1d-qt-ideal-solver: 1D Quantum Tunneling Solver

High-performance 1D quantum tunneling solver with absorbing boundary conditions. Implements split-operator Fourier method with Numba acceleration.

Physics

Solves the time-dependent Schrödinger equation:

$i\hbar\frac{\partial \psi}{\partial t} = \hat{H}\psi = \left[-\frac{\hbar^2}{2m}\nabla^2 + V(x,t)\right]\psi$

In natural units ($\hbar = 1$, $m = 1$), this simplifies to:

$i\frac{\partial \psi}{\partial t} = \left[-\frac{1}{2}\nabla^2 + V(x,t)\right]\psi$

Split-operator evolution with absorbing boundaries:

$\psi(x, t+\delta t) = \mathcal{M}_{\text{abs}} \cdot e^{-iV\delta t/2} \cdot \mathcal{F}^{-1}\left[e^{-ik^2\delta t/2}\mathcal{F}[\psi]\right] \cdot e^{-iV\delta t/2}$

Absorbing mask (cos⁴ profile):

$\mathcal{M}(x) = \begin{cases} 1 - s\left[1 - \cos^4\left(\frac{\pi i}{2n_b}\right)\right] & \text{left boundary} \ 1 & \text{safe zone} \ 1 - s\left[1 - \cos^4\left(\frac{\pi(N-i)}{2n_b}\right)\right] & \text{right boundary} \end{cases}$

Natural Units Convention:

• ℏ = 1 (reduced Planck constant)
• Electron mass m_e = 1
• Energy in eV, length in nm, time in fs

Observables:

• Transmission coefficient: T
• Reflection coefficient: R
• Absorbed probability: A
• Conservation: T + R + A ≈ 1

Features

• Absorbing boundary conditions prevent spurious reflections
• Adaptive time stepping
• Numba JIT compilation
• Stochastic environments with noise and decoherence
• Visualization with zone highlighting
• NetCDF4 output

Installation

# From PyPI
pip install 1d-qt-ideal-solver

# From source
git clone https://github.com/sandyherho/1d-qt-ideal-solver.git
cd 1d-qt-ideal-solver
pip install -e .

Quick Start

Command line:

# Run single case
qt1d-simulate case1

# Run both cases
qt1d-simulate --all

# Custom parameters
qt1d-simulate case1 --boundary-width 3.0 --cores 8

Python API:

from qt1d_ideal import QuantumTunneling1D, GaussianWavePacket

# Initialize solver
solver = QuantumTunneling1D(
    nx=2048, 
    x_min=-30.0,
    x_max=30.0,
    boundary_width=3.0,
    boundary_strength=0.03
)

# Create initial wavepacket
psi0 = GaussianWavePacket(x0=-8.0, k0=4.0, sigma=0.8)(solver.x)

# Define barrier
V = solver.rectangular_barrier(height=4.5, width=1.0)

# Solve
result = solver.solve(psi0=psi0, V=V, t_final=6.0, n_snapshots=200)

# Results
T = result['transmission_coefficient']
R = result['reflection_coefficient']
A = result['absorbed_probability']
print(f"T = {T:.4f}, R = {R:.4f}, A = {A:.4f}, T+R+A = {T+R+A:.4f}")

Test Cases

Case	Barrier Type	Height	Width	Domain
1	Rectangular	4.5 eV	1.0 nm	±30 nm
2	Gaussian	4.0 eV	0.8 nm	±30 nm

Configuration

Key parameters (see configs/ for examples):

# Spatial grid
nx = 2048
x_min = -30.0
x_max = 30.0

# Time integration
t_final = 6.0
n_snapshots = 200

# Absorbing boundaries
boundary_width = 3.0
boundary_strength = 0.03

# Environment
noise_amplitude = 0.0
decoherence_rate = 0.0

Output

Generated files:

• outputs/*.nc - NetCDF4 with wavefunction data
• outputs/*.gif - Animated visualization
• logs/*.log - Simulation diagnostics

Reading NetCDF data:

import netCDF4 as nc

data = nc.Dataset('outputs/case1_rectangular_barrier.nc')
x = data['x'][:]
t = data['t'][:]
psi_real = data['psi_real'][:, :]
psi_imag = data['psi_imag'][:, :]
probability = data['probability'][:, :]
potential = data['potential'][:]

T = data.transmission
R = data.reflection
A = data.absorbed

Citation

@article{qt1d_solver_2025,
  author = {Herho, Sandy H. S. and Kaban, Siti N. and Suwarman, Rusmawan and Anwar, Iwan P. and Trilaksono, Nurjanna J.},
  title = {{\texttt{1d-qt-ideal-solver}: 1D idealized quantum tunneling solver with absorbing boundaries}},
  journal = {xxxx},
  volume = {xxxx},
  pages = {xxxx},
  year = {xxxx},
  doi = {xxxx/xxxxx},
  url = {https://github.com/sandyherho/1d-qt-ideal-solver}
}

Authors

• Sandy H. S. Herho (sandy.herho@email.ucr.edu)
• Siti N. Kaban
• Rusmawan Suwarman
• Iwan P. Anwar
• Nurjanna J. Trilaksono

License

MIT License - See LICENSE for details.