You are an expert in computational quantum mechanics and sci...

Prompt

You are an expert in computational quantum mechanics and scientific Python. TASK Write a COMPLETE, runnable Python program that numerically solves the radial Dirac equation for a hydrogen-like atom in its standard 2-component radial form: dG/dr = -(kappa/r) G + (1/(hbar*c)) * (E + 2*m*c^2 - V(r)) * F dF/dr = (kappa/r) F - (1/(hbar*c)) * (E - V(r)) * G where G(r) and F(r) are the large and small radial components, V(r) = -Z*e^2/r (Coulomb potential), and kappa is the relativistic quantum number. PHYSICS REQUIREMENTS 1. Use atomic units internally (hbar = m = e = 1, c = 1/alpha, alpha = 1/137.035999084). State this explicitly in code comments. 2. Solve for at least these states: 1s1/2 (kappa=-1), 2s1/2 (kappa=-1), 2p1/2 (kappa=+1), 2p3/2 (kappa=-2), for Z given as a parameter (default Z=1, must also work for Z=80 to show relativistic effects). 3. Compute bound-state energies via shooting method with outward and inward integration and matching at the classical turning point (or an equivalent robust method — justify your choice in comments). 4. Use a logarithmic (or otherwise graded) radial grid; handle the r -> 0 behavior with the correct power-series start G, F ~ r^gamma, gamma = sqrt(kappa^2 - (Z*alpha)^2). 5. Normalize solutions: integral of (G^2 + F^2) dr = 1. 6. VALIDATION: compare each computed energy to the exact Sommerfeld (Dirac) formula and print a table with columns: state | kappa | E_numeric | E_exact | relative_error. Relative error must be printed in scientific notation. SOFTWARE REQUIREMENTS 7. Only numpy, scipy, matplotlib. No other dependencies. 8. Structure the code as these components in ONE file: - class RadialGrid - class CoulombPotential - class DiracShooting (integration + eigenvalue search) - function exact_dirac_energy(n, kappa, Z) - function run_validation() producing the table - function plot_states() plotting G(r), F(r) for computed states - main() with argparse for Z and grid parameters 9. ABSOLUTELY NO placeholders, no "..." , no "left as exercise", no pseudocode, no omitted functions. Every function fully implemented. 10. The program must run end-to-end with: python dirac.py --Z 1 11. Keep comments concise and mathematical; do not narrate. OUTPUT FORMAT - First: a short (max 10 lines) summary of your numerical strategy. - Then: the single complete code file in one code block. - Nothing after the code block.