| **Examples**: <br> 1. **Newton’s Laws of Motion**: <br> - If you apply a known force to an object, its acceleration and trajectory can be determined exactly. <br> - Example: $( x(t) = x_0 + v_0t + \frac{1}{2}at^2 )$ (kinematics equation). <br> <br> 2. **Ohm’s Law in Circuits**: <br> - $( V = IR )$ gives the exact voltage given current $( I )$ and resistance $( R )$. <br> <br> 3. **Schrödinger's Equation for Simple Systems**: <br> - The time-independent Schrödinger equation for a hydrogen atom yields **exact energy eigenvalues** for electron states. | **Examples**: <br> 1. **Quantum Mechanics (Wavefunction Collapse)**: <br> - The Schrödinger equation **deterministically** evolves a wavefunction, but upon measurement, the outcome is probabilistic. <br> - Example: Measuring the spin of an electron in a superposition state gives a **random outcome** (e.g., 50% spin-up, 50% spin-down). <br> <br> 2. **Radioactive Decay**: <br> - The decay of a single nucleus follows a **probability distribution**, not a deterministic function. <br> - We can only predict **half-life**, but not when a specific atom will decay. <br> <br> 3. **Chaotic Systems (Butterfly Effect)**: <br> - Some classical systems, like **weather models**, follow deterministic equations but are **highly sensitive to initial conditions**, making long-term predictions effectively non-deterministic. <br> <br> 4. **Monte Carlo Simulations**: <br> - Used in optimization and physics, these rely on **random sampling** to approximate solutions to complex problems. |
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