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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

Spin one-half and Pauli operators

Physics

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

Key knowledge

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.

Values and equations

S_i = (ħ/2)σ_i; σ_x = [[0,1],[1,0]]; σ_y = [[0,−i],[i,0]]; σ_z = [[1,0],[0,−1]]

S_i
Spin angular-momentum operator J s
ħ
Reduced Planck constant J s
σ_i
Dimensionless Pauli matrix
i
Imaginary unit in matrix entries; axis index in S_i/σ_i

Conditions and scope

  • Use the displayed ordered S_z basis and ordinary spin one-half Hilbert space.
  • Intrinsic spin is quantum angular momentum; no literal rotating-sphere model is asserted.
Sources · 2

References

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

  1. Feynman Lectures on Physics III: Pauli spin matricesRichard P. Feynman, Robert B. Leighton and Matthew Sands / CaltechTextbook chapterfeynmanlectures.caltech.eduRead source
  2. University Physics 3: Electron SpinSamuel J. Ling, Jeff Sanny and William Moebs / OpenStax, Rice UniversityTextbook chapteropenstax.orgRead source

Evidence and review

Feynman 11–1 and Table 11–1: matrices and ±ħ/2 basis; OpenStax equation 8.27 and s = 1/2.

2026-10-10 · Two-dimensional spin representation only; magnetic moments, g-factors and higher spins excluded.

Related reference entries

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Scientific relationships · 2 / 2

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

  1. 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.

  2. Related to
    Relationship sources · 3

    Linked statements

    • Hydrogenic bound states use n ≥ 1, l = 0,…,n−1, m_l = −l,…,l and electron m_s = ±½. Pauli exclusion permits at most one electron per spin-orbital, hence two opposite-spin electrons per spatial orbital.

    Electron spin projection supplies the two m_s labels in the stated orbital basis.

Entry detailsPermanent link to this entry ↗
Record identity
physics-spin-half
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university-reference.1
Structured reference record
University reference · University core expansion