Experiment 8: Classical vs Quantum Mechanics

Testing Complexity Hypothesis: Simple vs Complex Physics

Objective

Compare cage status between:

• Simple Physics: Classical harmonic oscillator (intuitive, analytical solution)
• Complex Physics: Quantum particle in a box (counterintuitive, discrete states)

Hypothesis: Quantum system (complex) should break the cage, while classical system (simple) should lock it.

Part A: Classical Harmonic Oscillator (Simple)

Physics

Equation: $x(t) = A \cos(\omega t + \phi)$

Where:

• $A$: Amplitude [0.1, 10.0] m
• $\omega$: Angular frequency [0.5, 5.0] rad/s
• $\phi$: Phase [0, $2\pi$] rad
• $t$: Time [0, 10] s

Simulator Implementation

class ClassicalHarmonicOscillator:
    def generate_dataset(self, n_samples=2000):
        np.random.seed(42)
        A = np.random.uniform(0.1, 10.0, n_samples)
        omega = np.random.uniform(0.5, 5.0, n_samples)
        phi = np.random.uniform(0, 2*np.pi, n_samples)
        t = np.random.uniform(0, 10.0, n_samples)
        
        # Truth: x(t) = A * cos(omega*t + phi)
        x = A * np.cos(omega * t + phi)
        
        X = np.column_stack((A, omega, phi, t))
        return X, x

Expected Results

• R²: > 0.99 (high accuracy)
• Cage Status: 🔒 LOCKED (correlation with A, omega, phi > 0.9)
• Reason: Intuitive physics, evolution prepared us for this

Part B: Quantum Particle in a Box (Complex)

Physics

Wave Function: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$

Probability Density: $|\psi_n(x)|^2 = \frac{2}{L} \sin^2\left(\frac{n\pi x}{L}\right)$

Where:

• $n$: Quantum number (1, 2, 3, …, 10) - DISCRETE
• $L$: Box width [1.0, 10.0] m
• $x$: Position [0, L] m

Key Complexity:

• Quantization (discrete n)
• Non-intuitive (probability, not position)
• No classical analog

Simulator Implementation

class QuantumParticleInBox:
    def generate_dataset(self, n_samples=2000):
        np.random.seed(42)
        n = np.random.randint(1, 11, n_samples)  # Discrete quantum number
        L = np.random.uniform(1.0, 10.0, n_samples)
        x = np.random.uniform(0, L, n_samples)  # Position within box
        
        # Truth: |psi|^2 = (2/L) * sin^2(n*pi*x/L)
        prob_density = (2.0 / L) * np.sin(n * np.pi * x / L)**2
        
        X = np.column_stack((n, L, x))
        return X, prob_density

Expected Results

• R²: > 0.95 (high accuracy)
• Cage Status: 🔓 BROKEN (correlation with n, L < 0.3)
• Reason: Counterintuitive physics, evolution didn't prepare us

Methodology

1. Data Generation

• Part A: 2000 samples, classical oscillator
• Part B: 2000 samples, quantum particle
• Same random seed for reproducibility

2. Models

• Baseline: Polynomial Regression (degree 4)
• Chaos Model: Optical Chaos (4096 features, brightness=0.001)

3. Evaluation

• Standard R²: Random train/test split (80/20)
• Cage Analysis:
  • Part A: Correlate features with A, omega, phi
  • Part B: Correlate features with n, L
• Extrapolation:
  • Part A: Train on t < 5, test on t > 5
  • Part B: Train on n ≤ 5, test on n > 5

4. Success Criteria

• Hypothesis confirmed if:
  • Part A: Cage LOCKED (correlation > 0.9)
  • Part B: Cage BROKEN (correlation < 0.3)
  • Both achieve high R² (> 0.95)

Implementation Checklist

• Implement ClassicalHarmonicOscillator simulator
• Implement QuantumParticleInBox simulator
• Create main experiment script with both parts
• Train baseline and chaos models on both parts
• Calculate R² scores
• Perform cage analysis (correlation with human variables)
• Test extrapolation
• Create visualizations comparing both parts
• Write benchmark script
• Document results in README

Files Structure

experiment_8_classical_vs_quantum/
├── experiment_8_classical_vs_quantum.py
├── benchmark_experiment_8.py
└── README.md