Quantization of EMR in charge acceleration

[DeepQ]

If I accelerate a charge, EM radiation is emitted. For a simple dipole model, the radiation propagates outward along a torus shape (see Griffiths, Intro to ED, 3ed., fig 11.4) - the acceleration field, with power according to Larmor formula.

Since EMR is quantized, in what way is the described dipole radiation quantized?

 

[Meir Achuz]

The emitted energy in the dipole radiation must be an integral multiple
of hbar\nu. The integer may be huge.

 

[DeepQ]

Yes... but, I've only found the classical descriptions no QM in Griffiths and Feynman (lecture notes). I cannot find it addressed by Sakurai, Shankar, or Zee (although it could be there somewhere). Dirac QM chap X (Radiation) comes at it from a hard-to-interpret perspective.

 

[Meir Achuz]

You can see the connection between classical EM and QM in chapter 16
of "Classical Electromagnetism" by Franklin.
Classical EM is really the one particle QM of the photon,
comparable to the Schordinger equation for the electron.
Since the photon is a Boson, strong fields can be obtained.
This is not possible for the electron, a Fermion.

 

[DeepQ]

That is an interesting description. Dirac and others rely on the model of a harmonic oscillator. But, that is just one case of acceleration. I am curious about a simple non-harmonic case such as this:

If I accelerate an electron by, say, a gravitational field, what do I get? Classic EM states I get an acceleration field that splits away and propagates to infinity (unlike the velocity field which moves with the charge).

If I assume that QED provides a superset of such classic theory, what defines the quantization in my acceleration example? This is not like an electron dropping to a lower energy state and emitting a photon. Such quantization from well defined states is easier to understand. But, I would like to understand the results of the example I describe above.

 

[Meir Achuz]

For an electron dropping in gravity wth acceleration g, the classical Larmor formula probably gives a good answer. Everything is quantum, but the classical limit works for many processes.
The rate of photon emission can be calculated as a bremsstrahlung problem.
Look this up in your quantum books.

 

[DeepQ]

For an electron dropping in gravity wth acceleration g, the classical Larmor formula probably gives a good answer. Everything is quantum, but the classical limit works for many processes.
The rate of photon emission can be calculated as a bremsstrahlung problem.
Look this up in your quantum books.

Yes, thanks. Before asking the question, I considered the Larmor and Bremsstrahlung approaches.

Problem has been Larmor is classic, so "Where's the quanta?" (Where's the beef?)

Bremsstrahlung as braking radiation is really more like a collision radiation resulting from sudden radical deceleration/re-acceleration. Ironically, I was planning to study that in detail next, just after I understood this very simple acceleration case.

 

[Meir Achuz]

Yes, thanks. Before asking the question, I considered the Larmor and Bremsstrahlung approaches.

Problem has been Larmor is classic, so "Where's the quanta?" (Where's the beef?)

Bremsstrahlung as braking radiation is really more like a collision radiation resulting from sudden radical deceleration/re-acceleration. Ironically, I was planning to study that in detail next, just after I understood this very simple acceleration case.

Larmor is classical. "Classical radiation" corresponds to the emission of so many photons that things look continuous. My examplee of g is probably too small an acceleration to be treated classically, But Xray production can be done with Larmor.
Bremsstrahlung need not be from a sudden collision.
In the QED calculation of bremsstrahlung, the photon emission operator is put between initial and final scattering wave functions. Be careful, Hans Bethe made a mistake the first time he did it.