README.md

Schrödinger Equation Dataset

Numerical solutions to the 1D time-dependent Schrödinger equation with harmonic oscillator potential.

Equation

Time-dependent Schrödinger equation:

iℏ ∂ψ/∂t = Ĥψ

where the Hamiltonian is:

Ĥ = -ℏ²/2m ∇² + V(x)

Harmonic oscillator potential:

V(x) = ½mω²x²

The complex wavefunction ψ = ψᵣ + iψᵢ is split into real and imaginary parts:

• Real part: ∂ψᵣ/∂t = (ℏ/2m)∇²ψᵢ - V(x)ψᵢ/ℏ
• Imaginary part: ∂ψᵢ/∂t = -(ℏ/2m)∇²ψᵣ + V(x)ψᵣ/ℏ

Variables

The dataset returns a dictionary with the following fields:

Coordinates

• spatial_coordinates: (Nx,) - 1D spatial grid points x ∈ [-Lx/2, Lx/2]
• time_coordinates: (time_steps,) - Time evolution points

Solution Fields

• psi_r_initial: (Nx,) - Real part of initial wavefunction
• psi_i_initial: (Nx,) - Imaginary part of initial wavefunction
• psi_r_trajectory: (time_steps, Nx) - Real part evolution
• psi_i_trajectory: (time_steps, Nx) - Imaginary part evolution
• state_trajectory: (time_steps, 2*Nx) - Concatenated [ψᵣ, ψᵢ] for ML
• probability_density: (time_steps, Nx) - |ψ|² probability density

Physical Quantities

• potential: (Nx,) - Harmonic oscillator potential V(x) = ½mω²x²
• total_energy: (time_steps,) - Total energy over time (conservation check)

Physical Parameters

• hbar: Reduced Planck constant
• mass: Particle mass
• omega: Harmonic oscillator frequency

Dataset Parameters

• Domain: x ∈ [-10, 10] (symmetric around origin for harmonic oscillator)
• Grid points: Nx = 256 (spectral resolution with Fourier basis)
• Time range: [0, 2.0] (sufficient to see wave packet oscillations)
• Spatial resolution: Δx ≈ 0.078 (domain length / grid points)
• Temporal resolution: Δt = 1e-3 (RK4 time stepping)

Physical Parameters

• Reduced Planck constant: ℏ = 1.0
• Particle mass: m = 1.0
• Harmonic oscillator frequency: ω = 1.0
• Boundary conditions: Periodic (suitable for localized wave packets)

Initial Conditions

• Wave packet type: Gaussian wave packets with random parameters
• Center position: x₀ ∼ Uniform([-5, 5])
• Wave packet width: σ ∼ Uniform([0.5, 2.0])
• Initial momentum: k₀ ∼ Uniform([-2.0, 2.0])
• Amplitude: A ∼ Uniform([0.5, 2.0]) (normalized after generation)

Physical Context

This dataset simulates quantum harmonic oscillator dynamics governed by the time-dependent Schrödinger equation. The equation models the quantum mechanical evolution of a particle in a harmonic potential well V(x) = ½mω²x².

Key Physical Phenomena:

• Wave packet oscillation: Gaussian wave packets oscillate back and forth in the harmonic potential
• Quantum tunneling: Wave function can extend into classically forbidden regions
• Energy quantization: Total energy is conserved and quantized in bound states
• Probability conservation: |ψ|² integrates to 1 at all times
• Phase evolution: Real and imaginary parts evolve with quantum phase relationships

Applications:

• Quantum mechanics education: Fundamental model system in quantum physics courses
• Atomic physics: Models trapped atoms in harmonic potentials (laser cooling, optical traps)
• Quantum optics: Describes coherent states and squeezed states of light
• Neural operator learning: Provides rich training data for physics-informed machine learning
• Bose-Einstein condensates: Mean-field dynamics in harmonic traps

Usage

from dataset import SchrodingerDataset

# Create dataset
dataset = SchrodingerDataset(
    Lx=20.0,                # Domain size
    Nx=256,                 # Grid resolution
    hbar=1.0, mass=1.0, omega=1.0,  # Physical parameters
    stop_sim_time=2.0,      # Simulation time
    timestep=1e-3
)

# Generate a sample
sample = next(iter(dataset))

# Access solution data
x = sample["spatial_coordinates"]           # Spatial grid
t = sample["time_coordinates"]             # Time points  
psi_r = sample["psi_r_trajectory"]         # Real part evolution
psi_i = sample["psi_i_trajectory"]         # Imaginary part evolution
state = sample["state_trajectory"]         # Combined [ψᵣ, ψᵢ] for ML
prob = sample["probability_density"]       # |ψ|² probability
energy = sample["total_energy"]            # Energy conservation

Visualization

Run the plotting scripts to visualize samples:

python plot_sample.py      # Static visualization
python plot_animation.py   # Animated evolution

Data Generation

Generate the full dataset:

python generate_data.py

This creates train/test splits saved as chunked parquet files in the data/ directory.