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:

First, we recall the Heisenberg uncertainty principle which relates the uncertainty in position $$\Delta x$$ to the uncertainty in momentum $$\Delta p$$ of a particle:

$$\Delta x \,\Delta p \;\ge\; \dfrac{\hbar}{2}.$$

When we want only the order of magnitude or a proportionality, we may write

$$\Delta x \;\propto\; \dfrac{\hbar}{\Delta p}.$$

So, to compare the positional uncertainties of different particles kept under identical conditions, we must compare the corresponding momentum uncertainties $$\Delta p$$.

Now, each gas is ideal and in thermal equilibrium at a common temperature $$T$$. For one translational degree of freedom, the equipartition theorem states that the mean translational kinetic energy is

$$\dfrac{1}{2} m \langle v^{2}\rangle \;=\;\dfrac{1}{2}k_{B}T.$$

Re-arranging this, we obtain the root-mean-square (rms) speed:

$$\langle v^{2}\rangle^{1/2} \;=\; \sqrt{\dfrac{k_{B}T}{m}}.$$

The momentum of a particle is $$p = mv$$, so its rms value is

$$p_{\text{rms}} \;=\; m \,\langle v^{2}\rangle^{1/2} \;=\; m \sqrt{\dfrac{k_{B}T}{m}} \;=\; \sqrt{m\,k_{B}T}.$$

The spread of momentum values in the gas, that is, the uncertainty $$\Delta p$$, is of the same order as this rms momentum, so we write

$$\Delta p\;\propto\;\sqrt{m\,k_{B}T}.$$

Because both gases are at the same temperature, the factor $$k_{B}T$$ is common and cancels out when we form a ratio. Hence,

$$\Delta p \;\propto\;\sqrt{m}.$$

Substituting this proportionality into the uncertainty-principle expression gives

$$\Delta x \;\propto\;\dfrac{1}{\Delta p} \;\propto\;\dfrac{1}{\sqrt{m}}.$$

Therefore, the positional uncertainty of a particle varies inversely as the square root of its mass.

Let $$\Delta x_e$$ denote the uncertainty for an electron of mass $$m_e$$ and $$\Delta x_p$$ for a proton of mass $$m_p$$. Using the above proportionality, we write

$$ \dfrac{\Delta x_e}{\Delta x_p} \;=\; \dfrac{1/\sqrt{m_e}}{1/\sqrt{m_p}} \;=\; \sqrt{\dfrac{m_p}{m_e}}. $$

This shows that the required ratio is proportional to $$\sqrt{m_p/m_e}$$.

Hence, the correct answer is Option A.