Editing Observable (section)

== 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}}