Question 40.6: Locating an Electron The speed of an electron is measured to......
Locating an Electron
The speed of an electron is measured to be 5.00 \times 10^{3} \mathrm{~m} / \mathrm{s} to an accuracy of 0.00300 \%. Find the minimum uncertainty in determining the position of this electron.
Conceptualize The fractional value given for the accuracy of the electron’s speed can be interpreted as the fractional uncertainty in its momentum. This uncertainty corresponds to a minimum uncertainty in the electron’s position through the uncertainty principle.
Categorize We evaluate the result using concepts developed in this section, so we categorize this example as a substitution problem.
Assume the electron is moving along the x axis and find the uncertainty in p_{x}, letting f represent the accuracy of the measurement of its speed:
\Delta p_{x}=m \Delta v_{x}=m f v_{x}
Solve Equation 40.23
\Delta x \Delta p_{x} \geq \frac{\hbar}{2 } (40.23)
for the uncertainty in the electron’s position and substitute numerical values:
\begin{aligned} \Delta x & \geq \frac{\hbar}{2 \Delta p_{x}}=\frac{\hbar}{2 m f v_{x}}=\frac{1.055 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}}{2\left(9.11 \times 10^{-31} \mathrm{~kg}\right)(0.0000300)\left(5.00 \times 10^{3} \mathrm{~m} / \mathrm{s}\right)} \\ & =3.86 \times 10^{-4} \mathrm{~m}=0.386 \mathrm{~mm} \end{aligned}