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Onicescu Energy

A Python package for computing Onicescu's Informational Energy, particularly useful in quantum physics and information theory applications.

Overview

The onicescu_energy package provides a simple function to calculate Onicescu's Informational Energy for a given probability distribution. This measure quantifies the concentration or dispersion within a distribution and is defined as:

$$ E = \sum_{i=1}^{n} p_i^2 $$

For continuous distributions, it adapts to:

$$ E = \int [\rho(x)]^2 , dx $$

This concept is particularly relevant in quantum mechanics, where it can be used to analyze the localization properties of quantum states.

Installation

Install the package via pip:

pip install onicescu_energy

Dependencies

• NumPy: A fundamental package for numerical computations in Python.

Usage

Import the necessary modules and the onicescu_energy function:

import numpy as np
from onicescu_energy import onicescu_energy

Computing Onicescu's Informational Energy

For a discrete probability distribution:

# Example discrete probability distribution
probabilities = np.array([0.2, 0.5, 0.3])

# Calculate Onicescu's Informational Energy
E = onicescu_energy(probabilities)

print(f"Onicescu's Informational Energy: {E}")

Computing for Quantum States

Suppose you have a quantum state represented by a wavefunction, and you want to compute the informational energy.

# Define the probability density function |(x)|²
def probability_density(x):
    # Example: |(x)|² = 2/L * sin²(nπx/L)
    L = 1.0  # Width of the potential well
    n = 1    # Energy level
    return (2 / L) * np.sin(n * np.pi * x / L)**2

# Generate a range of x values
x_values = np.linspace(0, 1, 1000)
dx = x_values[1] - x_values[0]

# Compute the probability densities
prob_densities = probability_density(x_values)

# For continuous distributions, approximate the integral
E = onicescu_energy(prob_densities) * dx

print(f"Onicescu's Informational Energy for the quantum state: {E}")

Function Definition

The onicescu_energy function computes the informational energy for a given array of probabilities or probability densities.

def onicescu_energy(probabilities):
    """
    Calculates Onicescu's Informational Energy.

    Parameters:
        probabilities (numpy.array): An array of probabilities or probability densities.

    Returns:
        float: The calculated informational energy.
    """
    return np.sum(probabilities**2)

Important Notes

• Ensure that the probabilities or probability densities provided are properly normalized.

  • Discrete distributions: Probabilities should sum up to 1.

  • Continuous distributions: The integral of the probability density over the entire space should be 1.

• When working with continuous distributions, multiplying by the differential element (e.g., dx) is necessary to approximate the integral.

License

This project is licensed under the MIT License.

Author

[Daia Alexandru]

Contribution

Contributions are welcome! Please feel free to submit issues or pull requests.

Contact

For any questions or inquiries, please contact [e-mail].

Acknowledgments

• Inspired by the work on Onicescu's Informational Energy in quantum physics.
• Special thanks to the open-source community for providing the tools that make this package possible.