Are electro-magnetic waves the same as the wave function?

[Herbascious J]

TL;DR Summary

How is the electro-magnetic wave description of light related to the wave function described by QM for a photon? Further how does this relate to electrons and the nature of their wave function?

When studying classical mechanics we are told that light is the propogation of electromagnetic waves. This makes perfect sense, as I can imagine these fields behaving this way, and in turn have an associated wave length. When learning about QM, I have heard that the wavelength of a (any) particle can be interpreted as a probability wave associated with the wave function. I am assuming this means that the probability of a particle being in any particular position is defined by a wave like description. That I can also accept, considering the two-slit experiment and it's results seem to show the the waves enforcing and canceling each other out and in turn more or less photons falling in those areas. My first question is, are these two different descriptions of a photon's 'wave' nature identical, or are these two, separate properties of the photon? I'm assuming the wavelengths are always identical, even if the descriptions are separate in nature.

Finally, if a photon's wave-like structure can be described by both a QM wave function and separately as a physical electromagnetic wave, then are there other particles which have a QM wave function identical to some other physical property, like for example an electron? I realize the photon is a force carrying particle, so perhaps the answer to my question is related there. I'm not sure. Any input is appreciated.

 

[hilbert2]

A photon wave function and a time-dependent electromagnetic field are completely different things. For instance, the electromagnetic field consists of two vectors, the electric field $\vec{E}$ and magnetic flux density $\vec{B}$. A wave function is a scalar complex valued function of position. Also, the electromagnetic field and photons have to be handled with relativistic quantum field theory, where the wave function formalism of non-relativistic QM isn't very useful.

 

[vanhees71]

Since there is no wave function for a photon, the question doesn't make sense to begin with.

That said, of course wave functions in non-relativistic quantum theory (where they make sense) are not restricted to scalar functions. E.g., in the non-relativistic limit an electron is described by a spinor-valued wave function.

For massive particles wave functions can make some sense also in the relativistic theory as long as you get not too relativistic, i.e., as long as you do not consider collisions of the particles under consideration with energies at or larger than their rest energies, because then you always have to possibility to create new particles and/or destroy the particles you started with. This cannot be described by a wave function, which can only be used when you consider processes where the particles stay the same all the time.

For photons there's no non-relativistic limit, because they are massless quanta. Note that I don't say massless particle, because photons don't have too much in common what you'd call "particle like". E.g., from the analysis of the symmetry group of the special-relativistic spacetime model it follows that you cannot define so easily a position operator for photons, i.e., you cannot localize them in a small volume. Among other things this is described adequately only with relativistic quantum field theory, and a photon is indeed a very special state (a socalled one-photon Fock state) of the electromagnetic quantum field.

What we observe as "light" is not such a one-photon Fock state but rather something which is very well (almost perfectly) described by a classical electromagnetic wave, at least if the intensity of the em. wave is not too small. Quantum-field theoretically that corresponds to a socalled coherent state, which is another specific kind of state of the electromagnetic quantum field. In terms of the Fock states, which desribe in general situations where you have a definite number of photons, it's a specific superposition of Fock states over all photon numbers. The coherent state doesn't describe a determined number of photons but the photon number is distributed according to a Poisson distribution, i.e., if you use a photon-number-couting detector to measure the number of photons you get a random sequence of results when repeating the experiment with a lot of coherent states, whose probability is described by a Poisson distribution with the mean value of the photon number describing the intensity of the electromagnetic radiation described by the coherent state.

 

[PeroK]

Finally, if a photon's wave-like structure can be described by both a QM wave function and separately as a physical electromagnetic wave, then are there other particles which have a QM wave function identical to some other physical property, like for example an electron? I realize the photon is a force carrying particle, so perhaps the answer to my question is related there. I'm not sure. Any input is appreciated.

Note that we have different theories, involving different fundamental concepts, that ultimately describe the same natural phenomena. Classical EM and QED (Quantum Electrodynamics) both describe light and electrons and at low energies make very similar predictions. This doesn't mean, however, there is a neat correspondence between the fundamental concepts that underpin the two theories. And, at high energies, the two theories diverge and, of course, classical EM fails to make correct predictions.

In fact, if you look at Rutherford scattering using QED, it's difficult to see anything that equates to the Coulomb inverse square force law of classical EM.