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Showing 211–240 of 1709 reference entries

PhysicsRC charging and discharge

A simple resistor–capacitor circuit has an exponential timescale.

Model
PhysicsSeries RLC impedance magnitude

Sinusoidal response combines resistance and net reactance.

Model
PhysicsFourier heat conduction

A temperature gradient drives conductive heat flux.

Model
PhysicsLatent heat of a phase transition

Energy changes phase at the specified coexistence conditions.

Model
PhysicsApproximate vapor-pressure relation

Vapor pressure changes with temperature through latent enthalpy.

Model
PhysicsBlackbody total emission

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

Model
PhysicsCanonical thermal ensemble

Fixed-temperature equilibrium weights states by energy.

Model
Physics · InformationGibbs entropy

Microstate probabilities quantify statistical entropy.

Definition
PhysicsQuantum observable expectations

A state and observable determine the mean of repeated outcomes.

Model
PhysicsRobertson uncertainty relation

A commutator constrains two outcome spreads in the same state.

Model
PhysicsSpin one-half and Pauli operators

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

Model
Physics · InformationPure and mixed density operators

A density operator represents quantum statistical states.

Definition
PhysicsIdeal projective measurement

Projectors determine outcome probabilities and conditional states.

Model
Physics · InformationComposite states and entanglement

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

Definition
PhysicsExchange symmetry of identical particles

Boson and fermion states respond differently to exchange.

Model
Physics · Mathematics & logicGeodesic motion

Free test motion follows the metric connection.

Model
PhysicsEinstein gravitational field equations

Spacetime curvature responds to stress and energy.

Model
PhysicsSchwarzschild static-observer redshift

Clocks at different radii measure different light frequencies.

Model
Physics · Earth & cosmosFriedmann cosmological evolution

Homogeneity and isotropy reduce gravitational expansion to two balances.

Model
MeasurementCalibration

Relate indications to standards, then infer measurement results.

Definition
MeasurementType A and Type B uncertainty evaluation

The classification describes how a component is evaluated.

Method
MeasurementExpanded uncertainty and coverage

A coverage factor scales standard uncertainty to a reported interval.

Method
Measurement · Mathematics & logicCovariance in uncertainty models

Shared input variation belongs in the joint uncertainty description.

Method
MeasurementMeasurement models

Specify how input quantities determine the measurand.

Method
Measurement · InformationMonte Carlo uncertainty propagation

Sample the joint input model to approximate the output distribution.

Method
MeasurementRepeatability and reproducibility

Precision depends on which measurement conditions are held or changed.

Definition
MeasurementResolution and sensitivity

Detectable change and response scale are different instrument properties.

Definition
MeasurementDynamic instrument response

Response depends on timescale, gain and phase.

Method
PhysicsHydrogen Coulomb energy levels

The ideal one-electron atom has discrete bound energies.

Model
Physics · ChemistryAtomic state labels and Pauli occupancy

Quantum labels specify orbitals and spin states.

Model
ModelSource checked

Ideal projective measurement

Physics

Projectors determine outcome probabilities and conditional states.

Key knowledge

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.

Values and equations

P_rP_s = δ_rs P_r; Σ_r P_r = I; p_r = ⟨ψ|P_r|ψ⟩; ψ_r = P_rψ/√p_r (p_r > 0)

P_r
Orthogonal outcome projector
I
Identity operator
δ_rs
Kronecker delta
p_r
Outcome probability
ψ, ψ_r
Initial and conditional normalized states

Conditions and scope

  • This is the ideal Lüders projective model, including degenerate outcome subspaces.
  • General measurement devices may require broader measurement operators; an outcome of zero probability has no defined conditional state here.
Sources · 1

References

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

  1. Introduction to Quantum Information: measurement operatorsTU Delft / QuTechCourse notesocw.tudelft.nlRead source

Evidence and review

Measurement-operator probability, completeness and normalization rules, restricted here to orthogonal projectors.

2026-10-10 · Pure-state ideal projective rule only; arbitrary device dynamics and general POVMs excluded.

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physics-projective-measurement
Revision
university-reference.1
Structured reference record
University reference · University core expansion