Q. 183.7( 15 Votes )

Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).

Answer :

we know the relation between energy and momentum P of a electromagnetic radiation having velocity C and energy E is


P = E/C


and we know energy of electromagnetic radiation is given by relation


E = h𝜈


Where E is the energy of Photon, 𝜈 is the frequency of electromagnetic radiation and h is Planck’s constant


So putting value of E in equation P = E/c


we get


P = h𝜈/c


We know relation between frequency wavelength and velocity of electromagnetic radiation as


c = 𝜈𝜆


Where 𝜈 is the frequency and 𝜆 is wavelength of electromagnetic radiation having velocity c


So re arranging we get


𝜈 = c/𝜆


Putting the value of 𝜈 in equation P = h𝜈/c


We get


P = hc/λc = h/𝜆


Or we can say the wavelength of electromagnetic radiation is given as


𝜆 = h/P


Where h is Planck’s constant and P is momentum of electromagnetic radiation


Now we know de Broglie wavelength of quantum (photon) can be given by relation


𝜆 = h/mv


Where 𝜆 is de Broglie wavelength of a photon having mass m and moving with velocity v (here the velocity of photon v = C)


But we know momentum is given by relation


P = mv.


So, substituting we get de Broglie wavelength of quantum (photon) as


𝜆 = h/P


This is same as wavelength of electromagnetic radiation


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