Step 1: Understanding the Concept:
According to de Broglie's hypothesis, a moving particle like an electron has a wave nature with an associated wavelength.
When an electron is accelerated from rest through a potential difference, it gains kinetic energy, which can be directly related to its de Broglie wavelength.
Step 2: Key Formula or Approach:
The specific formula for the de Broglie wavelength $\lambda$ of an electron accelerated through a potential difference $V$ (in volts) is given by $\lambda = \frac{1.227}{\sqrt{V}} \text{ nm}$.
Step 3: Detailed Explanation:
We are given the accelerating potential $V = 64 \text{ V}$.
Substitute this value directly into the empirical formula:
\[ \lambda = \frac{1.227}{\sqrt{64}} \text{ nm} \]
Calculate the square root of the denominator:
\[ \lambda = \frac{1.227}{8} \text{ nm} \]
Perform the final division:
\[ \lambda = 0.153375 \text{ nm} \]
Rounding to three decimal places to match the given options gives $0.153 \text{ nm}$.
Step 4: Final Answer:
The de Broglie wavelength associated with the electron is 0.153 nm.