A particle of mass M at rest decays into two masses m1 and m2with non-zero velocities. What is the ratio λ1/λ2 of de Broglie wavelengths of particles?

 We know that, according to de-Broglie’s hypothesis, the momentum of the particle is, p = hc/λ

Here,

• h is Planck’s constant
• c is the speed of light
• λ is the de Broglie wavelength.

According to the law of conservation of momentum, the momentum of a system remains conserved.

Therefore, we can write, Mv=m1v1 + m2v2

Here,

• v is the velocity of parent particle,
• v1 is the velocity m1 and m1 is the velocity of m2 .

Since the parent particle is at rest, the initial velocity v is zero. Therefore, the above equation becomes,

0=m1v1 + m2v2 ⇒ m1v1 =−m2v2

Therefore, from the above equation, the momentum of the particle of mass m1 and the momentum of the particle of mass m2 is equal.

So, we can write, p1=p2

⇒ hc/λ1 = hc/λ2

Planck’s constant h and speed of light c is constant for both particles.

Therefore, the wavelength of these particles is the same.

Therefore, we can write, ∴ λ1/λ2 = 1