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

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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 $ψ (x, t)$ or a state vector $| ψ ⟩$ .^[1]^[2]

The operator formalism is central to quantum theory, replacing classical variables with linear operators on a Hilbert space.

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: $\hat{x} = x$
Momentum operator: $\hat{p} = - i ℏ \frac{d}{d x}$
Energy (Hamiltonian): $\hat{H}$

These operators encode the measurable properties of the system and determine its evolution.^[3]

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:

$\hat{A} | ψ ⟩ = a | ψ ⟩$

where:

$\hat{A}$ 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:

$⟨ A ⟩ = ⟨ ψ | \hat{A} | ψ ⟩$

In wavefunction form:

$⟨ A ⟩ = \int ψ^{*} (x), \hat{A}, ψ (x), d x$

This connects the operator formalism to experimentally observable averages.^[1]

Commutation relations

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

$[\hat{A}, \hat{B}] = \hat{A} \hat{B} - \hat{B} \hat{A}$

A fundamental example is the position–momentum commutation relation:

$[x, p] = i ℏ$

Non-commuting operators correspond to observables that cannot be simultaneously measured with arbitrary precision, leading to uncertainty relations.^[3]

Hermitian operators

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

${\hat{A}}^{†} = \hat{A}$

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.^[3]

Table of contents (137 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum Observables and operators

Operators in quantum mechanics

Eigenvalues and measurement

Expectation values

Commutation relations

Hermitian operators

Physical significance

See also

Table of contents (137 articles)

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