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

Hint: The de-Broglie wavelength depends on the momentum of the particle.

Step 1: Conserve the momentum.

Given from conservation of momentum, 

|pC|=|pA|+|pB|

hλC=hλA+hλB         λ=hmv=hpp=hλhλC=hλB+hλAλAλBλCh=λAλBhλA+hλBλC=λAλBλA+λB

Step 2: Find the de-Broglie wavelength in each case.

Case l: Suppose both pA and pB are positive.

then, λC=λAλBλA+λB

Case ll: When both pA and pB are negative. 

then, λC=λAλBλA+λB

Case lll: When pA>0 pB<0 i.e., pA  is positive and pBis negative,

hλC=hλAhλB=(λBλA)hλAλBλC=λAλBλBλA

Case lV: pA>0 and pB<0 ie., pA  is negative and pB is positive, 

hλC=hλA+hλB=(λAλB)hλAλBλC=λAλBλAλB