Quantum Tunneling Simulation with Finite Potential Well

Introduction

This project simulates quantum tunneling through a finite potential barrier using Python. Quantum tunneling is a fundamental phenomenon in quantum mechanics where a particle has a probability to pass through a potential barrier, even if its energy is less than the height of the barrier. This project visualizes the evolution of a Gaussian wave packet interacting with a potential barrier over time.

The simulation numerically solves the time-dependent Schrödinger equation using eigenfunction expansion methods and visualizes the tunneling process.

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Installation

1. Clone the Repository:

   git clone https://github.com/Batuhan/Quantum-Tunneling-Simulation.git
   cd Quantum-Tunneling-Simulation

2. Install Required Dependencies:

   Make sure you have Python 3 installed. Then, install the necessary libraries:

   pip install numpy matplotlib numba

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Running the Simulation

To run the simulation and visualize the quantum tunneling process, execute the following command:

python quantum_tunneling.py

The script will generate wavefunction evolution plots that show the tunneling behavior over time.

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Code Overview

1. Gaussian Wave Packet

The initial wave function is defined as a Gaussian wave packet, given by the equation:

Where:

• $A$ is the normalization constant,
• $x_0$ is the initial position of the wave packet,
• $\sigma$ is the width (spread) of the packet,
• $k_0$ is the initial momentum (wave number).

This wave packet represents a localized particle with a certain momentum directed towards the potential barrier.

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2. Time-Dependent Schrödinger Equation

The evolution of the wave packet is governed by the time-dependent Schrödinger equation:

Where:

• $\hbar$ is the reduced Planck's constant (set to 1 in this simulation),
• $m$ is the mass of the particle (set to 1 for simplicity),
• $V(x)$ is the potential energy function representing the barrier.

The numerical solution involves diagonalizing the Hamiltonian and using eigenfunction expansion to compute the time evolution of the wave packet.

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3. Potential Barrier

The finite potential barrier is defined as:

Where:

• $V_0$ is the height of the potential barrier,
• $a$ is the starting position of the barrier,
• $w$ is the width of the barrier.

If the particle's energy is less than $V_0$, classical mechanics predicts total reflection. However, in quantum mechanics, there's a non-zero probability that the particle will tunnel through the barrier.

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4. Tunneling Probability

The probability of finding the particle in different regions is computed by integrating the squared modulus of the wave function:

• Left of the barrier (reflection probability):

• Inside the barrier:

• Right of the barrier (transmission probability):

  

These probabilities evolve over time, illustrating the quantum tunneling effect.

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Simulation Parameters

You can modify the following parameters in the wavepacket() function to explore different scenarios:

• N_grid: Number of grid points (spatial resolution).
• L: Length of the spatial domain.
• a: Starting position of the potential barrier.
• V0: Height of the potential barrier.
• w: Width of the potential barrier.
• x0: Initial position of the wave packet.
• k0: Initial momentum of the wave packet.
• sigma: Width of the Gaussian wave packet.
• t: Total simulation time.

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Sample Output

The following plot demonstrates how the Gaussian wave packet evolves over time and interacts with the potential barrier:

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License

This project is licensed under the MIT License.

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Feel free to contribute or open issues if you encounter any problems!