Animated Solution for Physics - Dual Nature of Matter and Radiation: A particle $A$ of mass $m$ and initial velocity $v$ collides with a particle $B$ of mass $\frac{m}{2}$ which is at rest. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths ${\lambda }_A$ to ${\lambda }_B$ after the collision is

The Collision

Imagine you are observing a microscopic billiard game. On the green felt of our quantum table, we have two players. Particle $A$, a robust sphere of mass $m$, is cruising along with an initial velocity $v$. Waiting patiently at rest is Particle $B$, a lighter sphere with exactly half the mass, $m/2$.

They are on a direct collision course. A head-on, perfectly elastic collision is about to occur. But we aren't just interested in where they go after the crash; we want to peer into their quantum nature. We want to find the ratio of their de-Broglie wavelengths, ${\lambda }_A$ to ${\lambda }_B$, after they bounce off each other.

Momentum and Restitution

Before we can talk about quantum waves, we need to solve the classical mechanics problem. Where do they go, and how fast?

Since our microscopic billiard table is frictionless and there are no external forces acting on the particles in the horizontal direction, the universe demands that the total linear momentum must be conserved.

Let's write down the master equation for momentum:

To make our lives easier, let's multiply the entire equation by 2 to clear the fraction:

We have one equation but two unknowns ($v_A$ and $v_B$). We need another piece of the puzzle. This is where the nature of the collision comes in. We are told it is perfectly elastic. In the language of physics, this means no kinetic energy is lost to heat or sound. Mathematically, it means the coefficient of restitution, $e$, is exactly $1$.

The coefficient of restitution is the ratio of the relative velocity of separation to the relative velocity of approach:

Substituting our known values ($e=1$, $u_A=v$, $u_B=0$):

Now we have a neat system of two linear equations. From equation (2), we can express $v_B$ as:

Let's substitute this into equation (1):

Now, plug this back into our expression for $v_B$:

We have successfully navigated the classical mechanics! Particle $A$ slows down to a third of its original speed, while Particle $B$ shoots off much faster.

Now, let's put on our quantum glasses. Louis de Broglie taught us that every moving particle has an associated wave. The wavelength of this matter wave is inversely proportional to the particle's momentum.

The legendary de-Broglie equation is:

We need the ratio of their wavelengths after the collision:

Notice how the Planck's constant $h$ beautifully cancels out! The ratio of their wavelengths is simply the inverse ratio of their momenta:

Let's substitute the expressions for their final momenta. Remember to use their respective masses!

So the ratio becomes:

The mass $m$ cancels out, leaving us with:

Finally, let's plug in the velocities we worked so hard to find ($v_A=v/3$ and $v_B=4v/3$):

The $v/3$ terms cancel out perfectly:

And there we have it! The de-Broglie wavelength of Particle $A$ is exactly twice that of Particle $B$ after the collision. A beautiful, clean integer emerging from the chaos of a quantum crash.