Two particles A and B of de-Broglie wavelength `lambda_(1)` and `lambda_(2)` combine to form a particle C.The process conserves momentum .Find the de-Broglie wavelength of the particle C.(The motion is one -dimensional).

Explanation

Here `vecPA||vecPB`,hence `P_(C)=p_(A)+p_(B)`….(1)
There are four possiblities:
(i)`p_(B)gt0,p_(A)ggt0` both are positive .
`therefore (h)/(lambda_(c))=(h)/(lambda_(A))+(h)/(lambda_(B))implies(1)/(lambdaC)=(1)/(lambdaA)+(1)/(lambda_(B))`
`therefore lambda_(C )=(lambda_(A)lambda_(B))/(lambda_(A)lambda_(B))`
(ii)`p_(A)lt0,p_(B)lt0` both are negative.
`therefore (h)/(lambdac)=(-h)/((lambda_(A)))+(-h)/(lambda_(B))`
`therefore lamba_(C)=-((lambda_(A)lambda_(B))/(lambda_(A)+lambda_(B)))`
(iii)`p_(A)gt0,p_(B)lt0p_(A)` ispositive and `p_(B)` is negative
`therefore (h)/(lambda_(C))=(h)/(lambda_(A))-(h)/(lambda_(B))implies(1)/(lambda_(C))=(1)/(lambda_(A))-(1)/(lambda_(B))`
`therefore lambda_(C)=(lambda_(A)lambda_(B))/(lambda_(B)-lambda_(A))`
(iv)`p_(A)lt0,p_(B)gt0p_(A)` is positive and `p_(B)` is negative
`(h)/(lambda_(C))=(-h)/(lambda_(A))+(h)/(lambda_(B))implies(1)/(lambda_(C))=(1)/(lambda_(B))=(1)/(lambda_(A))`
`therefore lambda_(C)=(lambda_(A)lambda_(B))/(lambda_(A)-lambda_(B))`