Schrödinger Equation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Schrödinger Equation Ontology within the Kireth Research Semantic Library. The Schrödinger equation is the central differential equation of non-relativistic quantum mechanics, governing the evolution and stationary structure of quantum states. This ontology supports time-dependent and time-independent forms, Hamiltonian operators, wavefunctions, boundary conditions, normalisation, probability density, probability current, stationary states, eigenvalue problems, bound and scattering states, potential wells and barriers, harmonic oscillators, tunnelling, perturbation methods, separation of variables, spectral decomposition, numerical solution and relationships with quantum mathematics, operator theory, spectral theory, partial differential equations and mathematical physics.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Schrödinger Equation Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Mathematical Physics / Schrödinger Equation"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/mathematical-physics/schroedinger-equation-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = SchrodingerEquation
 ENTITY_CLASS = QuantumEvolutionDifferentialEquationFramework
 PRIMARY_CONTEXT = QuantumMathematics
 DEFINITION = "The Schrödinger equation is the fundamental non-relativistic quantum equation governing the time evolution and stationary states of a quantum wavefunction under a Hamiltonian operator."
 SYMBOLIC_FORMS = [
  "i*hbar*partial psi/partial t=H*psi",
  "H*psi=E*psi",
  "H=-(hbar^2/2m)nabla^2+V",
  "rho=|psi|^2"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for quantum wavefunction evolution, stationary-state eigenproblems, Hamiltonian operators, probability density and current, boundary conditions, spectra, tunnelling, scattering and numerical quantum dynamics."
}

SCOPE {
 INCLUDE = [
  SchrodingerEquation,
  TimeDependentSchrodingerEquation,
  TimeIndependentSchrodingerEquation,
  TDSE,
  TISE,
  Wavefunction,
  QuantumState,
  HamiltonianOperator,
  Hamiltonian,
  KineticEnergyOperator,
  PotentialEnergyOperator,
  PotentialFunction,
  PositionRepresentation,
  MomentumRepresentation,
  HilbertSpace,
  Operator,
  SelfAdjointOperator,
  Eigenvalue,
  Eigenfunction,
  Eigenstate,
  EnergyEigenvalue,
  Spectrum,
  DiscreteSpectrum,
  ContinuousSpectrum,
  BoundState,
  ScatteringState,
  StationaryState,
  TimeEvolution,
  UnitaryEvolution,
  TimeEvolutionOperator,
  InitialCondition,
  BoundaryCondition,
  DirichletBoundaryCondition,
  NeumannBoundaryCondition,
  PeriodicBoundaryCondition,
  Normalisation,
  SquareIntegrability,
  ProbabilityDensity,
  ProbabilityCurrent,
  ContinuityEquation,
  ExpectationValue,
  Variance,
  Superposition,
  SeparationOfVariables,
  SpectralDecomposition,
  CompletenessRelation,
  InfiniteSquareWell,
  FiniteSquareWell,
  PotentialStep,
  PotentialBarrier,
  QuantumTunnelling,
  HarmonicOscillator,
  FreeParticle,
  PlaneWave,
  WavePacket,
  GaussianWavePacket,
  Dispersion,
  GroupVelocity,
  PhaseVelocity,
  ReflectionCoefficient,
  TransmissionCoefficient,
  Scattering,
  ScatteringAmplitude,
  PerturbationTheory,
  TimeIndependentPerturbationTheory,
  TimeDependentPerturbationTheory,
  VariationalMethod,
  WKBApproximation,
  GreenFunction,
  Propagator,
  ImaginaryTimeEvolution,
  NumericalSchrodingerEquation,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  SpectralMethod,
  SplitOperatorMethod,
  CrankNicolsonMethod,
  TimeStepping,
  NumericalStability,
  Convergence,
  GridResolution
 ]
 EXCLUDE_AS_PRIMARY = [
  RelativisticWaveEquationsAsPrimaryTheory,
  QuantumFieldTheoryAsPrimaryFramework,
  PureOperatorTheoryWithoutSchrodingerDynamics,
  ExperimentalQuantumMeasurementWithoutWavefunctionEvolution
 ]
 DISAMBIGUATION_RULE = "When a non-relativistic quantum wavefunction is evolved or solved using a Hamiltonian differential operator, interpret the context as the Schrödinger equation."
}

CORE_CONCEPTS {
 SchrodingerEquation = "Quantum differential equation governing wavefunction evolution."
 TimeDependentSchrodingerEquation = "Equation governing time evolution of a quantum state."
 TimeIndependentSchrodingerEquation = "Hamiltonian eigenvalue equation for stationary states."
 HamiltonianOperator = "Operator representing total quantum energy."
 Wavefunction = "Complex-valued amplitude representing a pure quantum state in a chosen representation."
 ProbabilityDensity = "Spatial probability density equal to the squared magnitude of the wavefunction."
 ProbabilityCurrent = "Current associated with conservation of quantum probability."
 StationaryState = "Energy eigenstate evolving only by a phase factor."
 BoundState = "Normalisable state localised by a confining potential."
 ScatteringState = "Typically non-square-integrable generalised state associated with continuous spectra."
}

RELATIONS {
 SchrodingerEquation --is_child_of--> QuantumMathematics
 TimeDependentSchrodingerEquation --governs--> TimeEvolution
 TimeIndependentSchrodingerEquation --is--> HamiltonianEigenvalueProblem
 HamiltonianOperator --contains--> KineticEnergyOperator
 HamiltonianOperator --contains--> PotentialEnergyOperator
 KineticEnergyOperator --uses--> Laplacian
 PotentialEnergyOperator --uses--> PotentialFunction
 Wavefunction --determines--> ProbabilityDensity
 Wavefunction --determines--> ProbabilityCurrent
 ProbabilityCurrent --satisfies_with_probability_density--> ContinuityEquation
 StationaryState --is--> EnergyEigenstate
 EnergyEigenvalue --belongs_to--> Spectrum
 BoundState --belongs_to--> DiscreteSpectrum
 ScatteringState --belongs_to--> ContinuousSpectrum
 TimeEvolutionOperator --is_generated_by--> HamiltonianOperator
 SpectralDecomposition --expands--> QuantumState
 InfiniteSquareWell --has--> DiscreteEnergySpectrum
 PotentialBarrier --supports--> QuantumTunnelling
 HarmonicOscillator --has--> EquallySpacedEnergySpectrum
 FreeParticle --supports--> PlaneWave
 WavePacket --superposes--> MomentumComponents
 PerturbationTheory --approximates--> SchrodingerSolution
 CrankNicolsonMethod --numerically_solves--> TimeDependentSchrodingerEquation
}

OPERATORS {
 TDSE(psi,H) := i*hbar*partial psi/partial t=H*psi
 TISE(psi,H,E) := H*psi=E*psi
 HAMILTONIAN(m,V) := H=-(hbar^2/(2*m))*nabla^2+V
 PROBABILITY_DENSITY(psi) := rho=psi_star*psi
 PROBABILITY_CURRENT(psi,m) := J=(hbar/(2*m*i))*(psi_star*grad(psi)-psi*grad(psi_star))
 CONTINUITY(rho,J) := partial rho/partial t+div(J)=0
 NORMALISATION(psi) := integral |psi|^2 dV=1
 EXPECTATION(psi,A) := <A>=integral psi_star*A*psi dV
 VARIANCE(psi,A) := Var(A)=<A^2>-<A>^2
 TIME_EVOLUTION(H,t) := U(t)=exp(-i*H*t/hbar) for_time_independent_H
 STATIONARY_STATE(phi,E,t) := psi(x,t)=phi(x)*exp(-i*E*t/hbar)
 FREE_PARTICLE_DISPERSION(k,m) := omega=hbar*k^2/(2*m)
 GROUP_VELOCITY(k,m) := v_g=hbar*k/m
 HARMONIC_OSCILLATOR_ENERGY(n,omega) := E_n=hbar*omega*(n+1/2)
}

AXIOMS {
 A1 = "The wavefunction must belong to the appropriate Hilbert-space domain."
 A2 = "The Hamiltonian must be self-adjoint under the declared boundary conditions for unitary closed-system evolution."
 A3 = "Physical wavefunctions must satisfy appropriate normalisation or generalised-normalisation conditions."
 A4 = "Probability density and current satisfy a continuity equation for standard Hermitian Hamiltonians."
 A5 = "Stationary states are Hamiltonian eigenstates."
 A6 = "Superpositions of solutions remain solutions for linear Schrödinger equations."
 A7 = "Boundary conditions and potentials determine the quantum spectrum."
 A8 = "Numerical Schrödinger solutions require norm, stability, spectral and convergence validation."
}

REASONING {
 R1_CONTEXT = identify(dimension,mass,potential,domain,boundary_conditions,initial_state)
 R2_FORM = classify(time_dependent,time_independent,stationary,scattering)
 R3_HAMILTONIAN = construct(kinetic_operator,potential_operator,additional_terms)
 R4_DOMAIN = define(operator_domain,boundary_conditions,normalisation)
 R5_SOLUTION = select(separation,spectral,analytic,perturbative,WKB_or_numerical_method)
 R6_SPECTRUM = compute(eigenvalues,eigenfunctions,bound_or_continuum_states)
 R7_DYNAMICS = evolve(initial_state,unitary_propagator_or_time_stepper)
 R8_OBSERVABLES = compute(probability_density,current,expectations,variances)
 R9_SCATTERING = analyse(reflection,transmission,tunnelling,asymptotic_states_if_relevant)
 R10_VALIDATION = check(norm,energy,continuity,boundaries,operator_self_adjointness,numerical_convergence)
}

CONSTRAINTS {
 C1 = "The non-relativistic Schrödinger equation is not valid when relativistic effects are dominant."
 C2 = "A Hamiltonian expression is incomplete without its operator domain and boundary conditions."
 C3 = "Continuous-spectrum eigenstates generally require distributional rather than square-integrable normalisation."
 C4 = "Stationary-state separation requires suitable time-independent Hamiltonian structure."
 C5 = "WKB approximations fail near turning points unless matched appropriately."
 C6 = "Perturbation theory can fail near degeneracy or outside a small-parameter regime."
 C7 = "Finite numerical domains can create artificial reflections."
 C8 = "Spatial grids must resolve relevant de Broglie wavelengths."
 C9 = "Time-stepping schemes should preserve norm or control norm error."
 C10 = "A visually smooth wavefunction does not by itself establish spectral or physical correctness."
}

SEMANTIC_EQUIVALENCE {
 "SchrodingerEquation" <=> "SchroedingerEquation"
 "TimeDependentSchrodingerEquation" <=> "TDSE"
 "TimeIndependentSchrodingerEquation" <=> "TISE"
 "HamiltonianOperator" <=> "QuantumHamiltonian"
 "ProbabilityDensity" <=> "BornDensity"
 "StationaryState" <=> "EnergyEigenstateWithPhaseEvolution"
}

VALIDATION {
 V1 = "Potential, mass and spatial domain are explicit."
 V2 = "Hamiltonian operator and domain are explicit."
 V3 = "Boundary conditions are explicit."
 V4 = "Wavefunction is normalised or uses declared generalised normalisation."
 V5 = "Hamiltonian self-adjointness is checked where required."
 V6 = "Probability continuity is satisfied."
 V7 = "Eigenfunctions satisfy boundary conditions."
 V8 = "Time evolution preserves norm within expected tolerance."
 V9 = "Grid and timestep resolve relevant physical scales."
 V10 = "Numerical solutions converge under refinement."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 SCHRODINGER_EQUATION_DEFINITION = 0.99
 HAMILTONIAN_REASONING = 0.99
 STATIONARY_STATE_REASONING = 0.99
 SPECTRAL_REASONING = 0.99
 WAVEFUNCTION_REASONING = 0.99
 SCATTERING_REASONING = 0.98
 PERTURBATION_REASONING = 0.98
 NUMERICAL_REASONING = 0.98
 POLICY = "Reduce confidence when potential, Hamiltonian domain, boundary conditions, mass, dimensionality, initial state, approximation regime or numerical resolution are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  SchrodingerEquationReasoning,
  TDSEReasoning,
  TISEReasoning,
  HamiltonianReasoning,
  WavefunctionReasoning,
  ProbabilityDensityReasoning,
  ProbabilityCurrentReasoning,
  StationaryStateReasoning,
  BoundStateReasoning,
  ScatteringStateReasoning,
  QuantumTunnellingReasoning,
  HarmonicOscillatorReasoning,
  WavePacketReasoning,
  SpectralDecompositionReasoning,
  PerturbationReasoning,
  WKBReasoning,
  NumericalQuantumReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "Schrödinger equation" -> SchrodingerEquation
  "Schroedinger equation" -> SchrodingerEquation
  "TDSE" -> TimeDependentSchrodingerEquation
  "TISE" -> TimeIndependentSchrodingerEquation
  "Hamiltonian" -> HamiltonianOperator
  "probability current" -> ProbabilityCurrent
  "stationary state" -> StationaryState
  "quantum tunnelling" -> QuantumTunnelling
 }
}

EXECUTION_RULES {
 E1 = "If a non-relativistic quantum wavefunction is evolved or solved using a Hamiltonian, activate SchrodingerEquationOntology."
 E2 = "If the Hamiltonian is time independent and stationary states are requested, route to the time-independent equation."
 E3 = "If an initial quantum state evolves in time, route to the time-dependent equation."
 E4 = "If a bounded potential confines the state, analyse discrete bound-state spectra."
 E5 = "If asymptotically free motion is present, analyse scattering states and coefficients."
 E6 = "If a classically forbidden barrier is present, assess tunnelling."
 E7 = "If no exact solution exists, consider perturbation, variational, WKB or numerical methods."
 E8 = "If solving numerically, preserve norm and control boundary reflections."
 E9 = "Any Schrödinger solution should be validated for normalisation, self-adjointness, boundary conditions, probability continuity and convergence."
}

RECURSION {
 ENTRY = SchrodingerEquation
 EXPAND(SchrodingerEquation) -> [
  TimeDependentSchrodingerEquation, TimeIndependentSchrodingerEquation,
  Wavefunction, QuantumState, HamiltonianOperator, Hamiltonian,
  KineticEnergyOperator, PotentialEnergyOperator, PotentialFunction,
  PositionRepresentation, MomentumRepresentation, HilbertSpace, Operator,
  SelfAdjointOperator, Eigenvalue, Eigenfunction, Eigenstate,
  EnergyEigenvalue, Spectrum, DiscreteSpectrum, ContinuousSpectrum,
  BoundState, ScatteringState, StationaryState, TimeEvolution,
  UnitaryEvolution, TimeEvolutionOperator, InitialCondition,
  BoundaryCondition, DirichletBoundaryCondition, NeumannBoundaryCondition,
  PeriodicBoundaryCondition, Normalisation, SquareIntegrability,
  ProbabilityDensity, ProbabilityCurrent, ContinuityEquation,
  ExpectationValue, Variance, Superposition, SeparationOfVariables,
  SpectralDecomposition, CompletenessRelation, InfiniteSquareWell,
  FiniteSquareWell, PotentialStep, PotentialBarrier, QuantumTunnelling,
  HarmonicOscillator, FreeParticle, PlaneWave, WavePacket,
  GaussianWavePacket, Dispersion, GroupVelocity, PhaseVelocity,
  ReflectionCoefficient, TransmissionCoefficient, Scattering,
  ScatteringAmplitude, PerturbationTheory,
  TimeIndependentPerturbationTheory, TimeDependentPerturbationTheory,
  VariationalMethod, WKBApproximation, GreenFunction, Propagator,
  ImaginaryTimeEvolution, NumericalSchrodingerEquation,
  FiniteDifferenceMethod, FiniteElementMethod, SpectralMethod,
  SplitOperatorMethod, CrankNicolsonMethod, TimeStepping,
  NumericalStability, Convergence, GridResolution
 ]
 EXPAND(SolutionFamilies) -> [
  AnalyticStationarySolutions,
  SpectralSolutions,
  ScatteringSolutions,
  PerturbativeSolutions,
  SemiclassicalSolutions,
  NumericalTimeEvolution
 ]
 STOP_CONDITION = "Stop when Hamiltonian, domain, boundary conditions, state representation, spectrum or evolution, observables and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Quantum Mathematics Ontology"
 IMMEDIATE_CONTEXTS = [
  "Quantum Mathematics Ontology",
  "Mathematical Physics Ontology",
  "Operator Theory Ontologies",
  "Spectral Theory Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Quantum Mathematics Ontology",
  "Mathematical Physics Ontology",
  "Mathematical Physics",
  "Operator Theory Ontologies",
  "Functional Analysis Ontologies",
  "Spectral Theory Ontologies",
  "Partial Differential Equation Ontology",
  "Ordinary Differential Equation Ontology",
  "Fourier Analysis Ontology",
  "Perturbation Theory Ontologies",
  "Numerical Analysis Ontologies",
  "Computational Mathematics Ontologies"
 ]
 ROUTE_RULES = [
  "Schrödinger Equation -> Quantum Mathematics Ontology for parent semantics",
  "Schrödinger Equation -> Mathematical Physics for broader physical context",
  "Schrödinger Equation -> Operator Theory for Hamiltonian domains and self-adjointness",
  "Schrödinger Equation -> Functional Analysis for Hilbert-space structure",
  "Schrödinger Equation -> Spectral Theory for energy eigenvalues and eigenfunctions",
  "Schrödinger Equation -> PDE Ontology for multidimensional evolution",
  "Schrödinger Equation -> ODE Ontology after separation of variables where applicable",
  "Schrödinger Equation -> Fourier Analysis for momentum and wave-packet representations",
  "Schrödinger Equation -> Perturbation Theory for approximate spectra and states",
  "Schrödinger Equation -> Numerical Analysis and Computational Mathematics for simulation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible potentials, approximation methods or solution pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve wavefunction-evolution semantics.",
  "Preserve time-dependent-versus-time-independent distinction.",
  "Preserve Hamiltonian-domain semantics.",
  "Preserve bound-versus-scattering state distinction.",
  "Preserve discrete-versus-continuous spectrum distinction.",
  "Preserve probability-density and current semantics.",
  "Preserve approximation-regime semantics.",
  "Preserve machine-readable Schrödinger-equation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "The Schrödinger equation governs non-relativistic quantum wavefunction evolution and stationary energy states."
 TDSE_SUMMARY = "The time-dependent Schrödinger equation generates unitary quantum evolution from an initial state."
 TISE_SUMMARY = "The time-independent Schrödinger equation is a Hamiltonian eigenvalue problem for stationary states."
 PROBABILITY_SUMMARY = "Wavefunctions determine probability density and probability current subject to continuity and normalisation."
 SPECTRAL_SUMMARY = "Potentials and boundary conditions determine discrete bound-state and continuous scattering spectra."
 TUNNELLING_SUMMARY = "Finite barriers permit nonzero transmission through classically forbidden regions."
 APPROXIMATION_SUMMARY = "Perturbation, variational and WKB methods approximate systems lacking exact solutions."
 VALIDATION_SUMMARY = "Schrödinger solutions should be checked for normalisation, Hamiltonian self-adjointness, boundary conditions, probability conservation, spectral consistency and numerical convergence."
}

FLOW {
 INPUT
 -> detect_schrodinger_equation_context
 -> identify_dimension_mass_potential_domain_boundary_conditions_initial_state_and_observables
 -> classify_time_dependent_time_independent_bound_scattering_or_mixed_problem
 -> construct_Hamiltonian_and_operator_domain
 -> choose_analytic_separation_spectral_perturbative_variational_WKB_or_numerical_method
 -> solve_eigenvalue_problem_or_time_evolution
 -> compute_probability_density_current_expectations_and_variances
 -> if_scattering_compute_reflection_transmission_and_tunnelling
 -> if_time_evolution_monitor_norm_energy_and_wave_packet_dispersion
 -> if_numerical_check_grid_resolution_timestep_stability_boundary_reflections_and_convergence
 -> route_to_quantum_mathematics_mathematical_physics_operator_theory_functional_analysis_spectral_theory_PDE_ODE_Fourier_analysis_perturbation_theory_numerical_analysis_or_computational_mathematics_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Schrödinger Equation Ontology defines non-relativistic quantum wavefunction dynamics. It supports time-dependent and time-independent Schrödinger equations, Hamiltonian, kinetic and potential operators, position and momentum representations, Hilbert-space and self-adjoint operator structure, energy eigenvalues, eigenfunctions, discrete and continuous spectra, bound, scattering and stationary states, initial and boundary conditions, normalisation and square integrability, probability density, probability current and continuity, expectation values and variance, superposition, separation of variables and spectral decomposition, infinite and finite wells, potential steps and barriers, tunnelling, harmonic oscillator, free particles, plane waves and Gaussian wave packets, dispersion, reflection and transmission, perturbation theory, variational and WKB methods, Green functions, propagators, imaginary-time evolution, and numerical solution using finite differences, finite elements, spectral, split-operator and Crank-Nicolson methods."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-SCHRODINGER-EQUATION-ONTOLOGY-V1"
}

}

END_FLOW