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---
title: Hand-wave Quantum Solver
emoji: 🌊
colorFrom: blue
colorTo: purple
sdk: streamlit
sdk_version: "1.28.0"
app_file: app.py
pinned: false
---
# ⚛️ Quantum Potential Solver
**Author:** Ahilan Kumaresan
**Institution:** Simon Fraser University
**Field:** Mathematical & Computational Physics
---
## 🎯 Overview
An advanced numerical solver for the **Time-Independent Schrödinger Equation** (TISE) using the Finite Difference Method. This project demonstrates rigorous computational physics methodology with comprehensive verification against analytical solutions and external libraries.
## 🔬 Key Features
### 1. **Accurate Numerical Solver**
- **Method:** 3-point Central Difference stencil for the Laplacian operator
- **Grid:** Adaptive resolution (1000-2000 points)
- **Units:** Hartree Atomic Units (ℏ=1, m=1)
- **Boundary Conditions:** Dirichlet (infinite walls)
### 2. **Multiple Potential Types**
- Infinite Square Well
- Harmonic Oscillator
- Double Well Potential
- Custom potentials via hand gestures (camera input)
### 3. **Interactive Visualizations**
- Plotly-based interactive charts
- Wavefunction probability densities
- Energy level diagrams
- Real-time solver performance metrics
### 4. **Rigorous Verification**
#### Analytical Benchmarks
| System | Max Error |
|--------|-----------|
| Infinite Square Well | < 0.003% |
| Harmonic Oscillator | < 0.02% |
| Half-Harmonic Oscillator | < 0.8% |
| Triangular Potential | < 0.003% |
#### External Library Comparison
- **Cross-verified with QMSolve** (Python quantum mechanics package)
- **Double Well Potential:** Agreement within 0.25%
- Demonstrates accuracy for systems without analytical solutions
## 📐 Mathematical Foundation
The solver discretizes the TISE:
$$\hat{H}\psi(x) = E\psi(x)$$
$$\left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) \right]\psi(x) = E\psi(x)$$
Using finite differences:
$$\frac{d^2\psi}{dx^2} \approx \frac{\psi_{i+1} - 2\psi_i + \psi_{i-1}}{\Delta x^2}$$
This transforms the problem into a matrix eigenvalue equation solved via `numpy.linalg.eigh`.
## 🛠️ Technical Implementation
- **Language:** Python 3.12
- **Core Libraries:** NumPy, SciPy
- **Visualization:** Plotly, Matplotlib
- **UI Framework:** Streamlit
- **Computer Vision:** MediaPipe (for hand gesture input)
## 📊 Verification Scripts
The project includes comprehensive verification:
- `verify_physics.py`: Analytical benchmarks
- `verify_comparison.py`: QMSolve cross-verification
- `Comparison_Notebook.ipynb`: Jupyter notebook with detailed analysis
## 🎓 Academic Applications
This project demonstrates:
1. **Numerical Methods:** Finite difference discretization of differential operators
2. **Linear Algebra:** Eigenvalue problems for Hermitian matrices
3. **Quantum Mechanics:** Stationary states and energy quantization
4. **Software Engineering:** Modular design, comprehensive testing, professional documentation
## 📖 Usage
### Interactive Simulator
1. Select a potential type from the sidebar
2. Adjust parameters using sliders
3. View real-time solutions with interactive plots
### Verification Dashboard
- Navigate to "Benchmarks & Verification" to see accuracy metrics
- Compare against analytical solutions and QMSolve
### Theory Section
- Detailed mathematical background
- Implementation methodology
- Code verification
## 🔗 Repository Structure
```
psi_solve2/
├── app.py # Main Streamlit application
├── functions.py # Physics engine (solver + potentials)
├── verify_physics.py # Analytical verification script
├── verify_comparison.py # QMSolve comparison script
├── Comparison_Notebook.ipynb # Jupyter analysis notebook
└── requirements.txt # Python dependencies
```
## 📝 Citation
If you use this solver in your research, please cite:
```
Kumaresan, A. (2024). Quantum Potential Solver: A Verified Finite Difference
Implementation for the Time-Independent Schrödinger Equation.
Simon Fraser University.
```
## 📧 Contact
**Ahilan Kumaresan**
Mathematical & Computational Physics
Simon Fraser University
---
*This project showcases advanced computational physics methodology suitable for graduate-level research in quantum mechanics and numerical analysis.*
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