Two identical non-re,aticivistic particles move at right angle to each other. Possesing de Broglie wavelength `lambda_(1)` and `lambda_(2)` Find the Broglie wavelength of each particle in the frame of their centre of inertia.

Explanation

If `vec(P)_(1),vec(P)_(2)` are the momenta of the two particles then their momenta in the `CM` frame will be `+-(vec(P)_(1)-vec(P)_(2))//2` as the particle are identical.
Hence their de Broglie wavelength will be
`lambda=(2piħ)/((1)/(2)|vec(P_(1))-vec(P)_(2)|)=(4piiħ)/(sqrt(P_(1)^(2)+P_(2)^(2)))` (beacuse`vec(P_(1))_|_vec(P_(2))`)
`=(2)/(sqrt((1)/(lambda_(2)^(2))+(1)/(lambda_(2)^(2))))=(2lambda_(1)lambda_(2))/(sqrt(lambda_(1)^(2)+lambda_(2)^(2)))`