Animated Solution for Physics - Dual Nature of Matter and Radiation: 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

Have you ever wondered what happens when the macroscopic world of thermodynamics collides with the microscopic realm of quantum mechanics? This problem is a beautiful illustration of exactly that. We are tasked with comparing two completely different gases—one made of incredibly light electrons, and the other made of relatively massive protons. Yet, they share a common macroscopic property: they are at the exact same temperature. From this single shared property, we must deduce a profound quantum mechanical consequence: the ratio of their position uncertainties.

This is not just a dry mathematical exercise; it is a journey into the heart of wave-particle duality and statistical mechanics. Let's break it down step by step, gear by gear, and uncover the elegant physics hidden within.

The Macroscopic Anchor

The problem begins by placing us in a familiar macroscopic setting. We have two separate ideal gases. One container holds electrons, and the other holds protons. We are told that both gases have the same number of particles and, crucially, they are at the same temperature, $T$.

What does temperature actually mean at the microscopic level? In the framework of the kinetic theory of gases, absolute temperature is not just a measure of 'hotness' or 'coldness'. It is a direct, proportional measure of the average translational kinetic energy of the particles in the gas.

The equipartition theorem tells us that for a monatomic ideal gas (and we can treat individual electrons and protons as monatomic particles), the average kinetic energy $K$ is given by:

where $k_B$ is the Boltzmann constant.

Notice what is missing from this equation: mass! The average kinetic energy of an ideal gas particle depends only on the temperature, regardless of whether the particle is a feather-light electron or a heavy proton.

Because both gases are at the same temperature $T$, we can immediately conclude our first major physical insight:

The average kinetic energy of the electrons is exactly equal to the average kinetic energy of the protons. This is our visual anchor, the foundation upon which the rest of the solution is built.

Bridging Energy and Mass

Now that we have established that their kinetic energies are equal, we need to connect this energy to a property that will eventually lead us to quantum uncertainty. That property is momentum.

In classical mechanics, kinetic energy $K$ is typically written as:

However, in quantum mechanics and advanced physics, it is almost always more useful to express kinetic energy in terms of momentum, $p$. Since momentum $p=mv$, we can rewrite the kinetic energy equation as:

This form is incredibly powerful because it directly links energy, mass, and momentum. Let's apply this to our two gases.

For the electron gas:

For the proton gas:

Since we already established that $K_e=K_p$, we can equate these two expressions:

This equation is the logic bridge we need. It tells us how the momenta of the two particles compare, given that they have the same energy but vastly different masses.

Our goal is to find a ratio, so let's rearrange this equation to isolate the momentum terms on one side and the mass terms on the other.

Multiplying both sides by $2$, we get:

Now, let's cross-multiply to group the momenta and the masses:

To find the ratio of the momenta themselves, we simply take the square root of both sides:

This is a profound intermediate result. It tells us that the ratio of their momenta is proportional to the square root of the ratio of their masses. Because a proton is roughly 1836 times more massive than an electron ($m_p\gg m_e$), the momentum of the proton is significantly larger than the momentum of the electron at the same temperature.

Physically, this makes sense. If a heavy truck (proton) and a tiny bicycle (electron) have the exact same kinetic energy, the truck must be moving much slower, but its massive weight means its overall momentum ($m\times v$) is still much greater than that of the bicycle.

The Uncertainty Principle

Now we shift gears from classical statistical mechanics to the strange and beautiful world of quantum mechanics. The problem asks for the ratio of the uncertainty in determining their positions.

Whenever you see the words 'uncertainty in position', your mind should immediately jump to Werner Heisenberg and his famous Uncertainty Principle.

Heisenberg's Uncertainty Principle states that it is fundamentally impossible to simultaneously know both the exact position and the exact momentum of a particle. The more precisely you know one, the less precisely you can know the other. Mathematically, this is expressed as:

where $\Delta x$ is the uncertainty in position, $\Delta p$ is the uncertainty in momentum, and $h$ is Planck's constant.

For the purpose of finding proportionalities and ratios, we can treat this relationship as an inverse proportionality:

But what is the uncertainty in momentum, $\Delta p$, for particles in a thermal gas? In a gas at thermal equilibrium, the particles are zipping around in all directions with a distribution of momenta (the Maxwell-Boltzmann distribution). The 'spread' or uncertainty in their momentum, $\Delta p$, is directly proportional to their root-mean-square average momentum, $p$.

Therefore, we can confidently state that:

Substituting this into our uncertainty proportionality, we get:

This is the crucial quantum leap! The uncertainty in a particle's position is inversely proportional to its momentum.

The Ratio of Uncertainties

We are now ready for the final atomic compute. We need to find the ratio of the uncertainty in the position of the electron ($\Delta x_e$) to that of the proton ($\Delta x_p$).

Using our inverse proportionality $\Delta x\propto 1/p$, we can write the ratio as:

Simplifying this complex fraction, we get:

Look at that! The ratio of their position uncertainties is exactly equal to the inverse ratio of their momenta.

But wait, we already calculated the ratio of their momenta in a previous step! We found that:

Substituting this into our uncertainty equation, we arrive at our final, elegant answer:

The Physical Meaning of the Result

Let's take a moment to appreciate what this final equation is telling us.

We found that $\frac{\Delta x_e}{\Delta x_p}=\sqrt{\frac{m_p}{m_e}}$. Since the mass of a proton ($m_p$) is much, much greater than the mass of an electron ($m_e$), this ratio is much greater than 1.

This means that $\Delta x_e\gg \Delta x_p$.

At the exact same temperature, the position of the electron is vastly more uncertain than the position of the proton. The electron is 'fuzzier'. Its quantum mechanical probability cloud is much more spread out.

This is a fundamental reason why, in atoms, the heavy nucleus (made of protons and neutrons) sits as a relatively well-localized, tiny dot in the center, while the light electrons form a vast, diffuse, and uncertain 'cloud' around it. The lighter the particle, the more pronounced its quantum wave nature becomes at a given energy.

By seamlessly connecting the macroscopic concept of temperature to the microscopic concept of kinetic energy, and then bridging that to quantum uncertainty via momentum, we have not just solved a physics problem—we have witnessed the beautiful, interconnected tapestry of the universe.