Questions: What is the uncertainty in the velocity of an electron if it is known to be located within 1.0 × 10^-10 m?

Transcript text: What is the uncertainty in the velocity of an electron if it is known to be located within $1.0 \times 10^{-10} \mathrm{~m} ?$

Solution

Solution Steps

Step 1: Identify the given values

Uncertainty in position ($\Delta x$): $1.0 \times 10^{-10}$ m
Planck's constant ($h$): $6.626 \times 10^{-34}$ Js
Reduced Planck's constant ($\hbar$): $\frac{h}{2\pi} = 1.055 \times 10^{-34}$ Js

Step 2: Use Heisenberg's Uncertainty Principle

Heisenberg's Uncertainty Principle states: \[ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \] where $\Delta p$ is the uncertainty in momentum.

Step 3: Calculate the uncertainty in momentum ($\Delta p$)

\[ \Delta p \geq \frac{\hbar}{2\Delta x} \] Substitute the given values: \[ \Delta p \geq \frac{1.055 \times 10^{-34}}{2 \times 1.0 \times 10^{-10}} \] \[ \Delta p \geq \frac{1.055 \times 10^{-34}}{2 \times 10^{-10}} \] \[ \Delta p \geq \frac{1.055 \times 10^{-34}}{2 \times 10^{-10}} = 5.275 \times 10^{-25} \text{ kg m/s} \]

Step 4: Relate momentum to velocity

Momentum ($p$) is given by: \[ p = m \cdot v \] where $m$ is the mass of the electron ($9.11 \times 10^{-31}$ kg).

Step 5: Calculate the uncertainty in velocity ($\Delta v$)

\[ \Delta v = \frac{\Delta p}{m} \] Substitute the values: \[ \Delta v = \frac{5.275 \times 10^{-25}}{9.11 \times 10^{-31}} \] \[ \Delta v = 5.79 \times 10^5 \text{ m/s} \]