Question

For the same kinetic energy, the de Broglie wavelengths associated with particles of different masses are

• A. directly proportional to their masses
• B. directly proportional to the square root of their masses
• C. inversely proportional to the square root of their masses
• D. inversely proportional to their masses

Hint:

If momentum were constant, \(\lambda\) would be independent of mass. If kinetic energy is constant, \(\lambda \propto 1/\sqrt{m}\). Always identify which physical quantity is being held constant.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The de Broglie wavelength of a particle is related to its momentum and kinetic energy.

Step 2: Key Formula or Approach:
\(\lambda = \frac{h}{p}\) and \(K = \frac{p^2}{2m} \implies p = \sqrt{2mK}\).
So, \(\lambda = \frac{h}{\sqrt{2mK}}\).

Step 3: Detailed Explanation:
From the derived equation \(\lambda = \frac{h}{\sqrt{2mK}}\):
Given that the kinetic energy (\(K\)) is the same for different particles.
The wavelength \(\lambda\) is proportional to \(1/\sqrt{m}\).
\[ \lambda \propto \frac{1}{\sqrt{m}} \]
Therefore, the de Broglie wavelength is inversely proportional to the square root of the particle's mass.

Step 4: Final Answer:
The wavelength is inversely proportional to the square root of their masses.