1.

Two electrons are moving with non-relativistic speeds perpendicular to each other. If corresponding de Broglie wavelengths are lamda_1 and lamda_2 , their de Broglie wavelength in the frame of reference attached to their centre of mass is:

Answer»

`lamda_(CM) = lamda_1 = lamda_2`
`(1)/(lamda_1)= (1)/(lamda_1) + (1)/(lamda_2)`
`lamda_(CM) = (2 lamda_1 lamda_2)/(sqrt(lamda_1^2 +lamda_2^2 ))`
`lamda_(CM) = ((lamda_1 + lamda_2)/(2))`

Solution :Momentum (p) of each electron `(h)/(lamda_1) HATI " and " (h)/(lamda_2) hatj`
velocity of CENTRE of mass
`V_(cm)= (h)/(2m lamda_1) hati + (h)/(2m lamda_2) hatj (because p = mv)`
velocity of 1st particle about centre of mass
`V_(1CM) = (h)/(2m lamda_1) hati - (h)/(2m lamda_2) hatj`
`lamda_(cm) = (h)/(sqrt( (h^2)/(4 lamda_1^2) + (h^2)/(4lamda_2^2) )) = (2lamda_1 lamda_2)/(sqrt(lamda_1^2 + lamda_2^2) ) ( becauselamda = h/p)`


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