README.md

metadata

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

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🎯 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:

$$\widehat{H}\psi (x)=E\psi (x)\backslash hat\{H\}\backslash psi(x)\; =\; E\backslash psi(x)$$

$$[-\frac{{\mathrm{\hslash }}^2}{2m}\frac{d^2}{d{x}^2}+V(x)]\psi (x)=E\psi (x)\backslash left[\; -\backslash frac\{\backslash hbar^2\}\{2m\}\backslash frac\{d^2\}\{dx^2\}\; +\; V(x)\; \backslash right]\backslash psi(x)\; =\; E\backslash psi(x)$$

Using finite differences:

$$\frac{d^2\psi }{d{x}^2}\approx \frac{{\psi }_{i+1}-2{\psi }_i+{\psi }_{i-1}}{\mathrm{\Delta }{x}^2}\backslash frac\{d^2\backslash psi\}\{dx^2\}\; \backslash approx\; \backslash frac\{\backslash psi\_\{i+1\}\; -\; 2\backslash psi\_i\; +\; \backslash psi\_\{i-1\}\}\{\backslash Delta\; x^2\}$$

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

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This project showcases advanced computational physics methodology suitable for graduate-level research in quantum mechanics and numerical analysis.