quantumprimes.py

import numpy as np
import matplotlib.pyplot as plt
from scipy.linalg import eigh_tridiagonal
from sympy import isprime

def potential_function(modulus):
    """Define the potential function V(x) for a given modulus."""
    return np.array([10 if (n % 2 == 0 or n % 3 == 0) else 1 for n in range(modulus)])

def solve_schrodinger(modulus, V, hbar=1.0, m=1.0):
    """Solve the Schrödinger equation for a given potential function V(x)."""
    N = len(V)
    dx = 1.0  # Assuming unit spacing for simplicity
    diagonal = np.full(N, -2.0)
    off_diagonal = np.full(N - 1, 1.0)
    # Solve the eigenvalue problem to get energies and wavefunctions
    energies, wavefunctions = eigh_tridiagonal(diagonal + V, off_diagonal)
    return energies, wavefunctions

def compute_prime_distribution(modulus):
    """Compute the distribution of primes across residue classes modulo a given modulus."""
    primes = [n for n in range(1, 1000) if isprime(n)]
    prime_distribution = np.zeros(modulus)
    for p in primes:
        prime_distribution[p % modulus] += 1
    return prime_distribution

def correlation_analysis(wavefunction, prime_distribution):
    """Calculate the correlation between the wave function's probability density and the prime distribution."""
    wave_prob_density = np.abs(wavefunction) ** 2
    correlation = np.corrcoef(wave_prob_density, prime_distribution)[0, 1]
    return correlation

def visualize_and_output_results(modulus, V, wavefunctions, prime_distribution, correlation, energies):
    """Visualize the results and output detailed analysis."""
    x = np.arange(modulus)
    wave_prob_density = np.abs(wavefunctions[:, 0]) ** 2
    
    # Plot the potential, wavefunction, and prime distribution
    plt.figure(figsize=(14, 8))
    plt.plot(x, V, label='Potential V(x)', color='red')
    plt.plot(x, wave_prob_density, label='Wavefunction Probability Density |ψ(x)|^2', color='blue')
    plt.bar(x, prime_distribution / prime_distribution.sum(), alpha=0.3, label='Normalized Prime Distribution', color='green')
    plt.xlabel('Position x (Residue Class)')
    plt.ylabel('Energy / Probability Density / Prime Count')
    plt.title(f'Quantum Model for Prime Distribution Modulo {modulus}')
    plt.legend()
    plt.grid(True)
    plt.show()
    
    # Output the results
    print(f"Modulus: {modulus}")
    print(f"Correlation between wavefunction and prime distribution: {correlation:.4f}")
    print(f"Energy levels (first 5): {energies[:5]}")
    print(f"Prime Distribution: {prime_distribution}")
    print(f"Wavefunction Probability Density: {wave_prob_density}")
    print("-" * 50)

def main():
    moduli = [6, 12, 18, 30]  # Moduli to analyze
    
    for modulus in moduli:
        V = potential_function(modulus)
        energies, wavefunctions = solve_schrodinger(modulus, V)
        prime_distribution = compute_prime_distribution(modulus)
        correlation = correlation_analysis(wavefunctions[:, 0], prime_distribution)
        visualize_and_output_results(modulus, V, wavefunctions, prime_distribution, correlation, energies)

main()