1D Time-Dependent Schrödinger Equation — Explicit Staggered Leapfrog

Graduate computational-physics final project. We numerically integrate the 1D TDSE

$$ i\hbar \partial_t \psi(x,t) = \hat H \psi(x,t) = \Big[-\tfrac{\hbar^2}{2m}\partial_x^2 + V(x,t)\Big]\psi(x,t) $$

with the explicit staggered-time leapfrog scheme due to P. B. Visscher (Computers in Physics 5, 596 (1991)) — the only known explicit finite-difference scheme for the TDSE that is both conditionally stable and conserves a quadratic invariant exactly. We compare it against three companion schemes: naïve Forward-Time Centered-Space (FTCS, unconditionally unstable), the implicit Crank–Nicolson scheme (unconditionally stable, exactly unitary), and an FFT-based split-operator Strang scheme (unitary, spectrally accurate in space, $O(N\log N)$ per step). The four-way comparison motivates the design choices in modern quantum solvers.

The numerical experiments progress from analytically solvable benchmarks (free Gaussian, infinite well, harmonic oscillator) through fully numerical scattering problems (rectangular and double barriers, double wells) to a thoroughly explored suite of time-dependent Hamiltonians — driven oscillator, Rabi oscillations, Landau–Zener transitions, adiabatic / sudden crossover, infinite-well revivals (quantum carpets), autocorrelation spectroscopy, photoionisation, and high-harmonic generation from a 1D soft-Coulomb atom. A dedicated breaking-points notebook exhibits the failure modes that every numerical PDE practitioner needs to recognize.

Every numerical experiment in this project includes a six-panel conservation diagnostics figure monitoring simultaneously the ordinary norm $\int|\psi|^2 dx$, the Visscher modified norm, the kinetic / potential / total energy decomposition $\langle T\rangle, \langle V\rangle, \langle H\rangle$, the position and momentum expectations $\langle x\rangle, \langle p\rangle$, and the position uncertainty $\sigma_x$. Each major time evolution is also saved as an animated GIF in figures/.

Contents

tdse/                              core simulation package
├── units.py                       SI scaling and display helpers
├── grid.py                        spatial / temporal grids
├── potentials.py                  V(x), V(x,t) library
├── boundaries.py                  hard wall, periodic, absorbing mask
├── solvers.py                     Visscher leapfrog, FTCS, Crank–Nicolson, FFT split-operator
├── observables.py                 <x>, <p>, <T>, <V>, <H>, norms, R/T, timeseries
├── analytical.py                  closed-form benchmarks + transfer-matrix T(E),R(E)
└── viz.py                         animations, space-time heatmaps, conservation panels

notebooks/
├── 00_theory_and_method.ipynb     derivation, stability, dispersion analysis
├── 01_free_and_bound_states.ipynb free Gaussian (+convergence-order plot), ISW, quantum SHO (eigenstate + coherent), finite well
├── 02_scattering.ipynb            step, barrier, double barrier, double well (transfer-matrix T(E))
├── 03_time_dependent.ipynb        NINE time-dependent experiments (see below)
└── 04_breaking_points.ipynb       seven failure modes + boundary comparison

figures/                           animated GIFs produced by the notebooks
├── 01_free_gaussian.gif           free Gaussian wave-packet spreading
├── 01_isw_beating.gif             infinite-well two-state beats
├── 01_coherent_state.gif          quantum SHO coherent state oscillation
├── 02_step.gif                    step-potential scattering
├── 02_tunnelling.gif              quantum tunnelling through a barrier
├── 02_double_barrier_resonant.gif on-resonance resonant tunnelling
├── 02_double_well.gif             double-well tunnel oscillations
├── 03_driven_resonant.gif         driven SHO on resonance
├── 03_driven_off_resonant.gif     driven SHO with detuning
├── 03_rabi_isw.gif                Rabi oscillation in a driven ISW
├── 03_landau_zener.gif            Landau-Zener avoided-crossing transfer
├── 03_adiabatic_sudden.gif        widening-well adiabatic/sudden demo
├── 03_quantum_carpet.gif          fractional-revival quantum carpet in an ISW
├── 03_photoionisation.gif         pulsed-laser ionisation from a finite well
└── 03_hhg.gif                     HHG in a 1D soft-Coulomb atom

tests/
├── conftest.py                    sys.path stub so the suite runs without `pip install -e .`
├── test_basic.py                  fast (<2 s) package + analytical-reference checks
├── test_solvers.py                pytest version of the old `smoke.py`
└── test_time_dependent.py         pytest version of the old `smoke_03.py` (marked `slow`)

tools/
├── build_notebooks.py             single-source generator for all five notebooks
└── build_report_figures.py        snapshot GIFs + regenerate analysis figures for the LaTeX report

report/
├── report.tex                     formal 15-20 page LaTeX write-up of theory + results
├── refs.bib                       bibliography
├── README.md                      build instructions
└── figures/                       PNG snapshots used by report.tex (regenerable)

.github/workflows/ci.yml           GitHub Actions: fast tests + slow tests + nb regen
pyproject.toml                     installable package metadata (`pip install -e .`)

Companion LaTeX report

In addition to the Jupyter notebooks, the project ships a fully typeset, ~17-page LaTeX report at report/report.tex. It contains:

• the full derivations from notebook 00 (Visscher leapfrog update, conserved-modified-norm proof, von-Neumann CFL bound, discrete dispersion analysis, Crank-Nicolson and FFT split-operator reference schemes);
• a curated "selected results" section that references the key figures from notebooks 01-04;
• a rigorous Appendix A justifying, system by system, why a one-dimensional treatment is sufficient (and in most cases exact) for every physics experiment in the project, including the dipole + single-active-electron + cylindrical-projection argument that underpins the use of the 1D soft-Coulomb model for HHG;
• an Appendix B with the full file map and reproducibility instructions.

To build the PDF:

python tools/build_report_figures.py    # regenerate the ~20 PNGs referenced by report.tex
cd report
latexmk -pdf report.tex                  # produces report/report.pdf

See report/README.md for details and a TeX-package list.

Contents of notebook 03 (the expanded time-dependent chapter)

§	system	physics / benchmark
1	driven SHO on resonance	classical envelope $\propto t$
2	driven SHO off resonance	Rabi-like beats at detuning
3	driven infinite square well	Rabi oscillations, population view, $\Omega_R$
4	two swept Gaussian wells	Landau-Zener $P = e^{-2\pi\Delta^2/(\hbar\alpha)}$
5	widening finite well	Adiabatic vs sudden approximation crossover
6	Gaussian packet in an ISW	Wavepacket revivals / quantum carpets
7	displaced SHO coherent-state sum	Autocorrelation spectroscopy ($C(t) \to E_n$)
8	finite well + Gaussian laser pulse	Photoionisation + photoelectron spectrum
9	1D soft-Coulomb atom + strong pulse	High-harmonic generation, $I_p + 3.17 U_p$ law

Every experiment reuses the same tdse/ package, the same six-panel conservation diagnostics, and the same space-time heatmap + GIF export pipeline as the earlier notebooks. Infrastructure added in support of §3-§9 (all in the main tdse/ package): imaginary-time relaxation (solvers.ground_state_imag_time) for ground-state preparation of arbitrary 1D potentials, sparse-Lanczos eigenstate finder (solvers.bound_eigenstates), and projection / autocorrelation / dipole- acceleration / power-spectrum observables (observables.populations, .autocorrelation, .dipole_acceleration, .power_spectrum).

Quick start

# Editable install -- pulls in numpy / scipy / matplotlib + notebook + test extras
python -m pip install -e ".[notebooks,test]"

python tools/build_notebooks.py            # regenerate notebook source
jupyter lab                                # open notebooks/00_theory_and_method.ipynb

requirements.txt is still provided for reproducibility but pip install -e . is now the canonical install: it makes the tdse package importable from any directory and configures the pytest test suite at the same time.

To re-execute every notebook end-to-end and refresh the GIFs:

jupyter nbconvert --to notebook --execute --inplace `
    notebooks/00_theory_and_method.ipynb `
    notebooks/01_free_and_bound_states.ipynb `
    notebooks/02_scattering.ipynb `
    notebooks/03_time_dependent.ipynb `
    notebooks/04_breaking_points.ipynb

Tests

pytest -m "not slow"            # 17 fast tests, ~2 s
pytest -m slow                  # 7 §3-§9 physics tests, ~10 s
pytest                          # full suite

The fast tier checks package imports, SI constants, grid math, ISW eigenstate orthonormality, the closed-form-vs-transfer-matrix barrier $T(E)$, observables on known states, imaginary-time relaxation of the SHO ground state, and the three solvers themselves (Visscher modified-norm machine-precision conservation, CN unitarity, FTCS divergence, and leapfrog $\leftrightarrow$ CN cross-check).

The slow tier exercises Rabi oscillations, Landau-Zener (fast vs slow sweep monotonicity), the sudden-approximation limit, the ISW half-revival, SHO autocorrelation spectroscopy, photoionisation, and HHG dipole-acceleration spectra -- one test per §3-§9 section of notebook 03.

CI runs both tiers across Python 3.10 / 3.11 / 3.12 on every push; see .github/workflows/ci.yml.

Convention for space-time plots

We follow the spacetime-diagram convention familiar from special relativity: time on the horizontal axis, position on the vertical axis. A free-particle group-velocity worldline therefore appears as a straight line of slope $v_g$ from bottom-left to top-right; a particle bouncing in a well appears as a zig-zag. Every space-time heatmap also has a small panel on the left showing $V(x)$, drawn at the same vertical scale as the heatmap so that potentials and dynamics are spatially aligned.

Units

We work internally in SI units (metres, seconds, joules, kilograms). For display we use nm, fs, eV, and electron mass $m_e$. The tdse.units module supplies the necessary conversion constants and ensures all numerical values stay in a numerically well-conditioned range despite $\hbar \approx 1.055\times10^{-34}\mathrm{Js}$.

Probability densities are plotted in 1/nm; momenta in $\hbar$/nm.

Validation strategy

The project relies on three independent ground-truth references:

1. Closed-form analytic solutions — tdse.analytical provides the exact free-Gaussian propagator, infinite/finite well eigenstates, harmonic oscillator eigenstates, step-potential reflection coefficient, and the rectangular-barrier transmission coefficient (Griffiths Eq. 2.171).
2. Transfer-matrix method — for arbitrary 1D piecewise-constant potentials (single barrier, double barrier, multilayer structures), analytical.transfer_matrix_TR(E, x, V) computes the exact $T(E)$ and $R(E)$ from the same grid the leapfrog uses. Wave-packet measurements are overlaid on the transfer-matrix curve in every scattering experiment.
3. Crank–Nicolson reference solver — implemented with sparse banded LU, exactly unitary in exact arithmetic. Used for cross-checks against the leapfrog scheme.
4. FFT split-operator solver — Strang splitting evaluated in momentum space for the kinetic operator and position space for the potential. Spectrally accurate in $\Delta x$, exactly unitary, and used as a fully independent fourth scheme for the §4/§8 cross-checks against leapfrog.

References

1. P. B. Visscher, A fast explicit algorithm for the time-dependent Schrödinger equation, Computers in Physics 5, 596 (1991).
2. A. Goldberg, H. M. Schey, J. L. Schwartz, Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena, Am. J. Phys. 35, 177 (1967).
3. W. van Dijk et al., Accurate numerical solutions of the time- dependent Schrödinger equation, Phys. Rev. E 75, 036707 (2007).