Question on Energy of emr from a range of frequencies

[Lou]

We are all familiar with the equations to quantify energy of EMR from a
single wavelength or frequency. But I have been unable to find any
reference on how to calculate energy from a *range* of wavelengths or
frequencies. In particular with reference to Doppler shifted emr For
instance: Can one calculate what difference in energy , if any, an
emitted source of emr between 100-200nm would have if it was redshifted
to 200-400nm?

[Richard Livingston]

Post by Lou
We are all familiar with the equations to quantify energy of EMR from a
single wavelength or frequency. But I have been unable to find any
reference on how to calculate energy from a *range* of wavelengths or
frequencies. In particular with reference to Doppler shifted emr For
instance: Can one calculate what difference in energy , if any, an
emitted source of emr between 100-200nm would have if it was redshifted
to 200-400nm?

At first I thought the answer was obvious, but then I thought about it a little
more and realized I needed to think about it more carefully.

It is easiest to think of this quantum mechanically, I think. The number of
photons does not change, but their rate is reduced proportional to the
frequency. This gives a factor of 1/2 to the rate of photons. In addition
each photon is now half the frequency, so the energy per photon is also
reduced by 1/2. So the total power in the band 200-400nm would be 1/4
the power in the 100-200nm band prior to being red shifted.

Doing this calculation in classical EM would require transforming the EM
tensor F^{\mu\nu} via the Lorentz transform to get the correct EM field
amplitudes, from which the energy density can be calculated. The QM
argument is much simpler.

Rich L.