Quantum Analysis Ontology

flow "QuantumAnalysisOntology" v1.0

# QUANTUM ANALYSIS ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/scientific-applications/quantum-analysis-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Scientific Applications"
ENTITY := "Quantum Analysis"

QUANTUM_ANALYSIS := {
 type: AppliedQuantumMathematicalFramework,
 domain: ScientificApplications,
 purpose: "Analyse quantum states, observables, operators, probability amplitudes, spectra, time evolution and measurement using calculus, linear algebra, functional analysis and differential equations."
}

CONCEPTS := {
 QUANTUM_STATE,
 WAVEFUNCTION,
 HILBERT_SPACE,
 OPERATOR,
 OBSERVABLE,
 HAMILTONIAN,
 SCHROEDINGER_EQUATION,
 EIGENSTATE,
 EIGENVALUE,
 PROBABILITY_AMPLITUDE,
 BORN_RULE,
 EXPECTATION_VALUE,
 COMMUTATOR,
 UNCERTAINTY,
 SUPERPOSITION,
 UNITARY_EVOLUTION,
 MEASUREMENT,
 ENTANGLEMENT
}

# ---------- QUANTUM STATE ----------
state:

|psi>
 -> QUANTUM_STATE

pure state
 -> ray_in(HILBERT_SPACE)

normalisation:

<psi|psi>
 =
1

GLOBAL_PHASE
 -> does_not_change(PHYSICAL_PURE_STATE)

# ---------- WAVEFUNCTION ----------
position representation:

psi(x,t)
 :=
<x|psi(t)>

probability density:

rho(x,t)
 =
|psi(x,t)|^2

WAVEFUNCTION
 -> COMPLEX_PROBABILITY_AMPLITUDE

# ---------- BORN RULE ----------
probability of region A:

P(A)
 =
integral_A(
 |psi(x,t)|^2
 dx
)

BORN_RULE
 -> maps(AMPLITUDE_TO_PROBABILITY)

# ---------- HILBERT SPACE ----------
H
 -> complete_inner_product_space

inner product:

<phi|psi>

norm:

||psi||
 =
sqrt(<psi|psi>)

QUANTUM_STATES
 -> represented_in(HILBERT_SPACE)

# ---------- OBSERVABLE ----------
observable A
 -> SELF_ADJOINT_OPERATOR(A_hat)

measurement values
 -> spectrum_of(A_hat)

EXPECTATION:

<A>
 =
<psi|A_hat|psi>

for normalised state

# ---------- EIGENVALUE EQUATION ----------
A_hat|a_n>
 =
a_n|a_n>

a_n
 -> EIGENVALUE

|a_n>
 -> EIGENSTATE

MEASUREMENT_OF_A
 -> possible_outcomes_from(SPECTRUM)

# ---------- SPECTRAL DECOMPOSITION ----------
for discrete nondegenerate illustrative case:

A_hat
 =
sum_n(
 a_n
 *
|a_n><a_n|
)

general case
 -> may_require(PROJECTORS_AND_CONTINUOUS_SPECTRAL_MEASURES)

SPECTRAL_THEORY
 -> foundation_of(QUANTUM_OBSERVABLE_ANALYSIS)

# ---------- SUPERPOSITION ----------
if

|psi_1>
and
|psi_2>

are states

then

alpha|psi_1>
+
beta|psi_2>

-> state_after_normalisation_if_nonzero

SUPERPOSITION
 -> linear_structure_of(QUANTUM_STATE_SPACE)

# ---------- SCHROEDINGER EQUATION ----------
i*hbar
*
d|psi>/dt
 =
H_hat|psi>

H_hat
 -> HAMILTONIAN

HAMILTONIAN
 -> generates(TIME_EVOLUTION)

# ---------- POSITION REPRESENTATION ----------
i*hbar
*
partial_psi/partial_t
 =
[
 -(hbar^2/(2*m))*LAPLACIAN
 +
V(x,t)
]
*
psi

for standard nonrelativistic spinless particle

# ---------- TIME-INDEPENDENT ----------
if Hamiltonian time independent:

H_hat|n>
 =
E_n|n>

stationary state:

|psi_n(t)>
 =
exp(
 -i*E_n*t/hbar
)
*
|n>

ENERGY_EIGENSTATE
 -> time_invariant_probability_density
    in position representation

# ---------- UNITARY EVOLUTION ----------
for time-independent self-adjoint H:

U(t)
 =
exp(
 -i*H*t/hbar
)

|psi(t)>
 =
U(t)|psi(0)>

U_dagger*U
 =
I

UNITARITY
 -> preserves(NORM_AND_INNER_PRODUCT)

# ---------- TIME-ORDERED EVOLUTION ----------
if

H=H(t)

then evolution
 -> TIME_ORDERED_EXPONENTIAL

U(t,t0)
 =
T_exp(
 -(i/hbar)
 *
integral_t0^t H(s) ds
)

TIME_DEPENDENT_HAMILTONIAN
 -> generally_requires(TIME_ORDERING)

# ---------- COMMUTATOR ----------
[A,B]
 :=
A*B-B*A

if

[A,B]=0

then
 -> compatible_simultaneous_spectral_structure
    under appropriate operator hypotheses

NONCOMMUTATIVITY
 -> central_quantum_structure

# ---------- UNCERTAINTY ----------
standard deviations:

Delta_A
 :=
sqrt(
 <A^2>-<A>^2
)

Delta_B
 :=
sqrt(
 <B^2>-<B>^2
)

Robertson relation:

Delta_A*Delta_B
 >=
1/2
*
|<[A,B]>|

# ---------- POSITION / MOMENTUM ----------
x_hat
 -> multiplication_by(x)

p_hat
 :=
-i*hbar*grad

canonical commutator:

[x_i,p_j]
 =
i*hbar*delta_ij

therefore

Delta_x*Delta_p
 >=
hbar/2

# ---------- PROBABILITY CURRENT ----------
for standard scalar potential:

J
 =
(hbar/m)
*
Im(
 psi_star*grad(psi)
)

continuity:

partial_rho/partial_t
+
div(J)
 =
0

PROBABILITY
 -> locally_conserved_under(HERMITIAN_CLOSED_SYSTEM_DYNAMICS)

# ---------- MEASUREMENT ----------
observable:

A_hat
 =
sum_a(
 a*P_a
)

probability of outcome a:

P(a)
 =
<psi|P_a|psi>

POST_MEASUREMENT_STATE
 -> depends_on(MEASUREMENT_MODEL)

PROJECTIVE_MEASUREMENT
 -> one_specific_measurement_framework

# ---------- DENSITY MATRIX ----------
pure state:

rho_hat
 =
|psi><psi|

mixed state:

rho_hat
 =
sum_i(
 p_i
 *
|psi_i><psi_i|
)

properties:

rho_hat
 -> positive_semidefinite
trace(rho_hat)=1

DENSITY_OPERATOR
 -> represents(PURE_OR_MIXED_STATE)

# ---------- EXPECTATION WITH DENSITY MATRIX ----------
<A>
 =
Tr(
 rho_hat*A_hat
)

DENSITY_MATRIX_FORMALISM
 -> supports(MIXED_STATES_AND_SUBSYSTEMS)

# ---------- VON NEUMANN EQUATION ----------
closed-system density evolution:

i*hbar
*
d rho_hat/dt
 =
[
 H_hat,
 rho_hat
]

VON_NEUMANN_EQUATION
 -> operator_form_of(QUANTUM_TIME_EVOLUTION)

# ---------- ENTANGLEMENT ----------
composite system:

H_AB
 =
H_A tensor_product H_B

state
 |psi_AB>

ENTANGLED
 if

|psi_AB>
 -/> factorises_as(
 |psi_A> tensor_product |psi_B>
)

ENTANGLEMENT
 -> nonseparable_quantum_correlation

# ---------- REDUCED STATE ----------
rho_AB
 -> composite_density_operator

rho_A
 =
Tr_B(
 rho_AB
)

PARTIAL_TRACE
 -> removes(SUBSYSTEM_B)

REDUCED_DENSITY_MATRIX
 -> local_state_of(SUBSYSTEM)

# ---------- ENTANGLEMENT ENTROPY ----------
for pure bipartite state:

S_A
 =
-
Tr(
 rho_A*log(rho_A)
)

VON_NEUMANN_ENTROPY
 -> measures(BIPARTITE_ENTANGLEMENT_FOR_PURE_STATE)

# ---------- TENSOR PRODUCT ----------
independent subsystems:

H_total
 =
H_A tensor_product H_B

operators:

A_on_A
 =
A tensor_product I_B

COMPOSITE_SYSTEM
 -> tensor_product_structure

# ---------- FREE PARTICLE ----------
H
 =
p^2/(2*m)

plane wave:

psi
 =
A*exp(
 i*(k dot x-omega*t)
)

p
 =
hbar*k

E
 =
hbar*omega
 =
hbar^2*k^2/(2*m)

FREE_PARTICLE
 -> DISPERSIVE_WAVE_DYNAMICS

# ---------- INFINITE WELL ----------
0<x<L

boundary:

psi(0)=0
psi(L)=0

eigenfunctions:

psi_n
 =
sqrt(2/L)
*
sin(n*pi*x/L)

energies:

E_n
 =
hbar^2*pi^2*n^2
/
(2*m*L^2)

BOUNDARY_CONDITIONS
 -> quantise(ENERGY_LEVELS)

# ---------- HARMONIC OSCILLATOR ----------
H
 =
p^2/(2*m)
+
1/2*m*omega^2*x^2

energies:

E_n
 =
hbar*omega*(n+1/2)

LADDER_OPERATORS
 -> generate(EIGENSTATES)

# ---------- TUNNELLING ----------
classically forbidden region:

E<V

wavefunction
 -> exponentially_decaying_component

finite barrier
 -> nonzero_transmission_possible

QUANTUM_TUNNELLING
 -> wave_boundary_effect

# ---------- PERTURBATION THEORY ----------
Hamiltonian:

H
 =
H_0
+
lambda*V

energy:

E_n
 =
E_n^(0)
+
lambda*E_n^(1)
+
...

first-order nondegenerate correction:

E_n^(1)
 =
<n^(0)|V|n^(0)>

PERTURBATION_THEORY
 -> approximate_spectrum_for(SMALL_INTERACTION)

# ---------- VARIATIONAL PRINCIPLE ----------
ground-state energy:

E_0
 <=
<psi_trial|H|psi_trial>
/
<psi_trial|psi_trial>

for admissible trial state

VARIATIONAL_METHOD
 -> upper_bound_to(GROUND_STATE_ENERGY)

# ---------- EHRENFEST ----------
d<A>/dt
 =
(i/hbar)
*
<[H,A]>
+
<partial_A/partial_t>

under suitable domain conditions

EHRENFEST_THEOREM
 -> expectation_value_dynamics

# ---------- CLASSICAL LIMIT ----------
quantum dynamics
 -> may_approach(CLASSICAL_BEHAVIOUR)

under appropriate:
 LARGE_ACTION
 DECOHERENCE
 SEMICLASSICAL
 WKB
 conditions

CLASSICAL_LIMIT
 -/> simply(hbar_equals_zero)

# ---------- NUMERICAL METHODS ----------
quantum problem
 -> may_require(
 FINITE_DIFFERENCE,
 FINITE_ELEMENT,
 SPECTRAL_METHOD,
 SPLIT_OPERATOR,
 MATRIX_DIAGONALISATION,
 MONTE_CARLO
)

NUMERICAL_QUANTUM_ANALYSIS
 -> approximate(STATES_AND_SPECTRA)

# ---------- SPLIT OPERATOR ----------
H
 =
T+V

short-time evolution:

exp(
 -i*H*Delta_t/hbar
)

-> approximated_by(SPLITTING_OF_KINETIC_AND_POTENTIAL_PROPAGATORS)

SPLIT_OPERATOR
 -> efficient_for(GRID_WAVEFUNCTION_EVOLUTION)

# ---------- VALIDATION ----------
numerical state
 -> check(
 NORMALISATION,
 ENERGY_BEHAVIOUR,
 BOUNDARY_CONDITIONS,
 GRID_CONVERGENCE,
 TIME_STEP_CONVERGENCE
)

EIGENSTATES
 -> verify(ORTHOGONALITY_AND_RESIDUAL)

QUANTUM_RESULT
 -> requires(NUMERICAL_AND_PHYSICAL_VALIDATION)

# ---------- APPLICATIONS ----------
QUANTUM_ANALYSIS
 -> applied_to(ATOMIC_PHYSICS)
 -> applied_to(MOLECULAR_PHYSICS)
 -> applied_to(QUANTUM_CHEMISTRY)
 -> applied_to(CONDENSED_MATTER)
 -> applied_to(SEMICONDUCTORS)
 -> applied_to(QUANTUM_OPTICS)
 -> applied_to(QUANTUM_INFORMATION)
 -> applied_to(NANOTECHNOLOGY)

# ---------- RELATION GRAPH ----------
QUANTUM_STATE
 -> lives_in(HILBERT_SPACE)

OBSERVABLE
 -> represented_by(SELF_ADJOINT_OPERATOR)

HAMILTONIAN
 -> generates(TIME_EVOLUTION)

WAVEFUNCTION
 -> determines(PROBABILITY_AMPLITUDE)

BORN_RULE
 -> determines(MEASUREMENT_PROBABILITY)

COMMUTATOR
 -> determines(COMPATIBILITY_AND_UNCERTAINTY)

TENSOR_PRODUCT
 -> constructs(COMPOSITE_SYSTEM)

PARTIAL_TRACE
 -> constructs(REDUCED_STATE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "quantum analysis" ~= "mathematical analysis of quantum states, operators, spectra, evolution and measurement",
 "wavefunction" ~= "complex probability amplitude representing a quantum state in a chosen basis",
 "Hamiltonian" ~= "energy operator generating quantum time evolution",
 "observable" ~= "self-adjoint operator associated with measurable quantities",
 "Born rule" ~= "rule converting quantum amplitudes into measurement probabilities",
 "density matrix" ~= "positive unit-trace operator representing pure or mixed quantum states"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 wavefunction_and_probability_density_are_distinct: true,
 operator_domains_are_required_for_rigorous_unbounded_operator_analysis: true,
 self_adjointness_is_required_for_standard_closed_system_unitary_evolution: true,
 global_phase_is_not_observable_for_isolated_pure_state: true,
 noncommuting_observables_require_uncertainty_and_measurement_context: true,
 projective_measurement_is_not_the_only_general_quantum_measurement_framework: true,
 numerical_quantum_results_require_normalisation_residual_and_convergence_checks: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 Hilbert_operator_relation: 0.99,
 Schroedinger_relation: 0.99,
 Born_measurement_relation: 0.99,
 uncertainty_relation: 0.99,
 density_entanglement_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(system_state_space_Hamiltonian_observables_and_units)
 -> choose(wavefunction_density_matrix_or_operator_representation)
 -> impose(normalisation_boundary_and_symmetry_conditions)
 -> solve(spectral_or_time_evolution_problem)
 -> compute(probabilities_expectations_currents_and_uncertainties)
 -> analyse(commutators_conservation_and_measurement_structure)
 -> analyse(composite_states_entanglement_or_reduced_states_if_relevant)
 -> choose(perturbative_variational_semiclassical_or_numerical_method)
 -> verify(normalisation_residuals_unitarity_and_convergence)
 -> validate(physical_interpretation_and_model_scope)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed quantum-analysis semantic graph",
 primary_entity: QUANTUM_ANALYSIS,
 preserve: [
  state,
  Hilbert_space,
  Hamiltonian,
  observables,
  basis,
  boundary_conditions,
  normalisation,
  units,
  measurement_context
 ],
 infer_through: [
  wavefunctions,
  operators,
  eigenstates,
  spectra,
  unitary_evolution,
  Born_probabilities,
  commutators,
  density_matrices,
  entanglement,
  perturbation_theory
 ],
 avoid: [
  interpreting_wavefunction_as probability_density_directly,
  ignoring_operator_domains,
  treating_non_self_adjoint_Hamiltonian_as standard_closed_system_generator,
  treating_global_phase_as measurable,
  assuming_all_measurements_are simple_projective_measurements,
  accepting_numerical_state_without convergence_checks
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(quantum_analysis_context)
 -> compare(existing_relations)
 -> preserve(state_operator_Hamiltonian_basis_normalisation_and_measurement_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + QUANTUM_ANALYSIS
 + QUANTUM_STATE
 + WAVEFUNCTION
 + HILBERT_SPACE
 + OPERATOR
 + HAMILTONIAN
 + SCHROEDINGER_EQUATION
 + BORN_RULE
 + EXPECTATION_VALUE
 + COMMUTATOR
 + UNITARY_EVOLUTION
 + MEASUREMENT
 + ENTANGLEMENT
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> QUANTUM_ANALYSIS
 -> SYSTEM_REPRESENTATION_IDENTIFICATION
 -> CONDITION_NORMALISATION_ANALYSIS
 -> SPECTRAL_EVOLUTION_ANALYSIS
 -> PROBABILITY_EXPECTATION_ANALYSIS
 -> COMMUTATOR_MEASUREMENT_ANALYSIS
 -> COMPOSITE_ENTANGLEMENT_ANALYSIS
 -> METHOD_SELECTION
 -> NUMERICAL_VERIFICATION
 -> PHYSICAL_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW