Two identical non-relativitic partcles A and B move at right angles to each othre, processing de Broglie wavelengths `lamda_1` and `lamda_2`, respectively. The de Broglie wavelength of each particle in their centre of mass frame of reference is

Options

A. `lambda_(1)+lambda_(2)`
B. `(lambda_(1)lambda_(2))/sqrt(lambda_(1)^(2)+lambda_(2)^(2))`
C. `(lambda_(1)lambda_(2))/(lambda_(1)+lambda_(2))`
D. `(2lambda_(1)lambda_(2))/sqrt(lambda_(1)^(2)+lambda_(2)^(2))`

Answer

D

Explanation

`1/lambda=h/(mv) "and let"bar(V_(1))=h/(mlambda_(1))i,bar(V_(2))=h/(mlambda_(2))j`and
`bar(V)_(c)=(mbar(V)_(1)+mbar(V)_(2))/(2m)=(barv_(1)+barv_(2))/2=h/(2mlambda_(1))i+h/(2mlambda_(2))j`
In Centre of mass frame of reference
`v_(1c)=barv_(1)-barv_(c)=h/(2mlambda_(1))i-h/(2mhlambda_(2))j` and
` barv_(2c)=barv_(2)-barv_(c)=(-h)/(2malambda_(1))i+h/(2mlambda_(2))j`
` i.e|barv_(1c)|=|barv_(2c)|=h/(2m)sqrt(1/lambda_(1)^(2)+1/lambda_(2)^(2))=h/(2m)sqrt(lambda_(1)^(2)+lambda_(2)^(2))/(lambda_(1)lambda_(2))` ltbr. De-Broglie's wavelength of one of the particles in the centre of mass frame of reference is
` lambda=h/(m|barv_(1c)|)=(2lambda_(1)lambda_(2))/sqrt(lambda_(1)^(2)+lambda_(2)^(2))`