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If the Compton wavelength of a particles is equal to the De-Broglie wavelength. Find the total energy of the particles. 
Asked by sahupramod650 | 03 Mar, 2020, 12:51: PM
answered-by-expert Expert Answer
Compton wavelength  λC of a particle of mass m is given as, λC = h/(mc) ................(1)
 
where h is planck's constant and c is speed of light
 
de-Broglie wavelength λd of a particle of mas m is given as, λd = h/p ...............(2)
 
where p is momentum of particle. We need to use relativistic formula for momentum in eqn.(2)
 
begin mathsize 14px style p space equals space fraction numerator m u over denominator square root of 1 minus begin display style open parentheses u squared divided by c squared close parentheses end style end root end fraction end style
 
hence , we get, de-Broglie wavelength λd  as, begin mathsize 14px style lambda subscript d space equals space fraction numerator h space square root of 1 minus space open parentheses u divided by c close parentheses squared end root over denominator m u end fraction end style......................(3)
If compton wavelength and de-Broglie wavelength are equal, then by comparing eqn.(1) and eqn.(3),
 
we get speed of particle u = ( c /√2 ) m/s
 
Total energy of relativstic particle , begin mathsize 14px style E space equals fraction numerator m u squared over denominator square root of 1 minus left parenthesis u divided by c right parenthesis squared end root end fraction space space equals space fraction numerator m left parenthesis space c to the power of 2 to the power of space end exponent divided by 2 right parenthesis over denominator square root of 1 minus left parenthesis 1 divided by 2 right parenthesis end root end fraction equals space 0.707 space m c squared end style
Answered by Thiyagarajan K | 03 Mar, 2020, 15:21: PM

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