QSPS — Quantum Structure of the Perinuclear Space

This repository contains the LaTeX manuscript and reproducible code for the project

“Quantum structure of the perinuclear space and spectral invariants of the Coulomb field in hydrogen-like atoms.”

The goal of the project is to analyse the radial potential and probability density in hydrogen–like atoms, to visualise the perinuclear region, and to study spectral invariants of the Coulomb field.

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Requirements

• Python ≥ 3.10
• pip and virtual environments (venv)
• A working LaTeX distribution (e.g. TeX Live 2025 / MacTeX 2025)

Python packages are listed in code/requirements.txt (typical contents: numpy, scipy, matplotlib).

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Quick start: reproduce the figures

git clone https://github.com/cryptoinsider1/QSPS.git
cd QSPS

python -m venv .venv
source .venv/bin/activate           # Windows: .venv\Scripts\activate

pip install -r code/requirements.txt

Radial probability density

python code/radial_density.py

Output: paper/figures/radial_density.png.

3D probability density

python code/density_3d.py

Output: paper/figures/3d_density.png.

Adjust numerical parameters (principal quantum number n, orbital quantum number l, nuclear charge Z) directly in the scripts to explore different hydrogen-like systems.

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Building the LaTeX manuscript

cd paper
pdflatex main.tex
pdflatex main.tex    # run twice to resolve references

Resulting file: main.pdf.

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Citation

If you use this work, please cite the preprint and the repository.

Preprint Vladimir Goncharov, Quantum structure of the perinuclear space and spectral invariants of the Coulomb field in hydrogen-like atoms, 2025. DOI: to be assigned arXiv: to be assigned.

A BibTeX template:

@article{Goncharov_QSPS_2025,
  author  = {Vladimir Goncharov},
  title   = {Quantum structure of the perinuclear space and spectral invariants of the Coulomb field in hydrogen-like atoms},
  year    = {2025},
  journal = {Preprint},
  note    = {Preprint, code and materials available at https://github.com/cryptoinsider1/QSPS},
  doi     = {10.0000/qsps.2025.00001},     
  eprint  = {arXiv:0009-0004-1430-3376},          
}

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License

• Source code in code/ is released under the MIT License (see LICENSE).
• The manuscript paper/main.tex, paper/main.pdf and figures in paper/figures/ are licensed under CC BY 4.0.