A particle A with a mass `m_(A)` is moving with a velocity v and hits a particle B (mass `m_(B)`) at rest (one dimensional motion). Find the change in the de-Broglie wavelength of the particle A. Treat the collision as elastic.

Explanation

As collision is elastic, hence laws of conservation of momentum and kinetic energy are obeyed.
According to law of conservation of momentum,
`" "1/2m_(A)v^(2)=1/2m_(A)v_(1)^(2)+1/2m_(B)v_(2)^(2)`
`implies" "m_(A)(v-v_(1))-m_(B)v_(2)`
According to law of conservation of kinetic energy,
`" "1/2m_(A)v^(2)=1/2m_(A)v_(1)^(2)+1/2m_(B)v_(2)^(2)`
`implies" "m_(A)(v-v_(1)^(2))=m_(B)v_(2)^(2)`
`implies" "m_(A)(v-v_(1))(v+v_(1))=m_(B)v_(2)`
Dividing Eq. (ii) by Eq. (i)
we get, `" "v+v_(1)=v_(2)orv=v_(2)-v_(1)`
Solving Eqs (i) and (iii), we get
`" "v_(1)=((m_(A)-m_(B))/(m_(A)+m_(B)))vand v_(2)=((2m_(A))/(m_(A)+m_(B)))v`
`" "lamda_("initial")=h/(m_(A)v)`
`" "lamda_("final")=h/(m_(A)v_(1))=(h(m_(A)+m_(B)))/(m_(A)(m_(A)-m_(B))v)`
`" "Deltalamda=lamda_("final")-lamda_("initial")=(h)/(m_(A)v)[(m_(A)+m_(B))/(m_(A)-m_(B))-1]`