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

`|vecp_(C)|=|vecp_(A)|+|vecp_(B)| or h/(lambda_(C))= h/(lambda_(A))+ h/(lambda_(B)).......(i)` Following cases may arise
Case (i) `p_(A)gt0, p_(B)gt0`, i.e., both `p_(A) and p_(B)` are positive.
From (i), `h/(lambda_(C))=(h(lambda_(A)+lambda_(B)))/(lambda_(A)lambda_(B)) or (lambda_(C))=((lambda_(A)lambda_(B)))/(lambda_(A)+lambda_(B))`
Case (ii) `p_(A)lt0, p_(B)lt0`, i.e., both `p_(A) and p_(B)` are negative.
From (i), `h/(lambda_(C))=-h/(lambda_(A))-h/(lambda_(B))=- (h(lambda_(A)+lambda_(B)))/(lambda_(A)lambda_(B)) or (lambda_(C))=-((lambda_(A)lambda_(B)))/(lambda_(A)+lambda_(B))`
Case (iii) `p_(A)gt0, p_(B)lt0`, i.e., both `p_(A)is +ve and p_(B)` -ve.
From (i), `h/(lambda_(C))=h/(lambda_(A))-h/(lambda_(B))= ((lambda_(B)-lambda_(A))h)/(lambda_(A)lambda_(B)) or (lambda_(C))=((lambda_(A)lambda_(B)))/(lambda_(B)-lambda_(A))`
Case (iv) `p_(A)lt0, p_(B)gt0`, i.e., both `p_(A)is -ve and p_(B)` +ve.
From (i), `h/(lambda_(C))=-h/(lambda_(A))+h/(lambda_(B))= ((lambda_(A)-lambda_(A))h)/(lambda_(A)lambda_(B)) or (lambda_(C))=((lambda_(A)lambda_(B)))/(lambda_(A)-lambda_(B))`