Quantum Mechanics Schrödinger Equation Solvers

A collection of numerical solvers for the Schrödinger equation in 1D quantum systems with visualisation and animation capabilities.

Overview

This project provides three complete implementations for solving and visualizing quantum mechanical systems:

1. Infinite Square Well - Particle confined in a box with infinite potential walls
2. Harmonic Oscillator - Quantum harmonic oscillator with parabolic potential
3. Time Evolution - Wave packet dynamics in arbitrary potentials

Features

• ✅ Numerical solution of time-independent Schrödinger equation using finite difference methods
• ✅ Energy eigenvalue and eigenfunction calculations
• ✅ Comparison with analytical solutions
• ✅ Time evolution animations showing wave packet dynamics
• ✅ Quantum superposition visualisation
• ✅ GIF export for animations
• ✅ Support for custom potentials

Quick Start

Installation

# Create conda environment
conda create -n quantum_physics python=3.9
conda activate quantum_physics

# Install dependencies
conda install numpy scipy matplotlib pillow

Run Examples

# Infinite square well
python infinite_well.py

# Harmonic oscillator
python harmonic_oscillator.py

# Time evolution examples
python time_evolution.py

Each script will generate static plots, animations, and save GIF files.

Usage Examples

Infinite Square Well

from infinite_well import InfiniteWellSolver

solver = InfiniteWellSolver(L=1.0, m=1.0, hbar=1.0, N=1000)
energies, wavefunctions = solver.solve(n_states=5)

solver.print_comparison(n_states=5)
solver.plot_results(n_states=4)

# Animate ground state
solver.animate_wavefunction_evolution(state_index=0, save_name='ground_state')

# Animate superposition
solver.animate_superposition(state_indices=[0, 1], coefficients=[1, 1],
                            save_name='superposition')

Harmonic Oscillator

from harmonic_oscillator import HarmonicOscillatorSolver

solver = HarmonicOscillatorSolver(omega=1.0, x_max=6.0, N=2000)
energies, wavefunctions = solver.solve(n_states=5)

solver.plot_results(n_states=5)
solver.animate_superposition(state_indices=[0, 1], coefficients=[1, 1],
                            save_name='harmonic_superposition')

Time Evolution

from time_evolution import TimeEvolutionSolver, free_potential, step_potential

# Free particle
solver = TimeEvolutionSolver(free_potential, x_min=-10, x_max=10, N=500)
psi0 = solver.gaussian_wavepacket(x0=-3, k0=1.5, sigma=1.0)
solver.time_evolve(psi0, dt=0.05, n_steps=300)
solver.animate_evolution(save_name='free_particle')

# Quantum tunneling
solver2 = TimeEvolutionSolver(lambda x: step_potential(x, height=5, position=0),
                              x_min=-10, x_max=10, N=500)
psi0_2 = solver2.gaussian_wavepacket(x0=-4, k0=2, sigma=1.0)
solver2.time_evolve(psi0_2, dt=0.05, n_steps=400)
solver2.animate_evolution(save_name='quantum_tunneling')

Output Files

Each script generates GIF animations in the gifs/ folder:

Infinite Well:

• infinite_well_energy_levels.gif - Animated energy level diagram
• infinite_well_ground_state.gif - Ground state (n=1) time evolution
• infinite_well_second_state.gif - Second state (n=2) time evolution
• infinite_well_superposition.gif - Superposition of n=1 and n=2

Harmonic Oscillator:

• harmonic_energy_levels.gif - Animated energy level diagram
• harmonic_ground_state.gif - Ground state (n=0) time evolution
• harmonic_excited_state.gif - First excited state (n=1) time evolution
• harmonic_superposition.gif - Superposition of n=0 and n=1

Time Evolution:

• free_particle.gif - Free particle wave packet spreading
• harmonic_oscillator_time.gif - Wave packet in harmonic potential
• quantum_tunneling.gif - Tunneling through step potential barrier

Total: 11 animated GIF files showcasing different quantum phenomena!

Documentation

• 📖 Theory & Background - Physics concepts and numerical methods
• 🎨 Visualisation Guide - Complete guide to all animations
• ⚙️ Customisation Guide - Extending and modifying the code

Project Structure

Schrodinger-eq/
├── README.md                 # This file
├── Files/
│   ├── Theory.md            # Physics and numerical methods
│   ├── Visualisation.md     # Complete guide to all 11 animations
│   └── Customisation.md     # How to extend the code
├──Notebooks/                 # Jupyter notebooks for interactive exploration
    └── gifs/                     # All generated animations (11 files)
        ├── infinite_well_energy_levels.gif
        ├── infinite_well_ground_state.gif
        ├── infinite_well_second_state.gif
        ├── infinite_well_superposition.gif
        ├── harmonic_energy_levels.gif
        ├── harmonic_ground_state.gif
        ├── harmonic_excited_state.gif
        ├── harmonic_superposition.gif
        ├── free_particle.gif
        ├── harmonic_oscillator_time.gif
        └── quantum_tunneling.gif
    ├── infinite_well.py          # Infinite square well solver
    ├── harmonic_oscillator.py    # Harmonic oscillator solver
    ├── time_evolution.py         # General time evolution solver    

Key Results

Infinite Square Well:

• Energy levels: E_n = (n²π²ℏ²)/(2mL²)
• Numerical error: < 0.001%

Harmonic Oscillator:

• Energy levels: E_n = ℏω(n + ½)
• Zero-point energy: E₀ = ½ℏω
• Numerical error: < 0.001%

Requirements

• Python 3.9+
• NumPy
• SciPy
• Matplotlib
• Pillow (for GIF export)

Educational Applications

Perfect for:

• Quantum mechanics courses
• Computational physics projects
• Visualizing abstract quantum concepts
• Understanding numerical methods
• Exploring quantum phenomena interactively

Validation

All solvers compare numerical results with analytical solutions where available. Errors can be reduced by increasing grid resolution (N parameter).

References

• Griffiths, D.J. (2018). Introduction to Quantum Mechanics (3rd ed.)
• Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.)
• Press, W.H. et al. (2007). Numerical Recipes (3rd ed.)

License

This project is open source and available for educational purposes.

Contributing

Feel free to extend this project with additional features. See CUSTOMISATION.md for ideas.

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Note: This is an educational implementation focused on clarity and visualisation. For production quantum simulations, consider specialized libraries like QuTiP, PyQuante, or PySCF.