Split-Step Fourier Diagnostics

Numerical experiments for a one-dimensional split-step Fourier solver for the time-dependent Schrödinger equation.

This repo is intentionally just the reproducibility side: Python code, generated data, regression tests, and result figures. It does not include paper source files, LaTeX build files, or submission notes.

What is here

The code studies the Strang split-step Fourier update on a periodic grid with a normalized FFT. The central check is simple but important: when the sampled potential is real and the FFT normalization is unitary, the update is a product of unitary operations. That means probability should be preserved up to floating-point error. The experiments then ask the more useful follow-up questions: where is the solution accurate, where do boundary effects show up, and how do other propagators behave under the same diagnostics?

The repository contains:

.
├── generate_figures.py          numerical experiments and figure generation
├── test_numerics.py             regression checks for core numerical behavior
├── data/                        generated JSON, text, and CSV summaries
├── figures/                     vector PDF outputs
├── assets/readme/               PNG previews used in this README
├── .github/workflows/tests.yml  CI smoke checks
├── requirements.txt
├── requirements-lock.txt
└── environment.yml

Result gallery

The images below are generated from the checked-in figure outputs. The PDF versions live in figures/.

Free Gaussian propagation. This is the clean reference case: a resolved packet moving on a periodic grid.	Finite barrier scattering. The diagnostics separate transmitted mass, reflected mass, and boundary-tail mass.
Harmonic oscillator center trajectory. The finite periodic box is treated as a modeling approximation and checked with tail diagnostics.	Method comparison. Unitary methods control norm drift; state error still depends on the method and step size.
Temporal self-convergence. The observed slope is about second order, as expected for Strang splitting.	Explicit finite-dimensional matrix check. The measured unitarity defect is at roundoff scale.

Headline diagnostics

These values come from data/results_summary.json, which is the best place to audit exact output.

Diagnostic	Current value
Free packet max norm error	3.305133944309091e-13
Free packet full-state error	3.958179842672649e-12
Free packet max center error	1.5116796703296131e-12
Barrier transmitted mass	0.04584802956943435
Barrier spectrum-weighted transmission estimate	0.045731053013186716
Barrier boundary-tail mass	0.0316741090225629
Harmonic oscillator max norm error	6.514788708500419e-13
Harmonic oscillator max center error	7.55860934482655e-05
Long-time norm error over 12000 steps	2.2341017924532025e-12
Temporal convergence slope	2.020702491197738
Explicit DFT unitarity Frobenius defect	1.0427091626200353e-13
Forward Euler final norm in comparison run	1.0732062337132835

Experiments

generate_figures.py can run the full suite or a single experiment.

Experiment key	What it checks
free	analytic Gaussian propagation, moments, tail mass, and full-state error
barrier	square-barrier scattering, transmitted/reflected mass, spectral estimate, and boundary tails
harmonic	oscillator center motion, variance, energy drift, and domain-tail behavior
norm_comparison	split-step norm preservation against forward Euler and RK4
method_comparison	SSF, Yoshida fourth order, Crank-Nicolson, RK4, and forward Euler on the same state
convergence	temporal self-convergence
spatial_convergence	spatial resolution against a high-resolution reference
long_time_norm	norm drift over a long closed-system run
unitarity_defect	explicit small-matrix unitarity defect
time_reversal	forward-then-backward recovery
precision_sensitivity	complex128 versus stored complex64 behavior

Setup

Python 3.11 or newer is recommended.

Using pip:

python -m venv .venv
python -m pip install -r requirements.txt

For an exact local environment snapshot, use requirements-lock.txt.

python -m pip install -r requirements-lock.txt

Using conda:

conda env create -f environment.yml
conda activate ssf-paper

Reproduce the outputs

Full run:

python generate_figures.py --root .

Quick smoke run:

python generate_figures.py --root . --quick

Single experiment:

python generate_figures.py --root . --experiment method_comparison

The full run refreshes:

• figures/*.pdf
• data/results_summary.json
• data/results_summary.txt
• data/time_convergence.csv
• data/spatial_convergence.csv

The script does not write TeX helper files by default. There is a --write-tex switch for local article-preparation workflows, but this repository is meant to stay focused on code and inspectable numerical outputs.

Tests

Run:

python test_numerics.py

The checks cover:

• norm preservation for the split-step update
• norm preservation for the fourth-order Yoshida composition
• norm growth for forward Euler on a generic state
• global phase alignment
• normalized DFT unitarity
• time-reversal recovery
• square-barrier transmission bounds
• consistency of the barrier spectral breakdown
• harmonic ground-state width stability
• stored complex64 update shape and dtype behavior

GitHub Actions runs these checks on Python 3.11 and 3.12, followed by a quick generation smoke test.

Notes on interpretation

Unitarity is not the same thing as accuracy. The split-step update can preserve norm while still showing phase error, boundary artifacts, or insufficient resolution. That is why the repository keeps several diagnostics side by side instead of relying on a single conservation plot.

The experiments use dimensionless units and periodic grids. Boundary-tail diagnostics are included because periodic boxes are a numerical model, not the infinite line.

Citation

If you use the code or generated diagnostics, cite this repository. Citation metadata is provided in CITATION.cff.

License

The code, tests, generated data, figures, and repository documentation are released under the MIT License. Manuscript text and journal-formatted publication assets should be reused according to the policy of the final publisher or preprint server.