Electric/magnetic fields vs energy of a photon

[Pierre007080]

If the energy of a photon ONLY depends on it's frequency, how does the SIZE of the electric/magnetic fields as "amplitude" of the waves affect the characteristics of the photon? Does a photon of a certain given frequency (certain energy)which has been generated by a larger field have fewer waves to the photon packet?? Be gentle guys!

 

[Born2bwire]

The electric and magnetic fields are a macroscopic observation of the photons. When we talk about electromagnetic fields in classical electrodynamics, we are talking about the fields produced by a statistically large number of photons. So what this means is that if we kept the amplitudes of the observed electric and magnetic fields the same and changed the frequency, the energy in the fields remains constant. What changes is the photon density of the fields. If we increase the frequency but keep the fields the same, then we are decreasing the number of photons that we would count at some detector for these fields since the energy per photon increases. This actually can be a problem because in very low power devices we can see a "grain" in the electromagnetic waves due to the density of the incoming photons becoming low.

But in terms of individual photons, we cannot really think of the fields in the same way because now the observed fields will have fluctuations in the amplitudes we measure due to the quantum effects becoming predominant.

 

[Pierre007080]

Hi Born2bwire,
Thanks for your reply. May I rephrase to make sure I follow: If we have a oscillating electric current at a fixed frequency and the increased the current, but maintained the oscillation frequency. Would the resultant changed radiation be measured as an increased number of photons of the original radiation per unit time?

 

[Born2bwire]

Hi Born2bwire,
Thanks for your reply. May I rephrase to make sure I follow: If we have a oscillating electric current at a fixed frequency and the increased the current, but maintained the oscillation frequency. Would the resultant changed radiation be measured as an increased number of photons of the original radiation per unit time?

Yes.

 

[Pierre007080]

Thanks again Born2bwire,
Does this increased energy from the increased number of photons behave as an increased amplitude? I get the feeling that if more photons that are in phase are present, that the resultant superposition could be interpreted and even calculated in the same way that an increased amplitude would be?

 

[Born2bwire]

Thanks again Born2bwire,
Does this increased energy from the increased number of photons behave as an increased amplitude? I get the feeling that if more photons that are in phase are present, that the resultant superposition could be interpreted and even calculated in the same way that an increased amplitude would be?

I'm not sure if you want to associate the two so strongly. You need to remember that the density of photons correlates to the amplitude of the observed fields. The larger the density the more this correlation becomes a linear relationship (that is, as we move further away from the quantum world the classical model becomes more accurate). First, the photons themselves do not directly have a phase. The photon's wavefunction has a phase. As we add more photons their wavefunctions will combine and the various phase interactions will give rise to interference and so on. And of course the wavefunction of a single photon can interfere with itself as we see in double slit experiments (this is why it is better to point out that the phase is with the wavefunction and not the photon itself. The photon is a particle and we generally think of them in the classical sense as being localized but this is more of a distinction between what we mean by a classical particle and a quantum particle). This interference means that there are areas of lower probability of finding a photon and areas of higher probability. For a system with a large number of photons this would correlate to identical regions having electromagnetic fields of lower amplitude and other areas having fields of higher amplitudes.

So yes, we do find that in problems where the interference of the photons results in areas of low and high density correlate to the same thing with the observed fields when we allow for a large number of photons in the system. If we have a small number of photons we may see the same interference patterns but we cannot make the same qualification on the observed fields for the same system though the mean results across a large number of experiments should show the proper correlation.

 

[Pierre007080]

Hi Born2wire,
I appreciate your thorough answer. I get the picture. May I impose on your knowledge further to help me to relate this "amplitude" of the field to the classic transverse wave function which relates Energy of a sinusoidal wave to both the frequency squared AND the amplitude squared. Can this comparison be made with EM waves?

 

[Born2bwire]

Hi Born2wire,
I appreciate your thorough answer. I get the picture. May I impose on your knowledge further to help me to relate this "amplitude" of the field to the classic transverse wave function which relates Energy of a sinusoidal wave to both the frequency squared AND the amplitude squared. Can this comparison be made with EM waves?

In classical electromagnetics, we have the electromagnetic fields as the primitives. These are the fields that we observe by measurement of forces on a test charge.

In quantum field theory, we have the photon which acts as the mediator of the electromagnetic force. The electromagnetic fields are no longer the primitives, the scalar and vector potentials are the primitives. Instead, the electromagnetic fields are the observables of system. One of the consequences of this is that the electromagnetic fields that we observe for identical quantum systems can have a variance in their values but the statistical mean should approach the values of the fields we would observe in the equivalent classical problem as we increase the number of photons in the system. So for systems with large numbers of photons, we expect to see a direct correlation between the classical rules for the electromagnetic fields and the rules for quantum field theory. So in quantum field theory, the photon is the mediator of the force and it also represents the energy in the fields. Thus, the energy density in a given volume is related to the density of photons in that volume. And since the classical energy density is related to the magnitude square of the fields, then increasing the magnitude of the fields represents an increase in the photon density in that volume and vice versa.

Now the wavefunction of the photon represents the probability to find the photon within a given volume. So areas where the wavefunction is large will mean that for a large number of photons in this system we expect to see a higher density of photons. Thus, since the density of photons is higher in this region we expect that the associated fields will be higher as the classical model predicts.

So yes, I would say in situations where the quantum model gives the same results as the classical model I think that we can say that there is a direct correlation between the magnitude squared of the wavefunction, the density of the photons, and the magnitude squared of the observed electromagnetic fields since all three should be directly proportional to the energy density.

I make use of this in Casimir energy calculations because it means that we can calculate the Casimir energy by counting up the energy in the photon modes, the energy represented in the wavefunction of the system's fields (a bit hand wavy description but we use path integrals for this) and by the energy in the equivalent classical electromagnetic fields (using the Maxwell Stress Tensor).

 

[Pierre007080]

Born2bwire,
you have certainly given me a thorough response and managed to keep it related to my original question. Well done and thank you. Regards.