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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Physics - Board Papers