Vern said:
Billy T; I know we can't apply statistical analysis to a one-cycle wave shape and get a meaningful result; all I can do is agree to the facts you present that show that Fourier analysis can't describe one. I don't think that excludes the existence of a one-cycle photon....
(1)I never mentioned "statistical analysis."
(2) As one CAN apply Fourier analysis to almost any F(x) I never said Fourier analysis did not apply to your one cycle long photon, so do not "agree" with me in what I did not say.
One of my many points is that a photon of E12 cycles has almost exactly the same frequency content as one of E13 cycles. Thus there energy is essentially the same. Certainly they different in energy by much less than 0.000,1% yet one (the E13 cycle one) is making 10 times more vacuum saturation points BY YOUR MODEL. - Do you not find that, at least, strange?
Most of my points were experimental evidence that shows (by the existence of interference patterns) that photons have considerable length, sometimes more than a meter, so they experimentally are known to have very many cycles, but you seem to be finally acknowledging these FACTS in your latest reply, so now the main objection is the fact that one of two photons, with less than 0.000,1% difference in energy between them, is making 10 times more vacuum saturation points.
-------------------------------------------------
later by edit:
I realize that I have been exaggerating. - No photon every observed by man is E16 cycles long. E16 is a very big number. A hundred meter long photon of 5000 Angstroms light (approximately green) has wavelength of 5xE(-7) meters so a 100 meter long photon line has only 2E8 cycles.
The longest photons come from isolated atoms. (They can be moving steadily in your reference frame so the energy you measure may not correspond exactly the difference in the energy levels of the radiating atom because of Doppler effect.) Any collision would thus change their steady velocity and you could consider the width of the photon due to Doppler effect, but Doppler effect is normally considered to be a statistical effect of many different photons from different atoms with different speeds. Not necesarily due to collisions changing the speed of the radiating atom. (Collisions only made a "thermal distribution" of the atom velocities.)
In true "collisional broadening" you could (I think more accurately) think that the energy levels themselves are dynamically changing during the collision so EACH photon has a broader range of frequencies if its radiating atom is in a dense source. - Solids being the extreme case of “continuum radiators”. Thus very long photons always come from low pressure sources.
Another requirement for a very long photon is that the life time of the upper state must be very long. This is directly related to the uncertainty principle but in ways perhaps impossible for humans to comfortably relate to. Putting it crudely and incorrectly, if it takes a long time for the electron to transition to the lower state, then the ‘delta t” of the uncertainty principle is large and consequently the “delta E” or energy spread / uncertainty of the photon frequency can be small. (A “sharp” spectral line, can be measured, but the "uncertainity" is fundamental to the photon, to "reality," not only the measurement. IN PRINCIPLE, IT HAS A FREQUENCY SPREAD)
I like to look at it with a more classically and surely a more wrong POV. - Namely that long lived upper state slowly “pumps out” the photon’s EM structure for a long time and the head of the photon has been traveling away for the radiating atom for some time. - All nonsense I am sure, but it makes me feel better, as if I understood, and it is consistent with Fourier analysis POV in that many cycles are required for the frequency to be sharply defined.
I think the longest photon ever observed is that of the “green line” from high atmospheric glow, excited by (and part of ) electron induced aurora. It is a “forbidden line” of the oxygen atom. (The transition violates quantum mechanical selection rules, probably is possible only by perturbation of passing electron or something.) As it is “forbidden” it has a very long life time for the upper state. It can not be produced by man, still, I think. I do not know, but bet it is a few hundred meters long. Each photon of it might have E9 cycles.
Perhaps the least number of cycles in a “line” photon is a few hundred. Thus even if they differ much less than 1% in energy, two of these extreme line photons could differ in the amount of vacuum saturation they produce by a factor of a million!