A particle of mass M at rest decays into two particles of masses m1 and m2 having non-zero velocitie

Q: Why do the particles move in opposite directions?

To conserve linear momentum. Since the initial momentum is zero, the final momentum vectors must add up to zero, which is only possible if they are equal in magnitude and opposite in direction.

Q: Does the mass of the particles affect their de-Broglie wavelength?

Not directly in this specific case! The de-Broglie wavelength depends entirely on momentum ($p = mv$). Even though their masses are different, their velocities adjust perfectly such that their momenta are exactly equal.

Q: What if the initial particle was moving before it decayed?

If the initial particle was moving, the total initial momentum would not be zero. The vector sum of the final momenta would equal that initial momentum, and their individual momenta (and thus de-Broglie wavelengths) would generally not be equal in the lab frame.

Q: Is kinetic energy conserved in this decay process?

No! The particle at rest has zero kinetic energy. The decayed particles have non-zero kinetic energy. This energy comes from the mass defect, where a tiny amount of rest mass is converted into kinetic energy according to $E = mc^2$.

Q: Why is the answer not dependent on the mass ratio $m_1 / m_2$?

Because the question asks for the ratio of de-Broglie wavelengths, which depends only on momentum. Since the momenta are forced to be equal by conservation laws, the mass ratio becomes completely irrelevant to the final wavelength ratio.

Question

A particle of mass $M$ at rest decays into two particles of masses $m_1$ and $m_2$ having non-zero velocities. The ratio of the de-Broglie wavelengths of the particles $\frac{\lambda_1}{\lambda_2}$ is

Options:

(A) $\frac{m_1}{m_2}$

(B) $\frac{m_2}{m_1}$

(C) $1$

(D) $\frac{\sqrt{m_2}}{\sqrt{m_1}}$

Sigma Percentile · Accepted Answer

### The Anatomy of a Decay

Imagine a heavy particle of mass $M$ floating completely at rest in the vacuum of space. Because it is stationary, its initial momentum is exactly zero.

Suddenly, an internal instability causes it to decay. It splits into two smaller fragments with masses $m_1$ and $m_2$.

These fragments fly apart with non-zero velocities. This explosive event might seem chaotic, but it is governed by one of the most unbreakable rules of the universe.

### The Invisible Hand of Momentum

Since no external forces triggered this explosion, the universe demands that the total linear momentum must remain conserved.

$$ \vec{p}_{initial} = \vec{p}_{final} $$

Because the initial momentum was zero, the vector sum of the final momenta of the two fragments must also be zero.

$$ 0 = \vec{p}_1 + \vec{p}_2 $$

For this to happen, the two fragments must shoot off in exactly opposite directions. More importantly, their momentum magnitudes must be perfectly equal.

$$ |\vec{p}_1| = |\vec{p}_2| = p $$

### The Quantum Connection

Now, we transition from classical mechanics to the quantum realm. In 1924, Louis de Broglie proposed that every moving particle has an associated matter wave.

The wavelength of this matter wave is inversely proportional to the particle's momentum, connected by Planck's constant $h$.

$$ \lambda = \frac{h}{p} $$

This elegant equation tells us that the de-Broglie wavelength doesn't care about the mass or velocity individually; it only cares about their product, the momentum.

### The Elegant Cancellation

We are asked to find the ratio of their de-Broglie wavelengths, $\frac{\lambda_1}{\lambda_2}$. Let's set up the fraction using our quantum formula.

$$ \frac{\lambda_1}{\lambda_2} = \frac{\frac{h}{p_1}}{\frac{h}{p_2}} $$

By simplifying the fraction, Planck's constant cancels out, leaving us with the inverse ratio of their momenta.

$$ \frac{\lambda_1}{\lambda_2} = \frac{p_2}{p_1} $$

But remember our crucial discovery from the conservation of momentum! The magnitudes of their momenta are identical ($p_1 = p_2 = p$).

$$ \frac{\lambda_1}{\lambda_2} = \frac{p}{p} = 1 $$

**The ratio is exactly 1.** Despite having different masses and different velocities, the universe perfectly balances their momenta, granting them identical quantum wavelengths.

Animated Solution for Physics - Dual Nature of Matter and Radiation: A particle of mass $M$ at rest decays into two particles of masses $m_{1}$ and $m_{2}$ having non-zero velocities. The ratio of the de-Broglie wavelengths of the particles $\frac{λ _{1}}{λ _{2}}$ is

Select Answer:

Visualized Solution

$I ni t ia l S t a t e$

$D ec a y E v e n t$

$C o n ser v a t i o n o f M o m e n t u m$

$M o m e n t u mV ec t or s$

$d e - B r o g l i e W a v e l e n g t h$

$W a v e l e n g t h R a t i o$

$F ina l A n s w er$

The Sigma Insight: Matter Waves and de Broglie Relation

The Anatomy of a Decay

The Invisible Hand of Momentum

The Quantum Connection

The Elegant Cancellation

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } InitialState

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