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{{Short description|Mathematical representation of measurable quantities in quantum mechanics using operators acting on state vectors}}
{{Short description|Mathematical representation of measurable quantities in quantum mechanics using operators acting on state vectors}}
 
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{{Quantum book backlink|Foundations}}
{{Quantum article nav|previous=Physics:Quantum Hilbert space|previous label=Hilbert space|next=Physics:Quantum mechanics|next label=Mechanics}}
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* <math>|\psi\rangle</math> is the eigenstate
* <math>|\psi\rangle</math> is the eigenstate


After measurement, the system collapses into the corresponding eigenstate.<ref name="SakuraiQM" />
After measurement, the system collapses into the corresponding eigenstate.<ref name="SakuraiQM">{{cite book
|last=Sakurai
|first=J. J.
|title=Modern Quantum Mechanics
|publisher=Addison-Wesley
|year=1994
|isbn=978-0201539295
}}</ref>


== Expectation values ==
== Expectation values ==
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<math>\langle A \rangle = \int \psi^*(x), \hat{A}, \psi(x), dx</math>
<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" />
This connects the operator formalism to experimentally observable averages.<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>


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


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


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

Latest revision as of 00:31, 24 May 2026

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Observables and operators in quantum mechanics, observables are physical quantities that can be measured, such as position, momentum, energy, and angular momentum. The operator formalism is central to quantum theory, replacing classical variables with linear operators on a Hilbert space. In quantum mechanics, observables are physical quantities that can be measured, such as position, momentum, energy, and angular momentum. An operator is a mathematical object that acts on a function or state vector to produce another function. Operators are generally linear and may be represented as differential operators, matrices, or more abstract mappings depending on the system. The measurement postulate of quantum mechanics states that when an observable is measured, the result is one of the eigenvalues of the corresponding operator.

Light-yellow schematic illustrating how quantum operators act on wavefunctions to yield measurable observables such as position, momentum, and energy.

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: x^=x
  • Momentum operator: p^=−iℏddx
  • Energy (Hamiltonian): H^

These operators encode the measurable properties of the system and determine its evolution.[1]

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:

A^|ψ⟩=a|ψ⟩

where:

  • A^ is the operator
  • a is the eigenvalue (measured value)
  • |ψ⟩ is the eigenstate

After measurement, the system collapses into the corresponding eigenstate.[2]

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:

⟨A⟩=⟨ψ|A^|ψ⟩

In wavefunction form:

⟨A⟩=∫ψ*(x),A^,ψ(x),dx

This connects the operator formalism to experimentally observable averages.[3]

Commutation relations

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

[A^,B^]=A^B^−B^A^

A fundamental example is the position–momentum commutation relation:

[x,p]=iℏ

Non-commuting operators correspond to observables that cannot be simultaneously measured with arbitrary precision, leading to uncertainty relations.[1]

Hermitian operators

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

A^†=A^

This property ensures:

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

These features make Hermitian operators essential for consistent physical interpretation.[2]

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.[1]

See also

Table of contents (217 articles)

Index

Full contents

References

  1. ↑ 1.0 1.1 1.2 Dirac, P. A. M. (1981). The Principles of Quantum Mechanics. Oxford University Press. 
  2. ↑ 2.0 2.1 Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley. ISBN 978-0201539295. 
  3. ↑ Griffiths, David J. (2005). Introduction to Quantum Mechanics (2nd ed.). Pearson. ISBN 978-0131118928. 
Author: Harold Foppele


Source attribution: Physics:Quantum Observables and operators

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