GLOBAL KNOWLEDGE ARK

Reference library

Foundations

Explore concepts, laws, equations and methods, with clear explanations of their use and sources. Choose an area of study or find a specific entry.

More filters

Reference entries

Showing 211–240 of 1709 reference entries

PhysicsRC charging and discharge

A simple resistor–capacitor circuit has an exponential timescale.

ModelPhysicsSeries RLC impedance magnitude

Sinusoidal response combines resistance and net reactance.

ModelPhysicsFourier heat conduction

A temperature gradient drives conductive heat flux.

ModelPhysicsLatent heat of a phase transition

Energy changes phase at the specified coexistence conditions.

ModelPhysicsApproximate vapor-pressure relation

Vapor pressure changes with temperature through latent enthalpy.

ModelPhysicsBlackbody total emission

Thermal equilibrium fixes an ideal absorber's total emitted power.

ModelPhysicsCanonical thermal ensemble

Fixed-temperature equilibrium weights states by energy.

ModelPhysics · InformationGibbs entropy

Microstate probabilities quantify statistical entropy.

DefinitionPhysicsQuantum observable expectations

A state and observable determine the mean of repeated outcomes.

ModelPhysicsRobertson uncertainty relation

A commutator constrains two outcome spreads in the same state.

ModelPhysicsSpin one-half and Pauli operators

A spin one-half state has two outcomes along a chosen axis.

ModelPhysics · InformationPure and mixed density operators

A density operator represents quantum statistical states.

DefinitionPhysicsIdeal projective measurement

Projectors determine outcome probabilities and conditional states.

ModelPhysics · InformationComposite states and entanglement

A joint pure state can fail to factor into subsystem states.

DefinitionPhysicsExchange symmetry of identical particles

Boson and fermion states respond differently to exchange.

ModelPhysics · Mathematics & logicGeodesic motion

Free test motion follows the metric connection.

ModelPhysicsEinstein gravitational field equations

Spacetime curvature responds to stress and energy.

ModelPhysicsSchwarzschild static-observer redshift

Clocks at different radii measure different light frequencies.

ModelPhysics · Earth & cosmosFriedmann cosmological evolution

Homogeneity and isotropy reduce gravitational expansion to two balances.

ModelMeasurementCalibration

Relate indications to standards, then infer measurement results.

DefinitionMeasurementType A and Type B uncertainty evaluation

The classification describes how a component is evaluated.

MethodMeasurementExpanded uncertainty and coverage

A coverage factor scales standard uncertainty to a reported interval.

MethodMeasurement · Mathematics & logicCovariance in uncertainty models

Shared input variation belongs in the joint uncertainty description.

MethodMeasurementMeasurement models

Specify how input quantities determine the measurand.

MethodMeasurement · InformationMonte Carlo uncertainty propagation

Sample the joint input model to approximate the output distribution.

MethodMeasurementRepeatability and reproducibility

Precision depends on which measurement conditions are held or changed.

DefinitionMeasurementResolution and sensitivity

Detectable change and response scale are different instrument properties.

DefinitionMeasurementDynamic instrument response

Response depends on timescale, gain and phase.

MethodPhysicsHydrogen Coulomb energy levels

The ideal one-electron atom has discrete bound energies.

ModelPhysics · ChemistryAtomic state labels and Pauli occupancy

Quantum labels specify orbitals and spin states.

Model
ModelSource checked

Quantum observable expectations

Physics

A state and observable determine the mean of repeated outcomes.

Key knowledge

A normalized quantum state ψ and self-adjoint observable A give expectation ⟨A⟩ = ⟨ψ|A|ψ⟩. When the second moment exists, its variance is (ΔA)² = ⟨A²⟩ − ⟨A⟩².

Values and equations

⟨A⟩ = ⟨ψ|A|ψ⟩; (ΔA)² = ⟨A²⟩ − ⟨A⟩²

ψ
Normalized state vector
A
Self-adjoint observable; its own physical unit
⟨A⟩, ΔA
Mean and standard deviation; same unit as A

Conditions and scope

  • For unbounded operators, ψ must lie in suitable domains and the required moments must be finite.
  • An expectation describes an ensemble mean; it need not be an allowed single-measurement eigenvalue.
Sources · 2

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Massachusetts Institute of Technology2 sources
    1. Quantum Physics I (2016), Lecture Notes 8: expectation valuesBarton Zwiebach / MITCourse notes · PDFocw.mit.eduRead source
    2. Quantum Physics I (2016), Lecture Notes 9: uncertaintyBarton Zwiebach / MITCourse notes · PDFocw.mit.eduRead source

Evidence and review

Lecture 8 equation 2.28; Lecture 9 equations 2.42–2.48: expectation and variance.

2026-10-10 · Normalized-state expectation and finite variance only; operator domain conditions added explicitly.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–8 of 8 relationships

Scientific relationships · 8 / 8

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Related to
    Relationship sources · 2

    Linked statements

    • A normalized quantum state ψ and self-adjoint observable A give expectation ⟨A⟩ = ⟨ψ|A|ψ⟩. When the second moment exists, its variance is (ΔA)² = ⟨A²⟩ − ⟨A⟩².

    The Born distribution is one representation used to compute normalized-state expectations.

  2. Uses definition
    Relationship sources · 1

    Linked statements

    • For self-adjoint observables A and B in a normalized state, with the products and finite variances defined, ΔA ΔB ≥ ½|⟨[A,B]⟩|, where [A,B] = AB − BA.

    Finite state variances and commutator expectation on a suitable common operator domain.

  3. Related to
    Relationship sources · 2

    Linked statements

    • In the S_z eigenbasis, a spin one-half angular momentum is represented by S_i = (ħ/2)σ_i. The Pauli matrices have eigenvalues ±1, so an axis measurement gives ±ħ/2.

    Spin components are observable operators in a two-dimensional state space.

  4. Related to
    Relationship sources · 1

    Linked statements

    • In finite dimensions, a density operator is Hermitian, positive semidefinite and has trace one. A pure state has ρ = |ψ⟩⟨ψ|; mixtures have ρ = Σ_j p_j|ψ_j⟩⟨ψ_j|.

    Pure and statistical quantum states determine observable outcome statistics.

  5. Related to
    Relationship sources · 1

    Linked statements

    • For orthogonal projectors P_r summing to identity, an ideal projective measurement in normalized state ψ gives p_r = ⟨ψ|P_r|ψ⟩ and, for p_r > 0, conditional state P_rψ/√p_r.

    The ideal projector outcome rule is normalized-state probability for an observable's subspaces.

  6. Related to
    Relationship sources · 2

    Linked statements

    • Quantum angular momentum obeys [J_i,J_j]=iħ ε_ijk J_k. Simultaneous eigenstates of J² and J_z have eigenvalues ħ²j(j+1) and ħm, with m=−j,…,j; J_± raises or lowers m by one with coefficient ħ√[j(j+1)−m(m±1)].

    Angular momentum components are noncommuting observables; simultaneous labels refer to J² and one component.

  7. Uses definition
    Relationship sources · 1

    Linked statements

    • For a self-adjoint Hamiltonian bounded below, every normalized trial state with a finite energy expectation satisfies ⟨ψ_trial|H|ψ_trial⟩≥E₀. Minimizing a chosen trial family's expectation provides an upper bound on the ground energy, not proof that the trial state is exact.

    Normalized trial-state Hamiltonian expectation bounds a lower-bounded ground energy.

  8. Uses definition
    Relationship sources · 1

    Linked statements

    • For a rotationally symmetric localized electronic orbital, the quadratic magnetic-field term in the nonrelativistic Hamiltonian gives δE=e²B²⟨r²⟩/(12m_e). The corresponding negative orbital susceptibility contribution for number density n of those electronic contributions is χ_d=−μ₀ne²⟨r²⟩/(6m_e).

    The localized orbital mean squared radius determines the bounded A² energy contribution.

Entry detailsPermanent link to this entry ↗
Record identity
physics-quantum-observables-expectation
Revision
university-reference.1
Structured reference record
University reference · University core expansion