1d-qt-ideal-solver: 1D Idealized Quantum Tunneling Solver with Absorbing Boundaries

TL;DR

The paper presents 1d-qt-ideal-solver, an open-source Python tool for simulating coherent, one-dimensional quantum tunneling by solving the TDSE with a Strang-split operator, FFT-based kinetic differentiation, and complex absorbing potentials to suppress reflections, all accelerated by Numba. The method is formalized via the 1D Hamiltonian $\hat{H} = -\frac{1}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)$ and unitary evolution $i\partial_t|\Psi\rangle = \hat{H}|\Psi\rangle$ in atomic units, with an initial Gaussian wave packet and optional dephasing to model decoherence. Validation occurs through two canonical barriers—a rectangular barrier and a Gaussian barrier—demonstrating machine-precision energy conservation ($|\Delta E/E| < 10^{-5}$) and accurate transmission/reflection coefficients, while a comprehensive analysis using Shannon entropy, Jensen-Shannon divergence, KL divergence, multiple non-parametric tests, and phase-space metrics reveals statistically significant yet practically modest differences between geometries in the over-barrier regime. The work positions the solver as an educational, qualitative benchmark and a modular platform for extensions to time-dependent potentials, multi-barrier structures, and open-system modeling, with MIT licensing enabling broad adoption.

Abstract

We present 1d-qt-ideal-solver, an open-source Python library for simulating one-dimensional quantum tunneling dynamics under idealized coherent conditions. The solver implements the split-operator method with second-order Trotter-Suzuki factorization, utilizing FFT-based spectral differentiation for the kinetic operator and complex absorbing potentials to eliminate boundary reflections. Numba just-in-time compilation achieves performance comparable to compiled languages while maintaining code accessibility. We validate the implementation through two canonical test cases: rectangular barriers modeling field emission through oxide layers and Gaussian barriers approximating scanning tunneling microscopy interactions. Both simulations achieve exceptional numerical fidelity with machine-precision energy conservation over femtosecond-scale propagation. Comparative analysis employing information-theoretic measures and nonparametric hypothesis tests reveals that rectangular barriers exhibit moderately higher transmission coefficients than Gaussian barriers in the over-barrier regime, though Jensen-Shannon divergence analysis indicates modest practical differences between geometries. Phase space analysis confirms complete decoherence when averaged over spatial-temporal domains. The library name reflects its scope: idealized signifies deliberate exclusion of dissipation, environmental coupling, and many-body interactions, limiting applicability to qualitative insights and pedagogical purposes rather than quantitative experimental predictions. Distributed under the MIT License, the library provides a deployable tool for teaching quantum mechanics and preliminary exploration of tunneling dynamics.

Figures (4)

• Figure 1: Temporal evolution of quantum tunneling through rectangular barrier at $t = 0$, 1.63, 2.08, and 6.0 fs.
• Figure 2: Temporal evolution of quantum tunneling through Gaussian barrier at $t = 0$, 1.87, 2.35, and 6.0 fs.
• Figure 3: Statistical comparison of probability density distributions between rectangular (blue) and Gaussian (red) barriers.
• Figure 4: Phase space distributions in $(\psi_{\text{Re}}, \psi_{\text{Im}})$ coordinates for (a) rectangular and (b) Gaussian barriers.