Helium Atom Simulation (3D, Time-Dependent)

A compact computational physics project for constructing and visualizing helium-like two-electron quantum states with:

• hydrogen-like orbital basis functions,
• spin-singlet/triplet coupling,
• exchange-symmetrized total wavefunctions,
• time-dependent superposition dynamics.

Project Highlights

1) Reference total-state density (2D slices + 3D cloud)

View Figure (PDF)

2) Two-state comparison (1s2p-like vs 1s4f-like)

View Figure (PDF)

3) Time-dependent transition animation

• Figure snapshot (PDF): View Transition Snapshot
• Animation exports: GIF

Report

• Full PDF report: docs/Report/main.pdf
• LaTeX source entry point: docs/Report/main.tex
• Report sections:
  • Abstract
  • Introduction
  • Theory
  • Method
  • Results
  • Discussion
  • Conclusion

To rebuild the report locally:

cd docs/Report
pdflatex -interaction=nonstopmode main.tex
bibtex main
pdflatex -interaction=nonstopmode main.tex
pdflatex -interaction=nonstopmode main.tex

Repository Structure

helium-atom/
├─ docs/
│  ├─ Roadmap.md
│  └─ Report/
│     ├─ main.tex
│     ├─ abstract.tex
│     ├─ sections/
│     ├─ figures/
│     └─ references.bib
├─ notebooks/
│  ├─ helium-system.ipynb
│  ├─ cool-superposition-states.ipynb
│  └─ plot_spherical_harmonics.ipynb
├─ src/helium_atom/
│  ├─ orbitals.py
│  ├─ spin.py
│  ├─ time_dependence.py
│  └─ plotting.py
├─ tests/
└─ results/
   └─ highlights/

Physics + Numerics Summary

• One-electron orbital model:
  • psi_nlm(r,theta,phi) = R_nl(r; Z_eff) * Y_lm(theta,phi)
• Two-electron combinations:
  • product state Psi_ab(1,2)
  • symmetric/antisymmetric spatial states Psi_+, Psi_-
  • spin singlet/triplet coupling enforcing fermionic antisymmetry
• Time dependence:
  • stationary phase factor exp(-i E t)
  • non-stationary normalized superpositions for visible density dynamics
• Visualization:
  • three orthogonal log-density slices (x-z, z-y, y-x)
  • thresholded 3D probability cloud
  • fixed axis/color scaling for stable animations

Quick Start

cd helium-atom
python -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt
pip install -e .
pytest -q tests

Core Modules

• src/helium_atom/orbitals.py
  • Coulomb potentials V_nuc = -Z/r, V_ee = 1/r12
  • spherical harmonics + hydrogenic radial basis
  • orbital constructor orbital(r, theta, phi, n, l, m, z_eff)
• src/helium_atom/spin.py
  • singlet and triplet spin basis states
• src/helium_atom/time_dependence.py
  • discrete inner products, normalization, phase evolution, linear combinations
• src/helium_atom/plotting.py
  • publication-style static layouts and animation plotting helpers