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The derivation of De Brogile's wave length:

We know that the energy of light E=mc^2

Again if the frequency of light is f

Then E=h*f [where h is planck constant]

So,

mc^2=h*f

or, mc^2=h*(c/l) [where c is the velocity of light,l is the wavelength ,& c=l*f]

or,l=(h*c)/mc^2

or,l=h/m*c

or,l=h/p.[ where p is the momentum=m*c]

this is the De Broglie's equation.

That derivation is completely wrong. The momentum of a photon is NOT given by m*c, as c=constant, m=0. Neither is the energy of light mc^2, as that also goes to zero. The derivation works by a miracle.

The proper derivation is this:

From a relativistic invariant we know that

E^2-p^2c^2=(mc^2)^2

For a photon, the mass = 0, so

E/c=p

For a photon,

E=hc/lambda

Therefore,

h/lambda=p

The first step works as a result of contra/covariant vectors, but can be checked using algebra. The relativistic Energy = m*c^2/sqrt(1-v^2/c^2) , momentum p = m*v/sqrt(1-v^2/c^2).

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