# The Bandwidth of the Photon ?

James Clerk Maxwell was the first person to connect light with other forms of electromagnetic radiation and in so doing he showed that the bandwidth of electromagnetic radiation extends beyond that of just light.   Since then the spectrum has been broadened to include radio waves, microwaves, infra-red, visible light, ultra violet radiation, X-rays and gamma rays.  Current thinking places no upper boundary on the frequency or energy of an individual photon and yet this cannot be the case.

In 1900 Max Planck showed that electromagnetic energy can only be absorbed or emitted in discrete amounts or quanta and found the relationship between the energy and frequency for such emissions; the higher the frequency, the higher the energy[1].   Later in 1905 Einstein showed that light itself was discrete in nature and consisted of a stream of particles, later called photons[2]. It is these two ideas which combine to give us our current picture of electromagnetic radiation as consisting of a stream of photons each of which carries a discrete amount of energy which is proportional to the frequency of the individual photon.

If the upper frequency limit of the photon were truly boundless then we could expect to see evidence of photons with truly massive energies and yet such photons are not seen.  This strongly suggests that there is a physical constraint on the maximum amount of energy that a single photon can carry and hence also that there must be a limit to the maximum frequency of the photon.  Which raises the obvious question as to where such a limit lies – and the less obvious one; as to what is the mechanism imposes such a limit.

In the classical model of the wave, the frequency of the wave, its wavelength and its velocity are related to one another by the simple equation:

 Equation 1

Where F is the frequency in Hz, v the velocity and λ the wavelength

It is almost certain that the energy carried by individual photons is limited in some way.  This could come about in one of two ways, either it is intrinsic; that is there is something about the physical nature of the photon which effectively prevents it from carrying more than a certain amount of energy, or it could be extrinsic; that is there are no processes in nature that can impart more than a certain amount of energy to an individual photon.

An extrinsic limit on the energy of individual photons would appear to be unlikely.  There is scant evidence for the existence of individual photons with energy levels of more than a few million eV and yet there are processes taking place inside stars and even here on earth where energies billions of times greater are involved.  It is far more likely therefore that the energy of the photon is limited in some way by its very nature.

If the photon energy is limited in some way then there must come a point where the wave equation breaks down.  The concept that the wavelength of the photon can extend all the way down to zero while the velocity of light remains constant for all wavelengths cannot hold true since to do so would imply that any such photons would have infinite energy.

This means that either: there is a minimum wavelength for the photon; a lower limit of wavelength below which the photon cannot exist, or that the velocity of light is not constant, but varies with the frequency and that it would have to do so in such a way that the velocity has a value of zero at zero wavelength.

[1] Planck, M. (1901)  On the Law of Distribution of Energy in the Normal Spectrum   Annalen der Physik
vol. 4, pp 553
[2] Einstein, Albert (1905), Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt (On a Heuristic Viewpoint Concerning the Production and Transformation of Light), Annalen der Physik 17 (6): 132–14

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