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{{Short description|Mathematical representation of measurable quantities in quantum mechanics using operators acting on state vectors}}

{{Quantum book backlink|Foundations}}

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In quantum mechanics, '''observables''' are physical quantities that can be measured, such as position, momentum, energy, and angular momentum. These quantities are represented mathematically by '''operators''' acting on the state of a system, typically described by a wavefunction <math>\psi(x,t)</math> or a state vector <math>|\psi\rangle</math>.<ref name="GriffithsQM">{{cite book
|last=Griffiths
|first=David J.
|title=Introduction to Quantum Mechanics
|edition=2nd
|publisher=Pearson
|year=2005
|isbn=978-0131118928
}}</ref><ref name="SakuraiQM">{{cite book
|last=Sakurai
|first=J. J.
|title=Modern Quantum Mechanics
|publisher=Addison-Wesley
|year=1994
|isbn=978-0201539295
}}</ref>

The operator formalism is central to quantum theory, replacing classical variables with linear operators on a Hilbert space.
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[[File:Quantum_operator_observable_yellow.jpg|thumb|280px|Light-yellow schematic illustrating how quantum operators act on wavefunctions to yield measurable observables such as position, momentum, and energy.]]
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== Operators in quantum mechanics ==

An operator is a mathematical object that acts on a function or state vector to produce another function. For example:

* Position operator: <math>\hat{x} = x</math>
* Momentum operator: <math>\hat{p} = -i\hbar \frac{d}{dx}</math>
* Energy (Hamiltonian): <math>\hat{H}</math>

These operators encode the measurable properties of the system and determine its evolution.<ref name="DiracQM">{{cite book
|last=Dirac
|first=P. A. M.
|title=The Principles of Quantum Mechanics
|publisher=Oxford University Press
|year=1981
}}</ref>

Operators are generally linear and may be represented as differential operators, matrices, or more abstract mappings depending on the system.

== Eigenvalues and measurement ==

The measurement postulate of quantum mechanics states that when an observable is measured, the result is one of the eigenvalues of the corresponding operator.

This is expressed through the eigenvalue equation:

<math>\hat{A} |\psi\rangle = a |\psi\rangle</math>

where:

* <math>\hat{A}</math> is the operator
* <math>a</math> is the eigenvalue (measured value)
* <math>|\psi\rangle</math> is the eigenstate

After measurement, the system collapses into the corresponding eigenstate.<ref name="SakuraiQM" />

== Expectation values ==

If the system is not in an eigenstate, measurements yield probabilistic results. The average value of many measurements is given by the expectation value:

<math>\langle A \rangle = \langle \psi | \hat{A} | \psi \rangle</math>

In wavefunction form:

<math>\langle A \rangle = \int \psi^*(x), \hat{A}, \psi(x), dx</math>

This connects the operator formalism to experimentally observable averages.<ref name="GriffithsQM" />

== Commutation relations ==

Operators in quantum mechanics do not always commute. The commutator of two operators is defined as:

<math>[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}</math>

A fundamental example is the position–momentum commutation relation:

<math>[x, p] = i\hbar</math>

Non-commuting operators correspond to observables that cannot be simultaneously measured with arbitrary precision, leading to uncertainty relations.<ref name="DiracQM" />

== Hermitian operators ==

Observable quantities are represented by '''Hermitian (self-adjoint) operators''', which satisfy:

<math>\hat{A}^\dagger = \hat{A}</math>

This property ensures:

* Real eigenvalues (physical measurements)
* Orthogonal eigenstates
* Completeness of the eigenbasis

These features make Hermitian operators essential for consistent physical interpretation.<ref name="SakuraiQM" />

== Physical significance ==

The operator–observable framework is one of the defining features of quantum mechanics. It provides:

* A direct link between mathematics and measurement
* A probabilistic interpretation of physical quantities
* The foundation for quantum dynamics via the Hamiltonian

This formalism generalizes naturally to more advanced theories, including quantum field theory, where fields themselves become operator-valued quantities.<ref name="DiracQM" />

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

=References=
{{reflist|3}}
{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum Observables and operators|1}}