QM explanation of electromagnetic waves

[J O Linton]

TL;DR

What is the standard quantum mechanical explanation for the propagation of an electromagnetic wave?

Just as General Relativity has superceeded Newtonian gravitation, so QED is claimed to have succeeded Maxwells laws of elecromagnetism. What then is the standard quantum mechanical explaition of the fact that a bunch of elecrons oscillating in a wire 100 miles away can make an electron in my TV aerial oscillate with the same frequency and appropriate phase?

 

[Dale]

It is the same as the explanation with Maxwell's equations. QED reduces to Maxwell's equations at the scale of a TV aerial etc.

 

[J O Linton]

That is not an answer. GR may reduce to Newtons laws at speeds less than c but the explanation of why planets move in ellipses is totally different under GR. I want an explanation in terms of photons and, perhaps, wave functions.

 

[renormalize]

Radio/TV carrier waves are so-called coherent (or Glauber) states of the quantized electromagnetic field:

(from https://www.rp-photonics.com/coherent_states.html).
A brief mathematical presentation of coherent states, with references, is available at:
https://arxiv.org/pdf/0903.5096v1.

 

[Dale]

In Maxwell's equations you have potentials and fields and sources, and in QED in this scenario you have potentials and fields and sources. In both cases the equations of motion are Maxwell's equations. The wave function is the thing that evolves according to Maxwell's equations.

There are not a definite number of photons because the state is a coherent state. So I don't think that photons are even relevant here. You only get a definite number of photons if you measure them, and regular antennas don't do that.

Edit: @renormalize gave a good image with a coherent state. It is just Maxwell's equations with some noise. And the quantum noise will be much smaller than the classical noise in your scenario.

 

[J O Linton]

Radio/TV carrier waves are so-called coherent (or Glauber) states of the quantized electromagnetic field:
View attachment 375107
(from https://www.rp-photonics.com/coherent_states.html).
A brief mathematical presentation of coherent states, with references, is available at:
https://arxiv.org/pdf/0903.5096v1.

Thank you for these very appropriate references

 

[bhobba]

If you want the gory technical detail, see:

https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=3211&context=physics_facpub

Be a bit wary, though, if you want to verify all the math. While a worthwhile exercise, some people have found it 'hard' going. You may need to post a question in the advanced homework help forum. Personally, these days, I tend to just accept math that is tedious to derive (not good practice, but I am 71 now and in an assisted living facility). You still get a lot out of reading papers like that.

Thanks
Bill

 

[Matterwave]

If you want the gory technical detail, see:

https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=3211&context=physics_facpub

Be a bit wary, though, if you want to verify all the math. While a worthwhile exercise, some people have found it 'hard' going. You may need to post a question in the advanced homework help forum. Personally, these days, I tend to just accept math that is tedious to derive (not good practice, but I am 71 now and in an assisted living facility). You still get a lot out of reading papers like that.

Thanks
Bill

The algebra in that paper and the algebra needed in QFT in general is truly quite a lot. I remember my QFT course in grad school was basically my professor grinding through expressions for the whole hour every lecture lol.

Unfortunately I got very little out of that course..

 

[bhobba]

The wave function is the thing that evolves according to Maxwell's equations.

Dale's answer is, as usual, excellent. However, after reading the link I posted a natural question is what is the wave function of a photon? That is quite deep:
https://arxiv.org/pdf/0708.0831

Thanks
Bill

 

[bhobba]

The algebra in that paper and the algebra needed in QFT in general is truly quite a lot. I remember my QFT course in grad school was basically my professor grinding through expressions for the whole hour every lecture lol.

Unfortunately I got very little out of that course..

When I was younger, I would grind through the details. I even taught myself Rigged Hilbert Spaces (it took some time) to understand Dirac's methods from his textbook and von Neumann's in his textbook, which was just an extension of Hilbert spaces I studied in my math degree. In hindsight, while my issues were resolved, its usefulness is debatable. Now I simply suggest people read a book on distributions (eg https://www.amazon.com.au/Theory-Distributions-Non-Technical-Introduction-Richards/dp/052137149X) and the first few chapters of Ballentine.

Thanks
Bill

 

[JimWhoKnew]

That is not an answer. GR may reduce to Newtons laws at speeds less than c but the explanation of why planets move in ellipses is totally different under GR. I want an explanation in terms of photons and, perhaps, wave functions.

Analyzing complex macroscopic multi-particle phenomena in terms of Feynman diagrams, may be extremely difficult (if possible at all).
However, from the Path Integral formalism of QFT, we expect macroscopic systems (i.e. when corrections of the order of are negligible, so it can be regarded as zero) to behave classically. We can ignore all paths, except those for which the classical action is extremal.

Edit:

However, after reading the link I posted a natural question is what is the wave function of a photon?

Cohen-Tannoudji et al. ("Photons and Atoms") think that wavefunctions (of the sort of non-relativistic QM) are inadequate for relativistic particles.
In a slightly different context, they hint that by superposing plane waves, it's impossible to construct a wave packet for a free photon moving in the $\hat z$ direction, say, which is both localized and an eigenstate of transverse polarization $~E\hat x~$. To localize the packet, one will need to include waves for which $\vec k$ has a non-vanishing $\hat x~$ component, so there will be some probability for either a longitudinal or $\hat y$ directed polarization.