Introduction to Quantum Probability

1. Imagine a wave function $\psi$ shaped like a bell curve centered at $x = 0$. 2. At $x = 0$, the value of $\psi$ is $0.5$. 3. At $x = 5$, the value of $\psi$ is $0.1$. 4. Using the Born Rule, the probability density at the center is $|0.5|^2 = 0.25$. 5. The probability density at $x = 5$ is $|0.1|^2 = 0.01$. 6. Conclusion: You are 25 times more likely to find the particle at the center than at $x = 5$.

1. An electron passes through a very narrow slit. By making the slit narrower, we decrease $\Delta x$ (we know exactly where it passed through). 2. According to $\Delta x \Delta p \ge \frac{h}{4\pi}$, as $\Delta x$ decreases, $\Delta p$ must increase to keep the inequality true. 3. This increase in momentum uncertainty causes the electron to 'kick' or spread out wildly after the slit. 4. Result: Trying to locate the electron precisely makes its future path completely unpredictable.

Calculate the minimum uncertainty in velocity ($\Delta v$) for an electron (mass $m \approx 9.11 \times 10^{-31}$ kg) if its position is known within the width of an atom ($1.0 \times 10^{-10}$ m). 1. Start with $\Delta x (m \Delta v) \ge \frac{h}{4\pi}$. 2. Rearrange for $\Delta v$: $\Delta v \ge \frac{h}{4\pi m \Delta x}$. 3. Substitute $h = 6.626 \times 10^{-34}$ J·s. 4. $\Delta v \ge \frac{6.626 \times 10^{-34}}{4 \times 3.14 \times 9.11 \times 10^{-31} \times 1.0 \times 10^{-10}}$. 5. $\Delta v \approx 5.8 \times 10^5$ m/s. 6. Even with a perfect 'atom-sized' measurement, the velocity is uncertain by over half a million meters per second!