Calculate the uncertainty in the momentum of an electron if it is confined to a linear region of length 1x 10^-8

Answer

To calculate the uncertainty in the momentum of an electron confined to a linear region of length \(1 \times 10^{-8}\) meters, we will use the Heisenberg Uncertainty Principle. The principle states that the uncertainty in position (\(\Delta x\)) and the uncertainty in momentum (\(\Delta p\)) are related by the equation:

\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]

Where:
- \(h\) is Planck's constant, approximately \(6.626 \times 10^{-34} \, \text{Js}\).

### Step-by-Step Solution:

**Step 1: Identify the given values.**
- Length of the confined region (\(\Delta x\)) = \(1 \times 10^{-8} \, \text{m}\)
- Planck's constant (\(h\)) = \(6.626 \times 10^{-34} \, \text{Js}\)

**Step 2: Use the uncertainty principle to find \(\Delta p\).**
From the uncertainty principle, we can express \(\Delta p\) as:

\[ \Delta p = \frac{h}{4\pi \Delta x} \]

**Step 3: Substitute the known values into the equation.**
Now, substituting the values of \(h\) and \(\Delta x\):

\[ \Delta p = \frac{6.626 \times 10^{-34}}{4 \cdot \pi \cdot (1 \times 10^{-8})} \]

**Step 4: Calculate the denominator.**
Calculating \(4\pi\):

\[ 4\pi \approx 12.566 \]

Now, substituting this back into the equation:

\[ \Delta p = \frac{6.626 \times 10^{-34}}{12.566 \times 10^{-8}} \]

**Step 5: Perform the division.**
Calculating the value:

\[ \Delta p \approx \frac{6.626 \times 10^{-34}}{12.566 \times 10^{-8}} \approx 5.28 \times 10^{-27} \, \text{kg m/s} \]

**Step 6: Final result.**
Thus, the uncertainty in the momentum of the electron is approximately:

\[ \Delta p \approx 5.28 \times 10^{-27} \, \text{kg m/s} \]

### Summary:
The uncertainty in the momentum of an electron confined to a linear region of length \(1 \times 10^{-8}\) meters is approximately \(5.28 \times 10^{-27} \, \text{kg m/s}\).