Hi all,
This is a question I have been wondering about for a long time (pure curiosity). I would like to know how one can deduce from an arbitrary electro-magnetic field the quantum characteristics of the photons making up the field. In the case of a laser, it is quite easy, there is a well defined momentum, energy, and polarization. The intensity of light is proportional to the flux of photons.
I am more interested in the case of a static electric/magnetic field. I think the quantum description of the force between charged particles is the exchange of virtual photons. I would like to know more about the mechanisms of this interaction. And why moving charged particles create a magnetic field according to quantum mechanics.
An other way to formulate the question is to say: I have a photon (possibly virtual) with state |psi> which would be a superposition of momentum, position and polarization states. Does it make sense to evaluate its contribution to the classical electro-magnetic field, and if yes, what are the rules ?
Subsidiary question: after correcting for the small effects (finite temperature etc.) responsible for the narrow spectrum of a laser, it looks like the momentum of the photons in the laser is perfectly defined. According to the uncertainty principle, it means that it should be totally delocalized in space. However, we can track down the beginning of the wave train of the laser, therefore it is not really perfectly delocalized. So, what is the solution to this paradox ?
Cf. any book on quantum field theory. A field is defined at every point in spacetime and average value of the square of its amplitude is proportional to the average number of particles in the state described by the field. The statistics of the distribution, beyond the average is determined by the higher moments and, since this is a field theory, it's possible to describe spatial correlations, also. All the questions in the text are addressed in textbooks, so it's a good idea to study them. In particular, how to describe wavepackets and the correlations in lasers. So the statement about the uncertainty principle can be shown, by direct calculation, to be related to the statement about the uncertainty in defining the beginning and end of a wavepacket-so the last statement is, simply, wrong. That's why there isn't any ``paradox''. This is a standard exercise in quantum mechanics-or even wave optics, since there does exist a corresponding ``uncertainty principle'' in wave optics.
Thx. QFT, that's just where I stopped learning Quantum Mechanics ;-). I've become a classical physicist (plasma) but I'm relearning QM for fun. QFT will be my next target. Can I have just one more hint in the meantime ? What is the state of the photons corresponding to a static pure electric (resp. magnetic) field ?
The answer to that is provided by the Fourier coefficients of the electric and magnetic fields: they are the quantities that describe configurations of well-defined vector number and frequency and the vector character describes the polarization states. Of course boundary conditions play a role. Static fields have zero frequency and infinitely long wavelength, absent boundaries. The polarization of physical photons is, of course, transverse. One doesn't need knowledge of quantum field theory to reach these conclusions; the properties of the solutions of Maxwell's equations suffice.
Yes, but if they have zero frequency, the energy should also be zero. But there is energy in the static electric field. How does this come up together ?
Zero and infinity are poorly defined in mathematics and physics. I don't think there is absolute zero. This is why zero point energy has a mass gap. To me zero means infinitesimally small value which can be ignored and infinity means something which cannot be explained with a finite number. I define infinity as one without a second. In other words, infinity does not have a boundary that you can define second something beyond that boundary. This is absolute infinity. When someone divides a finite number by zero, they get a very large number which is a pseudo infinity. There are so many pseudo infinities in physics and mathematics.
This simply shows that static fields are vacuum configurations for excitations of the electric and magnetic fields-that's where boundary conditions come into play. And to talk about energy means to describe the time evolution of probes, charges, else the statement that a static electric field has energy can't be given any meaning.
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