Question

The masses of two particles having same kinetic energies are in the ratio \(2 : 1\). Then their de Broglie wavelengths are in the ratio.

• A. \(2:1\)
• B. \(1:2\)
• C. \(\sqrt{2}:1\)
• D. \(1:\sqrt{2}\)

Hint:

Proportionalities for de Broglie wavelength \(\lambda\):
• For same velocity \(v\): \(\lambda \propto \frac{1}{m}\)
• For same kinetic energy \(K\): \(\lambda \propto \frac{1}{\sqrt{m}}\)
• For same accelerating potential \(V\): \(\lambda \propto \frac{1}{\sqrt{mq}}\)

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The de Broglie hypothesis states that every moving particle has an associated wave, termed a "matter wave".
The wavelength \( \lambda \) of this wave is inversely proportional to the momentum \( p \) of the particle.
This duality is a cornerstone of quantum mechanics.
For particles of different masses, the wavelength depends on how mass and velocity (or energy) are distributed.
In this problem, the kinetic energies of the two particles are equal, but their masses vary.
We need to establish a mathematical link between mass, kinetic energy, and the de Broglie wavelength.
Step 2: Key Formula or Approach:
The fundamental de Broglie equation is:
\[ \lambda = \frac{h}{p} \]
Where \( h \) is Planck's constant and \( p \) is momentum.
Kinetic energy \( K \) is related to momentum \( p \) and mass \( m \) as:
\[ K = \frac{p^2}{2m} \implies p = \sqrt{2mK} \]
Substituting the expression for \( p \) into the wavelength equation:
\[ \lambda = \frac{h}{\sqrt{2mK}} \]
Given that the kinetic energy \( K \) is the same for both particles, we see that:
\[ \lambda \propto \frac{1}{\sqrt{m}} \]
Step 3: Detailed Explanation:
Let the two particles be designated as 1 and 2.
Their masses are in the ratio \( m_1 : m_2 = 2 : 1 \).
This implies \( m_1 = 2m \) and \( m_2 = m \).
Their kinetic energies are identical, \( K_1 = K_2 = K \).
The ratio of their de Broglie wavelengths is:
\[ \frac{\lambda_1}{\lambda_2} = \frac{h/\sqrt{2m_1 K}}{h/\sqrt{2m_2 K}} \]
The constants \( h, 2, \) and \( K \) cancel out:
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{m_2}{m_1}} \]
Substitute the given mass ratio into this expression:
\[ \frac{\lambda_1}{\lambda_2} = \sqrt{\frac{1}{2}} \]
\[ \frac{\lambda_1}{\lambda_2} = \frac{1}{\sqrt{2}} \]
This means the lighter particle will have a longer de Broglie wavelength compared to the heavier particle when their kinetic energies are equal.
Specifically, the ratio is \( 1 : \sqrt{2} \).
Step 4: Final Answer:
The ratio of their de Broglie wavelengths is \( 1 : \sqrt{2} \).
The correct option is (D).