Wave Packet Ontology

FLOW ALGEBRA
WAVE PACKET ONTOLOGY
Version: 1.1
Library: Kireth Semantic Library
Source URL: https://kirethresearch.org/booksai-core/entity-graph/physical-entities/wave-entities/wave-packet-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Status: Independent scientific synthesis; source page not independently verified

01. PURPOSE
Model spatially or temporally localised superpositions of waves, their spectral composition, propagation, dispersion, energy transport and quantum interpretations.

02. SCOPE
Applies to linear wave theory, optics, acoustics, water waves and quantum wavefunctions where the governing assumptions are satisfied.

03. ENTITY TYPES
WavePacket : WaveEntity
WaveComponent : WaveEntity
CarrierWave : WaveComponent
Envelope : FieldFunction
Amplitude : ScalarOrComplexField
Phase : AngleField
WaveNumber : RealQuantity [rad m^-1]
AngularFrequency : RealQuantity [rad s^-1]
Spectrum : ComplexSpectralDistribution
DispersionRelation : Function k -> omega
GroupVelocity : Velocity
PhaseVelocity : Velocity
Medium : PhysicalEntity
Boundary : SpatialConstraint
WaveEquation : GoverningEquation
QuantumState : HilbertSpaceVector
Measurement : Observation

04. TYPED RELATIONSHIPS
COMPOSED_OF(WavePacket, WaveComponent)
HAS_ENVELOPE(WavePacket, Envelope)
HAS_SPECTRUM(WavePacket, Spectrum)
PROPAGATES_IN(WavePacket, Medium)
OBEYS(WavePacket, WaveEquation)
SATISFIES(WaveComponent, DispersionRelation)
HAS_GROUP_VELOCITY(WavePacket, GroupVelocity)
HAS_PHASE_VELOCITY(CarrierWave, PhaseVelocity)
INTERACTS_WITH(WavePacket, Boundary)
REPRESENTS(QuantumState, WavePacket) [context dependent]

05. CANONICAL SUPERPOSITION
For a one-dimensional linear field:
  psi(x,t) = integral A(k) exp(i[k*x - omega(k)*t]) dk
Physical real-valued displacement may be Re(psi).
A(k) is a complex spectral amplitude; normalisation convention must be declared.

06. FOURIER PAIR
Convention:
  A(k) = (1/sqrt(2*pi)) integral psi(x,0) exp(-i*k*x) dx
  psi(x,0) = (1/sqrt(2*pi)) integral A(k) exp(i*k*x) dk
These expressions require suitable function spaces or distributional interpretation.

07. CARRIER AND ENVELOPE
For a narrowband packet:
  psi(x,t) approximately a(x,t) exp(i[k0*x - omega0*t])
Envelope a varies slowly relative to the carrier when the slowly varying envelope approximation applies.

08. DISPERSION EXPANSION
  omega(k) = omega0 + omega'(k0)*(k-k0)
             + 0.5*omega''(k0)*(k-k0)^2 + ...
First derivative controls group translation; higher derivatives control dispersion.

09. PHASE VELOCITY
  v_phase = omega(k0)/k0, for k0 != 0
Phase velocity is not automatically the velocity of energy, information or a signal front.

10. GROUP VELOCITY
  v_group = d omega / dk at k0
For a narrowband packet in a differentiable dispersion relation, the envelope travels approximately at v_group.
Group velocity need not equal energy or information velocity in every medium.

11. NONDISPERSIVE CASE
If omega(k) = v*k over the relevant positive-k bandwidth, then v_group = v_phase = v.
An ideal packet translates without changing shape in the corresponding linear model.

12. DISPERSIVE CASE
If omega''(k0) != 0, spectral components acquire different relative phases and the packet can broaden or chirp.
The precise evolution depends on initial spectral phase and higher dispersion orders.

13. GAUSSIAN INITIAL PACKET
  psi(x,0) = C exp[-(x-x0)^2/(4*sigma_x^2)] exp(i*k0*x)
Here |psi|^2 has spatial standard deviation sigma_x after probability normalisation.
For the Fourier convention above, sigma_k = 1/(2*sigma_x).

14. FOURIER UNCERTAINTY
For square-integrable normalised amplitudes with finite variances:
  sigma_x * sigma_k >= 1/2
Equality is achieved by an unchirped Gaussian under the stated convention.
This is a Fourier localisation bound, not by itself a measurement-disturbance claim.

15. QUANTUM CONNECTION
With p = hbar*k, a normalised quantum wavefunction satisfies:
  sigma_x * sigma_p >= hbar/2
Probability density is |psi(x,t)|^2 in the position representation.
A classical mechanical wave amplitude is not automatically a probability amplitude.

16. SCHRODINGER PACKET
For a free nonrelativistic particle:
  omega(k) = hbar*k^2/(2*m)
  v_group = hbar*k0/m = p0/m
  v_phase = hbar*k0/(2*m) for k0 != 0
This phase velocity is not particle transport velocity.

17. FREE GAUSSIAN SPREADING
For a minimum-uncertainty initial free-particle Gaussian:
  sigma_x(t) = sigma_x(0)*sqrt(1 + [hbar*t/(2*m*sigma_x(0)^2)]^2)
Assumes one spatial dimension and a free nonrelativistic particle.

18. ENERGY AND INTENSITY
Energy density and flux require the relevant governing theory.
Do not infer universal energy density solely from |psi|^2 for arbitrary wave media.

19. INTERFERENCE
  psi_total = psi_1 + psi_2
  |psi_total|^2 = |psi_1|^2 + |psi_2|^2 + 2*Re(psi_1*conj(psi_2))
Interference requires compatible coherent contributions and appropriate observables.

20. REFLECTION AND TRANSMISSION
At an interface, impose field-specific boundary conditions.
Reflection and transmission coefficients depend on impedance, frequency, angle and medium.
Energy conservation checks require flux-normalised coefficients.

21. GROUP DELAY
For a transfer phase phi(omega), group delay is convention-dependent:
  tau_g = -d phi/d omega when transfer convention is H=|H|exp(-i*phi).
Always record Fourier and phase sign conventions.

22. CHIRP
Instantaneous angular frequency is omega_inst(t) = d theta(t)/dt for temporal phase theta.
A chirped packet has time-dependent instantaneous frequency.

23. COHERENCE
Coherence length and time depend on spectrum and coherence definition.
A rough narrowband scaling is tau_coh ~ 1/Delta_f, not a universal exact identity.

24. DIMENSIONAL VALIDATION
k : rad/m; omega : rad/s; omega/k : m/s; d omega/d k : m/s.
Check all derivatives and numerical inputs for consistent units.

25. INFERENCE RULES
R1: NARROWBAND(packet) and DIFFERENTIABLE(omega,k0)
    -> APPROX_GROUP_SPEED(packet, omega'(k0)).
R2: LINEAR_SYSTEM and SUPERPOSITION_VALID
    -> SUM_OF_SOLUTIONS_IS_SOLUTION.
R3: NONZERO_SECOND_DISPERSION and FINITE_BANDWIDTH
    -> DISPERSION_EFFECT_POSSIBLE, not guaranteed monotonic broadening.
R4: NORMALISED_QUANTUM_STATE and POSITION_REPRESENTATION
    -> POSITION_DENSITY = |psi|^2.
R5: GAUSSIAN_MINIMUM_UNCERTAINTY
    -> sigma_x*sigma_k = 1/2.

26. WORKED EXAMPLE: TWO NEARBY WAVES
  cos(k1*x-omega1*t) + cos(k2*x-omega2*t)
  = 2*cos([(k1-k2)*x-(omega1-omega2)*t]/2)
      *cos([(k1+k2)*x-(omega1+omega2)*t]/2)
The first factor describes beating; the second is the carrier.
For close components, envelope speed approaches d omega/d k.

27. WORKED EXAMPLE: GAUSSIAN UNCERTAINTY
Choose sigma_x = 2.0e-3 m.
Then a minimum-uncertainty Gaussian has sigma_k = 250 m^-1.
Product = 0.5, consistent with the stated Fourier convention.

28. WORKED EXAMPLE: FREE ELECTRON
For mean momentum p0 and electron mass m_e:
  v_group = p0/m_e
Valid when p0 << m_e*c so the nonrelativistic approximation is appropriate.

29. SEMANTIC EQUIVALENCE
WavePacket ~= LocalisedWaveSuperposition [in linear wave contexts]
Envelope != CarrierWave
PhaseVelocity != GroupVelocity [generally]
WavePacket != QuantumParticle [ontology types differ]
SpectralBandwidth != SpatialWidth [related by Fourier transform, not identical]

30. ONTOLOGY BOUNDARIES
Distinguish classical waves, probability amplitudes, solitons, pulses and wave packets.
Solitons require nonlinear/dispersive balance and are not interchangeable with arbitrary linear packets.

31. ASSUMPTIONS
Record linearity, homogeneity, isotropy, dimensionality, dispersion law, bandwidth,
boundary conditions, normalisation, measurement convention and medium properties.

32. VALIDATION TESTS
T01: Fourier inversion reconstructs input within tolerance.
T02: Normalised quantum packet integrates |psi|^2 to 1.
T03: sigma_x*sigma_k >= 0.5 under declared convention.
T04: omega(k)=v*k gives constant group speed v.
T05: Numerically evolved free Gaussian matches analytic width.
T06: Energy-flux checks use medium-specific equations.
T07: Units of all wave parameters are consistent.
T08: No universal identification of group speed with information speed.

33. CONFIDENCE
High: Fourier superposition, group velocity derivative, Gaussian uncertainty under assumptions.
Conditional: envelope approximations, pulse broadening, energy transport and coherence estimates.
Not established: model-specific conclusions without medium and boundary parameters.

34. PROVENANCE
Source page URL recorded above.
Document is an independent scientific synthesis, not a verified transcription of the webpage.
Every numeric empirical claim should retain its measurement source, date and uncertainty.

35. FLOW_HASH
Algorithm: SHA-256 over canonical UTF-8 body, excluding mutable hash value.
Canonicalisation: LF line endings, Unicode NFC, stable field order.
Value: NOT_COMPUTED (compute after final content freeze).

36. UPDATE_MODEL
On revision: update semantic version, date, changed definitions, dependent rules and validation results.
Recompute FLOW_HASH after canonicalisation and flag affected linked ontologies.

37. AI_CONSUMPTION
Parse entity types and typed relations first.
Apply inference rules only if preconditions and physical assumptions hold.
Distinguish theorem, approximation, empirical observation and hypothesis.
Return units, uncertainty, provenance and relevant limitations with conclusions.

38. CROSS-ONTOLOGY LINKS
Parent: Wave Entities
Related: Wave Ontology; Travelling Wave Ontology; Standing Wave Ontology;
Electromagnetic Wave Ontology; Mechanical Wave Ontology; Quantum State Ontology;
Fourier Analysis; Dispersion; Interference; Group Velocity.

39. COMPLETENESS CHECK
Required sections: 01 through 39 inclusive.
Required metadata: FLOW ALGEBRA, version, source, authors, confidence, FLOW_HASH,
UPDATE_MODEL, AI_CONSUMPTION, validation and closing marker.

END FLOW ALGEBRA — WAVE PACKET ONTOLOGY — COMPLETE