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Physics · MeasurementElectric current

Current measures charge crossing a chosen surface per unit time.

DefinitionPhysicsElectric field and force

An electric field assigns force per unit test charge.

DefinitionPhysicsMagnetic force on a moving charge

Magnetic force depends on charge motion relative to the field.

ModelPhysicsOhm's law and its domain

A linear current-voltage relation requires suitable conditions.

Empirical findingPhysicsFirst law of thermodynamics

Keep the heat and work sign convention explicit in an energy balance.

ModelPhysicsSecond law: isolated-system entropy

Apply entropy increase to the whole isolated system.

ModelPhysics · ChemistryIdeal-gas equation of state

A simple gas model relates pressure, amount, volume and temperature.

ModelPhysicsPhoton energy and frequency

One photon's energy is proportional to its frequency.

ModelPhysics · Mathematics & logicInvariant spacetime interval

Lorentz transformations preserve a combined space-time separation.

ModelPhysicsNewton's third law

Interaction forces act on different bodies in equal and opposite pairs.

ModelPhysicsAngular momentum and torque

Torque changes angular momentum about a specified origin.

ModelPhysicsHydrostatic pressure

Static fluid pressure rises with depth under gravity.

ModelPhysicsArchimedes' buoyancy principle

The hydrostatic upward force equals displaced-fluid weight.

ModelPhysicsMaxwell equations in free space

Four field equations connect electric and magnetic fields with charge and current.

ModelPhysicsSchrödinger time evolution

A quantum wavefunction evolves according to its Hamiltonian.

ModelPhysicsBorn position probability

A normalized wavefunction assigns probabilities to position intervals.

ModelPhysics · MeasurementElementary charge

The elementary charge has an exact SI numerical value.

DefinitionPhysics · MeasurementBoltzmann constant

The Boltzmann constant has an exact SI numerical value.

DefinitionMeasurement · Chemistry · PhysicsAvogadro constant

The mole is linked to a specified number of entities.

DefinitionMeasurement · PhysicsSI defining luminous efficacy

A specified frequency fixes the photometric SI reference.

DefinitionChemistryAtoms and elemental identity

The proton count of an atomic nucleus identifies its chemical element.

DefinitionChemistryIsotopes and mass number

Isotopes share a proton count but have different neutron counts.

DefinitionChemistryPeriodic organization of the elements

Atomic-number order reveals recurring patterns in electron structure and chemical properties.

ModelChemistryChemical bonds

Bonding describes interactions that bind atoms into molecules or extended structures.

DefinitionChemistry · MeasurementMole and amount of substance

The mole counts specified entities using the exact Avogadro constant.

DefinitionChemistryReaction stoichiometry

Balanced coefficients relate the amounts consumed and produced by a reaction.

MethodChemistryBalancing atoms and electric charge

A chemical equation must preserve each element's atom count and the total charge.

MethodChemistryDynamic chemical equilibrium

At equilibrium, opposing reactions continue without net composition change.

ModelChemistryBrønsted acids and bases

In the Brønsted description, an acid donates a proton and a base accepts it.

DefinitionChemistryGibbs energy and thermodynamic direction

Gibbs-energy change distinguishes thermodynamic favourability from reaction speed.

Model
ModelSource checked

Born position probability

Physics

A normalized wavefunction assigns probabilities to position intervals.

Key knowledge

For a normalized one-dimensional position wavefunction ψ(x,t), the Born rule assigns probability density |ψ(x,t)|². The probability of a position measurement in [a,b] is the integral of that density over the interval.

Values and equations

∫_domain |ψ(x,t)|² dx = 1; P(a ≤ x ≤ b; t) = ∫_a^b |ψ(x,t)|² dx

ψ
Normalized position wavefunction m⁻¹ᐟ²
|ψ|²
Position probability density m⁻¹
P
Dimensionless position-interval probability
x
Position coordinate m
a, b
Interval endpoints with a ≤ b within the allowed domain m
t
Time s

Conditions and scope

  • The wavefunction describes a single-particle pure state and is square-integrable and normalized over its allowed position domain.
  • Density has units of inverse length; its value at a point is not an interval probability.
  • Other observables and mixed states require their corresponding measurement rules; no classical pre-existing trajectory is asserted.
Sources · 3

References

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

  1. University Physics, volume 3: Wave FunctionsSamuel J. Ling, Jeff Sanny and William Moebs / OpenStax, Rice UniversityTextbook chapteropenstax.orgRead source
  2. 8.04 Quantum Physics I (2016), Lecture 6: normalization and probability currentBarton Zwiebach — Massachusetts Institute of TechnologyCourse notes · PDFocw.mit.eduRead source
  3. Quantum Mechanics · wavefunctions and one-dimensional dynamicsDavid Tong · University of CambridgeCourse notes · PDFdamtp.cam.ac.ukRead source

Evidence and review

OpenStax 7.1: modulus-squared density, interval integration and normalization; Zwiebach Lecture 6 equations 1.3–1.4 and 3.12, page 6 units.

2026-10-10 · Position probabilities for a normalized one-dimensional pure state only; no ontological interpretation or general measurement calculus is certified.

Foundational review

2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.

Sources from other places

Agreement within this scope

A normalized pure-state wavefunction gives position probability by integrating its squared modulus over the region.

In one dimension this is a density per length, not the probability of an exact point; unnormalized plane waves require separate treatment.

Related reference entries

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

Showing 1–4 of 4 relationships

Scientific relationships · 4 / 4

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

  1. Related to
    Relationship sources · 1

    For the stated real-potential evolution and suitable boundaries, the normalized wavefunction's total position probability is preserved.

  2. 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.

  3. Uses definition
    Relationship sources · 2

    Linked statements

    • A nonrelativistic point-Coulomb two-body bound state separates as ψ_nlm=R_nl(r)Y_lm(θ,φ). Its radial probability density is r²|R_nl|², not the spatial density at a point. The normalized hydrogenic 1s spatial amplitude is exp(−r/a)/(√π a³ᐟ²); its radial density 4r²exp(−2r/a)/a³ peaks at r=a, whereas its spatial density peaks at r=0.

    The radial marginal integrates the spatial Born density over solid angle with its r² volume factor.

  4. Related to
    Relationship sources · 1

    Linked statements

    • A finite-outcome POVM is a family of positive operators E_a whose sum is the identity. Outcome a has probability Tr(ρE_a), allowing nonorthogonal and unsharp measurements. The effects alone specify probabilities; they do not specify which state the instrument leaves after the outcome.

    This generalizes measurement probabilities without assigning a unique state update.

Entry detailsPermanent link to this entry ↗
Record identity
physics-born-position-probability
Revision
core.1
Structured reference record
Essential core · first collection · University core expansion