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

Explanation

Given from conservation of momentum,
`|P_(C)|=|P_(A)|+|P_(B)|`
`implies" "h/(lamda_(C))=h/(lamda_(A))+h/(lamda_(B))" "[becauselamda~~h/(mv)=h/pimpliesp=h/lamda]`
`implies" "h/(lamda_(C))=(hlamda_(B)+hlamda_(A))/(lamda_(A)lamda_(B))`
`implies" "(lamda_(C))/(h)=(lamda_(A)lamda_(B))/(hlamda_(A)+hlamda_(B))implieslamda_(C)=(lamda_(A)lamda_(B))/(lamda_(A)+lamda_(B))`
Case I Suppose both `p_(A) and p_(B)` are positive, then `" "lamda_(C)=(lamda_(A)lamda_(B))/(lamda_(A)+lamda_(B))`
Case II when both `p_(A) and p_(B)` are negative , then `" "lamda_(C)=(lamda_(A)lamda_(B))/(lamda_(A)+lamda_(B))`
Case III When `p_(A)ggt0,p_(B)lti.e., p_(A)` is positive and `p_(B)` is negative,
`" "h/(lamda_(C))=h/(lamda_(A))-h/(lamda_(B))=((lamda_(B)-lamda_(A))h)/(lamda_(A)lamda_(B))`
`implies" "lamda_(C)=(lamda_(A)lamda_(B))/(lamda_(B)-lamda_(A))`
Case IV `p_(A)lt0,p_(B)gt0,i.e.,p_(A)` is negative and `p_(B)` is positive,
`therefore" "h/(lamda_(C))=(-h)/(lamda_(A))+h/(lamda_(B))`
`implies" "=((lamda_(A)-lamda_(B))h)/(lamda_(A)lamda_(B))implieslamda_(C)=(lamda_(A)lamda_(B))/(lamda_(A)-lamda_(B))`