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 61–90 of 1709 reference entries

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

Schrödinger time evolution

Physics

A quantum wavefunction evolves according to its Hamiltonian.

Key knowledge

For one nonrelativistic particle of mass m in one spatial dimension with a scalar potential energy V(x,t), the wavefunction obeys iℏ ∂ψ/∂t = −(ℏ²/2m) ∂²ψ/∂x² + Vψ, where ℏ = h/(2π).

Values and equations

iℏ ∂ψ(x,t)/∂t = −(ℏ²/2m) ∂²ψ(x,t)/∂x² + V(x,t)ψ(x,t); ℏ = h/(2π)

ψ
One-dimensional position wavefunction m⁻¹ᐟ²
i
Imaginary unit, i² = −1
ℏ
Reduced Planck constant J s
h
Planck constant J s
m
Particle mass kg
x
Position coordinate m
t
Time s
V
Scalar potential energy J

Conditions and scope

  • This form omits spin and magnetic vector potentials; relativistic particles and many-particle states need other formulations.
  • Specify the initial state and boundary conditions, with the derivatives defined in the chosen domain.
  • For probability-preserving closed-system evolution, use a real potential and appropriate boundary conditions; on the full line the source requires vanishing boundary probability flux.
Sources · 4

References

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

  1. University Physics, volume 3: The Schrödinger EquationSamuel J. Ling, Jeff Sanny and William Moebs / OpenStax, Rice UniversityTextbook chapteropenstax.orgRead source
  2. Massachusetts Institute of Technology2 sources
    1. 8.04 Quantum Physics I (2016), Lecture 5: Schrödinger equationBarton Zwiebach — Massachusetts Institute of TechnologyCourse notes · PDFocw.mit.eduRead 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 7.3, equation 7.23; Zwiebach Lecture 5 page 4, equation 2.3; Lecture 6 pages 1–6: normalization and boundary flux.

2026-10-10 · One-dimensional scalar-potential nonrelativistic equation and normalization conditions only; measurement-state update and general Hamiltonians are outside scope.

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

Scope or convention differs

The nonrelativistic scalar-potential Schrödinger equation and probability conservation agree for real, time-independent V and suitable boundaries.

ARK also permits a real time-dependent scalar potential; the chapter examples use V(x) and omit spin and magnetic vector potentials.

Related reference entries

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

Showing 1–11 of 11 relationships

Scientific relationships · 11 / 11

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

  1. Uses definition
    Relationship sources · 2

    The one-dimensional scalar-potential time-evolution equation uses ℏ = h/(2π).

  2. Related to
    Relationship sources · 1

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

  3. Derived from
    Relationship sources · 2

    Linked statements

    • For nonrelativistic hydrogen with a fixed point nucleus, Coulomb bound levels have E_n = −m_e k_e²e⁴/(2ħ²n²), n = 1,2,… . Photon emission between allowed levels has hν = E_initial − E_final > 0.

    Nonrelativistic fixed-nucleus Coulomb bound-state model with corrections excluded.

  4. Related to
    Relationship sources · 1

    Linked statements

    • A cross section relates incident flux to interaction rate, and a differential cross section resolves outgoing solid angle. For the stated elastic nonrelativistic potential-scattering convention, dσ/dΩ = |f(θ,φ)|².

    The displayed elastic potential-scattering amplitude is obtained from the specified stationary quantum wave model.

  5. Related to
    Relationship sources · 2

    Linked statements

    • For a periodic one-electron Hamiltonian, Bloch states have ψ_nk(r) = e^(ik·r)u_nk(r), with u_nk(r+R) = u_nk(r) for lattice vectors R. Their energies E_n(k) form bands.

    The periodic effective one-electron Hamiltonian selects Bloch states and bands.

  6. Related to
    Relationship sources · 1

    Linked statements

    • A closed quantum system evolves by iħ d|ψ⟩/dt=H(t)|ψ⟩. For a time-independent self-adjoint Hamiltonian, U(t,t₀)=exp[−iH(t−t₀)/ħ] preserves the state norm; time-dependent noncommuting Hamiltonians require time ordering.

    The scalar position equation is a representation of the closed-system Hamiltonian evolution, with its own restricted domain.

  7. Derived from
    Relationship sources · 1

    Linked statements

    • A real one-dimensional potential conserves probability current. For a rectangular barrier of height V₀ and width a with equal zero-potential exterior regions and 0<E<V₀, the transmitted flux fraction is [1+V₀² sinh²(κa)/(4E(V₀−E))]⁻¹, κ=√[2m(V₀−E)]/ħ; reflection and transmission fractions sum to one.

    Stationary elastic finite-step matching and conserved probability flux, with equal exterior speeds.

  8. Approximates
    Relationship sources · 1

    Linked statements

    • Away from turning points, a slowly varying one-dimensional potential admits leading stationary WKB waves ψ≈[A e^(i∫k dx)+B e^(−i∫k dx)]/√k in an allowed region and ψ≈[C e^(∫κ dx)+D e^(−∫κ dx)]/√κ in a forbidden region, with k=√[2m(E−V)]/ħ and κ=√[2m(V−E)]/ħ. The local amplitude varies with the inverse square root of the local wavenumber.

    The stationary one-dimensional scalar equation admits the bounded slow-variation expansion only away from turning points.

  9. Related to
    Relationship sources · 2

    Linked statements

    • A nonrelativistic spinless particle of signed charge q in electromagnetic potentials has H=(−iħ∇−qA)²/(2m)+qφ and kinetic momentum Π=p−qA. The gauge transformation A′=A+∇χ, φ′=φ−∂_tχ, ψ′=exp(iqχ/ħ)ψ preserves the Schrödinger equation and electromagnetic observables.

    The external charged-particle Schrödinger Hamiltonian replaces canonical derivatives by kinetic covariant derivatives.

  10. Derived from
    Relationship sources · 1

    Linked statements

    • For stationary three-dimensional scattering by a short-range real potential, a unit-amplitude incident plane wave has asymptotic form ψ≈exp(ikz)+f(θ,φ)exp(ikr)/r. The differential cross section is dσ/dΩ=|f(θ,φ)|² when incident and outgoing speeds are equal.

    A real short-range stationary scalar problem with an outgoing condition has the stated asymptotic scattering solution.

  11. Related to
    Relationship sources · 1

    Linked statements

    • For a free-particle plane-wave component in a stated frame, momentum p=ħk and nonzero momentum magnitude gives wavelength λ=h/|p|. Massive particles can show interference/diffraction; their wavefunction is not a classical material string. A localized wave packet contains a spread of momenta and does not have one exact wavelength.

    The two records describe related physical or metrological models or distinctions; the present claim does not require every hypothesis or result in the linked record.

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