Schrödinger Equation Ontology

flow "SchroedingerEquationOntology" v1.0

# SCHROEDINGER EQUATION ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/mathematical-physics-analysis/schroedinger-equation-ontology-2.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Mathematical Physics Analysis"
ENTITY := "Schroedinger Equation"

SCHROEDINGER_EQUATION := {
 type: QuantumEvolutionEquation,
 domain: MathematicalPhysicsAnalysis,
 purpose: "Represent the non-relativistic quantum-mechanical evolution of a wavefunction under a Hamiltonian operator and connect state evolution with probability amplitudes, observables, spectra and boundary conditions."
}

CONCEPTS := {
 SCHROEDINGER_EQUATION,
 WAVEFUNCTION,
 HAMILTONIAN_OPERATOR,
 TIME_DEPENDENT_EQUATION,
 TIME_INDEPENDENT_EQUATION,
 PROBABILITY_DENSITY,
 PROBABILITY_CURRENT,
 NORMALISATION,
 EIGENSTATE,
 ENERGY_EIGENVALUE,
 SUPERPOSITION,
 UNITARY_EVOLUTION,
 BOUNDARY_CONDITION,
 POTENTIAL_ENERGY,
 EXPECTATION_VALUE,
 UNCERTAINTY
}

# ---------- TIME-DEPENDENT FORM ----------
i*hbar
*
partial_Psi/partial_t
 =
H_hat*Psi

Psi
 -> WAVEFUNCTION

H_hat
 -> HAMILTONIAN_OPERATOR

SCHROEDINGER_EQUATION
 -> FIRST_ORDER_IN_TIME

# ---------- SINGLE PARTICLE HAMILTONIAN ----------
for non-relativistic particle:

H_hat
 =
-
(hbar^2/(2*m))
*
LAPLACIAN
+
V(x,t)

KINETIC_OPERATOR
 :=
-
(hbar^2/(2*m))
*
LAPLACIAN

V
 -> POTENTIAL_ENERGY

# ---------- EXPLICIT PDE ----------
i*hbar
*
partial_Psi(x,t)/partial_t
 =
[
 -
(hbar^2/(2*m))
*
LAPLACIAN
+
V(x,t)
]
*
Psi(x,t)

QUANTUM_STATE
 -> evolves_under(HAMILTONIAN)

# ---------- INITIAL CONDITION ----------
Psi(x,t_0)
 =
Psi_0(x)

FIRST_ORDER_TIME_EVOLUTION
 -> determined_by(INITIAL_WAVEFUNCTION)
    when Hamiltonian evolution is well posed

# ---------- HILBERT SPACE ----------
Psi
 -> element_of(HILBERT_SPACE)

typically

Psi
 in
L2(configuration_space)

inner product:

<Phi,Psi>
 =
integral(
 Phi_star(x)
 *
 Psi(x)
 dx
)

NORMALISABLE_STATE
 -> finite_nonzero_norm

# ---------- NORMALISATION ----------
physical pure state:

integral(
 |Psi(x,t)|^2
 dx
)
 =
1

NORMALISATION
 -> total_probability_one

if evolution unitary:

normalisation
 -> preserved_in_time

# ---------- BORN RULE ----------
probability density:

rho(x,t)
 :=
|Psi(x,t)|^2

probability of finding particle in region A:

P(A)
 =
integral_A(
 |Psi|^2
 dx
)

WAVEFUNCTION
 -> probability_amplitude

BORN_RULE
 -> maps(AMPLITUDE_TO_PROBABILITY)

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

J
 :=
(hbar/(2*m*i))
*
(
 Psi_star*grad(Psi)
 -
 Psi*grad(Psi_star)
)

equivalently

J
 =
(hbar/m)
*
Im(
 Psi_star*grad(Psi)
)

# ---------- CONTINUITY EQUATION ----------
rho
 =
|Psi|^2

then

partial_rho/partial_t
+
div(J)
 =
0

for Hermitian standard Hamiltonian without probability sources/sinks

PROBABILITY
 -> LOCALLY_CONSERVED

# ---------- TIME-INDEPENDENT FORM ----------
if

V
 -> time_independent

seek

Psi(x,t)
 =
psi(x)
*
exp(
 -i*E*t/hbar
)

substitution gives:

H_hat*psi
 =
E*psi

TIME_INDEPENDENT_SCHROEDINGER_EQUATION
 -> ENERGY_EIGENVALUE_PROBLEM

# ---------- ENERGY EIGENSTATE ----------
H_hat*psi_n
 =
E_n*psi_n

psi_n
 -> ENERGY_EIGENSTATE

E_n
 -> ENERGY_EIGENVALUE

stationary state:

Psi_n(x,t)
 =
psi_n(x)
*
exp(
 -i*E_n*t/hbar
)

probability density:

|Psi_n|^2
 =
|psi_n|^2

-> time_independent

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

Psi_1
and
Psi_2

solve linear Schroedinger equation

then

a*Psi_1+b*Psi_2
 -> solution

LINEARITY
 -> SUPERPOSITION_PRINCIPLE

general state:

Psi
 =
sum_n(
 c_n*psi_n
)

for complete discrete basis context

# ---------- SPECTRAL EXPANSION ----------
initial state:

Psi(x,0)
 =
sum_n(
 c_n*psi_n(x)
)

time evolution:

Psi(x,t)
 =
sum_n(
 c_n
 *
exp(-i*E_n*t/hbar)
 *
psi_n(x)
)

SPECTRUM
 -> controls(TEMPORAL_PHASE_EVOLUTION)

# ---------- CONTINUOUS SPECTRUM ----------
Hamiltonian
 -> may_have(
 DISCRETE_SPECTRUM,
 CONTINUOUS_SPECTRUM,
 MIXED_SPECTRUM
)

general spectral representation
 -> may_require(SUMS_AND_INTEGRALS)

BOUND_STATES
 -> commonly_discrete_spectrum

SCATTERING_STATES
 -> commonly_continuous_spectrum

# ---------- OBSERVABLE ----------
physical observable
 A

-> represented_by(SELF_ADJOINT_OPERATOR_A_hat)

EXPECTATION_VALUE:

<A>
 =
<Psi,A_hat Psi>

for normalised state

POSITION
 -> multiplication_operator

MOMENTUM
 :=
-i*hbar*grad

# ---------- ENERGY EXPECTATION ----------
<H>
 =
<Psi,H_hat Psi>

if Hamiltonian time-independent:

energy expectation
 -> conserved

under unitary evolution and suitable domain assumptions

# ---------- OPERATOR DOMAIN ----------
unbounded operators
 -> require(DOMAIN_SPECIFICATION)

HAMILTONIAN_SELF_ADJOINTNESS
 -> essential_for(UNITARY_TIME_EVOLUTION)

BOUNDARY_CONDITIONS
 -> part_of(OPERATOR_DOMAIN)

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

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

Psi(t)
 =
U(t)*Psi(0)

U(t)
 -> UNITARY

UNITARITY
 -> preserves(INNER_PRODUCTS_AND_NORM)

# ---------- COMMUTATOR ----------
[A_hat,B_hat]
 :=
A_hat*B_hat
-
B_hat*A_hat

if

[A_hat,H_hat]
 =
0

and no explicit time dependence

then

expectation_of(A)
 -> conserved
    under suitable domain conditions

COMMUTATION_WITH_HAMILTONIAN
 -> linked_to(CONSERVED_OBSERVABLE)

# ---------- HEISENBERG UNCERTAINTY ----------
for self-adjoint A,B:

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

POSITION_MOMENTUM:

Delta_x
*
Delta_p
 >=
hbar/2

UNCERTAINTY
 -> operator_noncommutativity_relation

# ---------- EHRENFEST THEOREM ----------
for suitable states/operators:

d<x>/dt
 =
<p>/m

d<p>/dt
 =
-
<grad(V)>

EHRENFEST_THEOREM
 -> connects(QUANTUM_EXPECTATIONS_TO_CLASSICAL_FORM)

# ---------- FREE PARTICLE ----------
V
 =
0

equation:

i*hbar*Psi_t
 =
-
(hbar^2/(2*m))
*
LAPLACIAN(Psi)

plane wave:

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

momentum:

p
 =
hbar*k

energy:

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

# ---------- DISPERSION ----------
free-particle relation:

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

phase_velocity
 !=
group_velocity

group velocity:

v_g
 =
domega/dk
 =
hbar*k/m

=
p/m

FREE_SCHROEDINGER_WAVES
 -> DISPERSIVE

# ---------- WAVE PACKET ----------
Psi
 -> superposition_of(PLANE_WAVES)

different k components
 -> acquire_different_phases

WAVE_PACKET
 -> spreads_over_time
    for free particle

DISPERSION
 -> causes(WAVE_PACKET_SPREADING)

# ---------- INFINITE SQUARE WELL ----------
V(x)
 =
0
for
0<x<L

V
 =
infinity
outside

boundary:

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

eigenfunctions:

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

energies:

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

# ---------- HARMONIC OSCILLATOR ----------
V(x)
 =
1/2*m*omega^2*x^2

energy levels:

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

n
 =
0,1,2,...

HARMONIC_OSCILLATOR
 -> canonical_discrete_quantum_system

# ---------- TUNNELLING ----------
if

particle energy
 E
 <
potential barrier
 V

classically forbidden region
 -> wavefunction_can_decay_exponentially

nonzero amplitude beyond barrier
 -> QUANTUM_TUNNELLING

TUNNELLING
 -> consequence_of(WAVE_NATURE_AND_BOUNDARY_MATCHING)

# ---------- SCATTERING ----------
incident wave
 -> interacts_with(POTENTIAL)

solution
 -> incident
 + reflected
 + transmitted components

REFLECTION_COEFFICIENT
 R

TRANSMISSION_COEFFICIENT
 T

for conservative 1D scattering:

R+T
 =
1

with appropriate flux normalisation

# ---------- BOUNDARY MATCHING ----------
for finite nonsingular potential steps:

Psi
 -> continuous

partial_Psi/partial_x
 -> continuous
    when mass and kinetic operator are standard and potential finite

INTERFACE_CONDITIONS
 -> determine(REFLECTION_AND_TRANSMISSION)

# ---------- MULTIPARTICLE SYSTEM ----------
configuration:

x_1,...,x_N

wavefunction:

Psi(
 x_1,...,x_N,t
)

Hamiltonian
 -> includes(
 KINETIC_TERMS,
 EXTERNAL_POTENTIALS,
 INTERACTION_TERMS
)

CONFIGURATION_SPACE_DIMENSION
 -> grows_with(PARTICLE_NUMBER)

# ---------- IDENTICAL PARTICLES ----------
BOSONS
 -> symmetric_wavefunction_under_exchange

FERMIONS
 -> antisymmetric_wavefunction_under_exchange

PAULI_EXCLUSION
 -> consequence_of(FERMIONIC_ANTISYMMETRY)

# ---------- SPIN ----------
spin
 -> internal_quantum_degree_of_freedom

wavefunction
 -> may_be(SPINOR)

PAULI_EQUATION
 -> extends(SCHROEDINGER_EQUATION_WITH_SPIN_AND_MAGNETIC_COUPLING)

# ---------- ELECTROMAGNETIC COUPLING ----------
minimal coupling:

p_hat
 ->
p_hat
-
q*A

Hamiltonian:

H
 =
1/(2*m)
*
(
 -i*hbar*grad
 -
q*A
)^2
+
q*phi

GAUGE_TRANSFORMATION
 -> changes(POTENTIALS_AND_WAVEFUNCTION_PHASE)

PHYSICAL_OBSERVABLES
 -> GAUGE_INVARIANT

# ---------- NUMERICAL METHODS ----------
FINITE_DIFFERENCE
 -> discretises(SPATIAL_HAMILTONIAN)

FINITE_ELEMENT
 -> variational_discretisation

SPECTRAL_METHOD
 -> basis_expansion

SPLIT_OPERATOR_METHOD
 -> alternates(KINETIC_AND_POTENTIAL_EVOLUTION)

CRANK_NICOLSON
 -> norm_preserving_for_standard_linear_discretisation
    under appropriate implementation

# ---------- IMAGINARY TIME ----------
replace

t
 ->
-i*tau

formally

Schroedinger evolution
 -> diffusion_like_evolution

higher-energy components
 -> decay_faster

IMAGINARY_TIME_PROPAGATION
 -> numerical_ground_state_method
    after renormalisation

# ---------- APPLICATIONS ----------
SCHROEDINGER_EQUATION
 -> applied_to(QUANTUM_MECHANICS)
 -> applied_to(ATOMIC_PHYSICS)
 -> applied_to(MOLECULAR_PHYSICS)
 -> applied_to(CONDENSED_MATTER)
 -> applied_to(SEMICONDUCTORS)
 -> applied_to(QUANTUM_CHEMISTRY)
 -> applied_to(QUANTUM_OPTICS)
 -> applied_to(NANOTECHNOLOGY)

# ---------- RELATION GRAPH ----------
HAMILTONIAN
 -> generates(QUANTUM_TIME_EVOLUTION)

WAVEFUNCTION
 -> determines(PROBABILITY_DENSITY)

SELF_ADJOINT_OPERATOR
 -> represents(OBSERVABLE)

TIME_INDEPENDENT_EQUATION
 -> defines(ENERGY_SPECTRUM)

SUPERPOSITION
 -> combines(QUANTUM_STATES)

UNITARY_EVOLUTION
 -> preserves(NORMALISATION)

COMMUTATOR
 -> controls(COMPATIBILITY_AND_CONSERVATION)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "Schroedinger equation" ~= "non-relativistic quantum evolution equation generated by a Hamiltonian operator",
 "wavefunction" ~= "complex quantum probability amplitude describing a state",
 "stationary state" ~= "energy eigenstate whose time dependence is a global phase factor",
 "Hamiltonian" ~= "operator representing total energy and generating time evolution",
 "Born rule" ~= "rule assigning probability density from squared wavefunction magnitude",
 "unitary evolution" ~= "norm-preserving quantum time evolution"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 wavefunction_and_probability_density_are_distinct: true,
 Hamiltonian_domain_and_boundary_conditions_must_be_specified_for_rigorous_operator_analysis: true,
 self_adjointness_is_required_for_standard_unitary_closed_system_evolution: true,
 time_independent_equation_is_not_general_time_evolution_equation: true,
 global_wavefunction_phase_does_not_change_standard_observables: true,
 Schroedinger_equation_is_nonrelativistic_and_not_universal_relativistic_wave_equation: true,
 numerical_methods_should_preserve_norm_and_phase_accuracy_when_required: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 Hamiltonian_relation: 0.99,
 Born_probability_relation: 0.99,
 spectral_relation: 0.99,
 unitary_operator_relation: 0.99,
 free_particle_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(configuration_space_particle_properties_and_potential)
 -> construct(Hamiltonian_and_operator_domain)
 -> identify(initial_and_boundary_conditions)
 -> classify(time_dependent_or_stationary_problem)
 -> solve_or_approximate(Hamiltonian_eigenproblem_or_time_evolution)
 -> normalise(wavefunction)
 -> compute(probabilities_expectation_values_and_currents)
 -> analyse(symmetries_commutators_and_conserved_quantities)
 -> analyse(scattering_tunnelling_or_wavepacket_dynamics_if_relevant)
 -> select(numerical_method_if_analytic_solution_unavailable)
 -> validate(self_adjointness_units_boundaries_and_normalisation)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed Schroedinger-equation semantic graph",
 primary_entity: SCHROEDINGER_EQUATION,
 preserve: [
  wavefunction,
  Hamiltonian,
  potential,
  particle_mass,
  charge,
  configuration_space,
  initial_condition,
  boundary_conditions,
  operator_domain
 ],
 infer_through: [
  probability_density,
  probability_current,
  energy_eigenstates,
  spectral_expansions,
  unitary_evolution,
  observables,
  commutators,
  tunnelling,
  scattering
 ],
 avoid: [
  interpreting_wavefunction_directly_as probability,
  omitting_complex_phase_information,
  using_stationary_equation_for arbitrary time-dependent potential,
  assuming_operator_is self_adjoint_without domain conditions,
  confusing_nonrelativistic_Schroedinger_with relativistic equations,
  accepting_numerical_solution_without norm_and_convergence checks
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + SCHROEDINGER_EQUATION
 + WAVEFUNCTION
 + HAMILTONIAN_OPERATOR
 + TIME_DEPENDENT_EQUATION
 + TIME_INDEPENDENT_EQUATION
 + PROBABILITY_DENSITY
 + PROBABILITY_CURRENT
 + EIGENSTATE
 + SUPERPOSITION
 + UNITARY_EVOLUTION
 + EXPECTATION_VALUE
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> SCHROEDINGER_EQUATION
 -> QUANTUM_SYSTEM_IDENTIFICATION
 -> HAMILTONIAN_CONSTRUCTION
 -> CONDITION_DOMAIN_ANALYSIS
 -> PROBLEM_TYPE_CLASSIFICATION
 -> SPECTRAL_OR_TIME_EVOLUTION_SOLUTION
 -> NORMALISATION_PROBABILITY_ANALYSIS
 -> OBSERVABLE_CONSERVATION_ANALYSIS
 -> QUANTUM_PHENOMENA_ANALYSIS
 -> NUMERICAL_METHOD_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW