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