Question

Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to :

• A. $\sqrt{\frac{m_e}{m_p}}$
• B. $\frac{m_p}{m_e}$
• C. $\sqrt{\frac{m_p}{m_e}}$
• D. $(\frac{m_p}{m_e})^{3/2}$

Hint:

Remember that for thermal particles at the same temperature, de-Broglie wavelength and position uncertainty follow the same scaling: both are inversely proportional to the square root of the mass (\( \lambda \propto 1/\sqrt{m} \)).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The Heisenberg Uncertainty Principle states that the product of uncertainty in position ($\Delta x$) and uncertainty in momentum ($\Delta p$) is at least of the order of $h/4\pi$. For particles in an ideal gas at a given temperature, the momentum is related to their thermal energy.
Step 2: Key Formula or Approach:
1. Uncertainty Principle: \( \Delta x \cdot \Delta p \approx \frac{h}{4\pi} \implies \Delta x \propto \frac{1}{\Delta p} \).
2. Thermal Momentum: Kinetic energy \( K = \frac{p^2}{2m} = \frac{3}{2} k_B T \implies p = \sqrt{3 m k_B T} \).
Assuming the uncertainty in momentum \(\Delta p\) is proportional to the average momentum \(p\) of the particles.
Step 3: Detailed Explanation:
Since the temperature $T$ is the same for both gases:
For electrons: \( \Delta p_e \propto \sqrt{m_e} \)
For protons: \( \Delta p_p \propto \sqrt{m_p} \)
From the uncertainty principle:
\( \Delta x_e \propto \frac{1}{\sqrt{m_e}} \)
\( \Delta x_p \propto \frac{1}{\sqrt{m_p}} \)
Taking the ratio:
\[ \frac{\Delta x_e}{\Delta x_p} = \frac{1/\sqrt{m_e}}{1/\sqrt{m_p}} = \sqrt{\frac{m_p}{m_e}} \]
Step 4: Final Answer:
The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to \(\sqrt{\frac{m_p}{m_e}}\).