# If we apply the concept of the de Broglie wave to a photon, then what will happen?

In 1923, a French physicist Prince Louis de Broglie found the concept of matter wave, according to some book I read.

He deduced an equation which relates the wave length of the matter wave with its momemtum.

Let that matter wave length be denoted by L, and let the Planck constant and the momentum be denoted by h and p respectively.

Then his equation can be written as,

L=h/p          (1)

This is called as the de Broglie wave length of a matter moving with a momentum p.

How about if we consider a photon to apply the concept of de Broglie?

Let the electromagnetic wave length of the photon be L'.

By definition of electromagnetism, it follows that

L'=c/v     (2)

where the symbol v means the frequency (regard this symbol as Greek letter neu) and c is the light speed.

The usual textbooks tell us that the momentum p' of a photon having frequency v is given by an equation

p'=hv/c     (3)

Now, let us proceed to calculate the de Broglie wave length L" of this photon.

The calculation goes as follows.,

L"=h/p'=h/(hv/c)=c/v     (4)

Oh, what a surprise!  The equation (2) and the equation (4) coincide, that is, L'=L" !

The former equation comes from a simple definition of electromagnetism, while the latter comes from the calculation based on the concept of the de Broglie wave.

I wonder why do they coincide.

Perhaps almost of the readers of this blog will not be interested in such a tiny finding.

It may even seems a tautolgical calculation. But for me, it seems to be a thrilling finding.

I don't have yet understood this completely.