Towards Quantum Mechanics: Wave-Particle Duality

In this example, we calculate the uncertainty in position for a macroscopic object: a golf ball with a mass of $40 \text{ g}$ ($0.04 \text{ kg}$) moving at a speed of $45 \text{ m/s}$ with a $2\%$ accuracy. First, we determine the uncertainty in the speed, $\Delta v$. Since the speed is accurate to $2\%$, we find $\Delta v = 45 \times (2/100) =$ $\text{m/s}$. Next, we apply the Heisenberg uncertainty principle equation, rearranging it to solve for position: $\Delta x = \frac{h}{4\pi m \Delta v}$. Plugging in the known values, including Planck's constant ($6.626 \times 10^{-34} \text{ J s}$), we get $\Delta x = \frac{6.626 \times 10^{-34}}{4 \times 3.1416 \times 0.04 \times 0.9} =$ $\text{m}$. This resulting value is nearly $10^{18}$ times smaller than the diameter of a typical atomic nucleus. Because this uncertainty is so insignificantly small, we conclude that the uncertainty principle sets no meaningful limit to the precision of measurements for objects.