Question

Two particles A and B of de Broglie wavelengths λ₁ and λ₂ combine to form a particle C. The process conserves momentum. Find the de Broglie wavelength of the particle C. (The motion is one dimensional).

Answer 1

By the law of conservation of momentum,

|P_C| = |P_A| + |P_B|

Let us first take the case I when both P_A and P_B are positive, PA

then `λ_C = (λ_Aλ_B)/(λ_A + λ_B)`

In the second case when both P_A, and P_B are negative,

then `λ_C = (λ_Aλ_B)/(λ_A + λ_B)`

In case III when P_A > P_B < 0 i.e., P_A is positive and P_B is negative,

`h/λ_C = h/λ_A - h/λ_B = ((λ_B - λ_A)h)/(λ_Aλ_B)`

⇒ λC = `(λ_Aλ_B)/(λ_B - λ_A)`

And In the case of IV when P_A < 0, P_B > 0, i.e., P_A is negative and P_B is positive.

∴ `h/λ_C = (-h)/λ_A + h/λ_B`

⇒ `((λ_A - λ_B)h)/(λ_Aλ_B)`

⇒ `λ_C = (λ_Aλ_B)/(λ_A - λ_B)`