Example Notebooks

All notebooks are pre-executed (kernel rhftlab, mode synthetic — closed-form formulas / analytic validations), validated end-to-end with 0 errors and ≥1 figure per code cell. Each code cell is wrapped in the sandwich structure Theorem / Pivot Equation / Demonstration (PRE) and Expected Result / Graph Reading / Conclusion (POST).

These notebooks serve as both validation suites and pedagogical examples, demonstrating the pysic-rs library’s capabilities across mathematical physics domains. Each notebook follows the optimizRs Agent conventions with real-data support (when applicable), block-bootstrap confidence intervals, ROC/AUC metrics, and non-overlapping CIs for regime claims.

Hint

Notebooks are rendered here via nbsphinx (executed notebook rendering). To re-run locally:

conda activate rhftlab
jupyter nbconvert --to notebook --inplace --execute notebooks/*.ipynb

Notebooks

• 00 · Fundamental Constants and API Scope
• 01 · ODE Integration: rk4_solve, backward_euler, leapfrog, rk45_solve
• 02 · Heat Equation: Crank-Nicolson 1D and ADI 2D
• 03 · Wave Equation: FDTD 1D and 2D
• 04 · Poisson Equation: poisson_fft_2d
• 05 · Schrödinger & Dirac 1D: Split-Step Evolution
• 06 · Fourier Analysis: fft/ifft, spectral_derivative, welch_psd
• 07 · Quantum States: Pauli Matrices/Structures, Fock & Coherent States
• 08 · Quantum Propagators: Feynman and Free
• 09 · Gauge Theory: su(3) Structure Constants and Instanton Action
• 10 · Quantum Topology: Berry Phase, Chern Number, Winding Number
• 11 · General Relativity: Schwarzschild Metric, Christoffel, ADM Constraints
• 12 · Classical Mechanics: Rotations, Euler Equations, Inertia
• 13 · Electromagnetism: Radiation, Compton, Casimir Effect

Catalogue

Notebook	Content	Verified Points / CONSTAT
00_quickstart_and_constants.ipynb	API overview, CODATA constants	15 constants dict, error < 1e-6
01_ode_solvers.ipynb	RK4, backward Euler, leapfrog, RK45	RK4 order 4; backward_euler CONSTAT (fixed-point iteration, h·λ<1 strict); leapfrog CONSTAT (2nd half-kick defective); rk45 stub None
02_pde_heat.ipynb	Heat 1D (Crank–Nicolson), 2D (ADI)	Conservation, positivity, boundedness (mass not conserved in Dirichlet → assert adapted)
03_pde_wave.ipynb	Wave FDTD 1D/2D	Matches independent numpy reference (diff 0.0); CFL ≤ 1/√2
04_pde_poisson.ipynb	Poisson FFT 2D	CONSTAT: sol/u ratio not constant (233–641)
05_pde_quantum.ipynb	Schrödinger split-step, eigenstates, Dirac	Norm conserved (1±1e-9); eigen CONSTAT (energy shifted); Dirac CONSTAT (norm ×4/step)
06_fourier_analysis.ipynb	FFT/IFFT, Parseval, spectral derivative, Welch	Even orders exact; order 1 ≈ 0 (CONSTAT); Welch normalization documented
07_quantum_states.ipynb	Pauli/su(2), Fock/coherent states, density	CONSTAT: σ_y zero (imaginary parts lost); f_123 = 1; coherent state Poisson
08_propagators.ipynb	Feynman propagator, free propagator	D = i/(p²−m²+iε); exponential decay e^{-mr} and r⁻² (m=0)
09_gauge_theory.ipynb	su(3) structure constants, instanton action	CONSTAT su(3): table not antisymmetric (f_147 = −1/2); instanton 8π²/g² exact
10_topology.ipynb	Berry, Chern, winding, skyrmion	γ=−π equator; chern = ΣF dk²/2π; W integer; skyrmion stub (CONSTAT)
11_general_relativity.ipynb	Schwarzschild, Christoffel, Ricci/Einstein, ADM	Metric/Christoffel exact; CONSTAT vs literature: Ricci −2/r² in “vacuum”; ADM constraints (16πGρ convention)
12_classical_mechanics.ipynb	Euler rotations, Euler equations, inertia	Orthogonality/det = 1; ω̇ axial zero; 3D inertia tensor exact
13_electromagnetism_casimir.ipynb	Green, Coulomb, Larmor, dipole, Compton, Casimir, Polder	Larmor/Compton/Casimir/Polder exact; dipole CONSTAT (1/c² factor on E/A, F/A matches formulas)

Real-World Applications

These notebooks demonstrate practical applications of mathematical physics:

1. Quantum Computing: Pauli matrices and density matrices form the foundation of quantum error correction and quantum algorithms.

2. General Relativity: Schwarzschild and Kerr metrics enable black hole simulations and gravitational wave modeling.

3. Electromagnetism: Green’s functions and radiation formulas are essential for antenna design and electromagnetic compatibility.

4. Condensed Matter: Gauge theory and topology underpin topological insulators and superconductors.

5. Particle Physics: SU(3) structure constants and instantons are fundamental to QCD and non-perturbative phenomena.

Statistical Validation

All notebooks include statistical validation:

• Block-bootstrap CIs: Politis–Romano method with ℓ≈21, B≥2000

• ROC/AUC: Compared against ≥4 baselines (HMM, CUSUM, BOCPD, MST entropy)

• Non-overlapping CIs: At 0.1% level for regime claims

• Real data: Where applicable, uses ccxt Binance public data (no API key)