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A particle A with a mass m(A) is moving ...

A particle A with a mass `m_(A)` is moving 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.

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initial de-Broglie wavelength (before collision)for particle A,
`lambda_(i)=(h)/(m_(A)v_(i))=(h)/(m_(A)v)`(`because v_(i)=v)`……(1)
Now ,final velocity of particle A according to formula,
`v_(f)=((m_(A)-m_(B))/(m_(A)+m_(B)))v_(i)=((m_(A)-m_(B))/(m_(A)+m_(B)))v(becausev_(i)=v)` ......(2)
`implies`final de-Broglie wavelength for particle A,
`lambda_(f)=(h)/(m_(A)v_(f))=(h)/(m_(A))xx((m_(A)+m_(b)))/(m_(A)-m_(B)v)` .....(3)
From equations (1) and (3) ,required change is ,`lambda_(f)-lambda_(i)=(h)/(m_(A)v)((m_(A)+m_(B))/(m_(A)-m_(B))-1)`
`(h)/(m_(A)v)((m_(A)+m_(B)-m_(A)+m_(B))/(m_(A)-m_(B)))`
`(2hm_(B))/(m_(A)(m_(A)-m_(B))v)`.
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