# Schrödinger Equation Dataset

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

![Sample Plot](sample_plot.png)

## 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

```python
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:

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

## Data Generation

Generate the full dataset:

```bash
python generate_data.py
```

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