electromagnetic field]]s and eventually reading a value off some gauge.  In systems governed by [[classical mechanics]], any [[experiment]]ally observable value can be shown to be given by a [[real number|real]]-valued [[function (mathematics)|function]] on the set of all possible system states.
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Physically meaningful observables must also satisfy [[transformation law]]s which relate observations performed by different [[observation|observer]]s in different [[frames of reference]]. These transformation laws are [[automorphism]]s of the state space, that is [[bijective]] [[Transformation (mathematics)|transformation]]s which preserve some mathematical property. 
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


== Quantum mechanics ==
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:<math forcemathmode="png">E=mc^2</math>


In [[quantum physics]], the relation between system state and the value of an observable requires some basic [[linear algebra]] for its description.  In the [[mathematical formulation of quantum mechanics]], states are given by non-zero [[vector (geometry)|vector]]s in a [[Hilbert space]] ''V'' (where two vectors are considered to specify the same state if, and only if, they are scalar multiples of each other) and observables are given by [[self-adjoint operator]]s on ''V''. However, as indicated below, not every self-adjoint operator corresponds to a physically meaningful observable. For the case of a system of [[Elementary particle|particle]]s, the space ''V'' consists of functions called [[wave function]]s or [[state vector]]s.
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In the case of transformation laws in quantum mechanics, the requisite automorphisms are [[unitary operator|unitary]] (or [[antiunitary]]) linear transformations of the [[Hilbert space]] ''V''. Under [[Galilean relativity]] or [[special relativity]], the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.
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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|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 [[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.
==Demos==


In quantum mechanics each dynamical variable (e.g. position, translational momentum, orbital angular momentum, spin, total angular momentum, energy, etc.) is associated with a [[Hermitian operator]] that acts on the [[Quantum state|state]] of the quantum system and whose [[eigenvalues]] correspond to the possible values of the dynamical variable. For example, suppose <math>|a\rangle</math> is an eigenket ([[eigenvector]]) of the observable <math>\mathbf{A}</math>, with eigenvalue <math>a</math>, and exists in a d-dimensional [[Hilbert space]]. Then
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


:<math>\mathbf{A}</math><math>|a\rangle</math> = <math>a</math> <math>|a\rangle.</math>


This eigenket equation says that if a [[measurement]] of the observable <math>\scriptstyle \mathbf{A}</math> is made while the system of interest is in the state <math>\scriptstyle |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>\scriptstyle |\phi\rangle\in\mathcal{H}</math>, then the eigenvalue <math>a</math> is returned with probability <math>\scriptstyle |\langle a|\phi\rangle|^2</math> ([[Born rule]]). One must note that the above definition is somewhat dependent upon our convention of choosing real numbers to represent real physical quantities. Indeed, just because dynamical variables are "real" and not "unreal" in the metaphysical sense does not mean that they must correspond to real numbers in the mathematical sense.  
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To be more precise, the dynamical variable/observable is a (not necessarily bounded) Hermitian operator in a Hilbert Space and thus is represented by a Hermitian matrix if the space is finite-dimensional. In an infinite-dimensional Hilbert space, the observable is represented by a [[Symmetric operator#Symmetric operators|symmetric operator]], which may not be ''defined everywhere'' (i.e. its [[Domain (mathematics)|domain]] is not the whole space - there exist some states that are not in the domain of the operator). The reason for such a change is that in an infinite-dimensional Hilbert space, the operator becomes [[unbounded]], which means that it no longer has a largest eigenvalue. This is not the case in a finite-dimensional Hilbert space, where every operator is bounded - it has a largest eigenvalue. For example, if we consider the position of a point particle moving along a line, this particle's position variable can take on any number on the real-line, which is [[uncountably]] infinite. Since the eigenvalue of an observable represents a real physical quantity for that particular dynamical variable, then we must conclude that there is no largest eigenvalue for the position observable in this uncountably infinite-dimensional Hilbert space, since the [[Field (mathematics)|field]] we're working over consists of the real-line. Nonetheless, whether we are working in an infinite-dimensional or finite-dimensional Hilbert space, the role of an observable in quantum mechanics is to assign real numbers to outcomes of ''particular measurements''; this means that 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.
==Test pages ==


==Incompatibility of observables in quantum mechanics==
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A crucial difference between classical quantities and quantum mechanical observables is that the latter may not be simultaneously measurable. This is mathematically expressed by non-[[commutativity]] of the corresponding operators, to the effect that
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:<math>\mathbf{A}\mathbf{B} - \mathbf{B}\mathbf{A} \neq \mathbf{0}.</math>
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This inequality expresses a dependence of measurement results on the order in which measurements of observables <math>\scriptstyle \mathbf{A}</math> and <math>\scriptstyle \mathbf{B}</math> are performed. Observables corresponding to non-commutative operators are called ''incompatible''.
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
==See also==
* [[Observable universe]]
* [[Observer (quantum physics)]]
 
== Further reading ==
{{refbegin}}
* S. Auyang, ''How is Quantum Field Theory Possible'', Oxford University Press, 1995.
* G. Mackey, ''Mathematical Foundations of Quantum Mechanics'', W. A. Benjamin, 1963.
* V. Varadarajan, ''The Geometry of Quantum Mechanics'' vols 1 and 2, Springer-Verlag 1985.
* Leslie E. Ballentine, "Quantum Mechanics: A Modern Development", World Scientific, 1998
* R. Blume-Kohout, "Lecture 14:  <math>L^2(\reals)</math> and Hilbert space.  Wavefunctions, unbounded operators, and rigged Hilbert space.", www.am473.ca, 10/26/08
{{refend}}
 
[[Category:Quantum mechanics]]
 
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Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

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