Photon and Wave

Question

There are some aspects of light that can be easily demonstrated by using the concept of wave. However I really want to know what it would be like in term of photon point of view. So I have some question.

• I know that real light is a combination plane wave in different wavelength, a Gaussian wave which can be expanded to a Fourier series. Each plane wave in the Fourier series has a quanta, which is a photon with corresponding frequency and energy. Then what is the relationship of the Gaussian wave with those infinite quantas?

• In a medium, there is phase velocity and group velocity. What are they in the photon point of view? (I think a photon may be absorbed by an atom then be released after some times so the time is delayed, therefore the velocity is decreased. I think the velocity in this sense is the phase velocity. If that is true, what is the group velocity then?)

It would be great if you can link me some papers to read about. I can understand quantum physics as long as it's not too mathematically complicated.

Answer 1

I would recommend the first chapter of Scully and Zubairy, "Quantum Optics".

Maxwell's equations are to the photon what the Dirac equation is to the electron. Indeed, one can write down Maxwell's equations in a form that is identical to a zero mass Dirac equation. In quantum optics the electric and magnetic field vectors become vector valued quantum observables, whose Cartesian co-ordinates fulfill the canonical commutation relationships. In a one-photon state, the mean values of the field observables can be shown to propagate exactly following Maxwell's equations. A one-photon state is particularly simple, insofar that it is wholly defined by these means. This is somewhat analogous to the Poisson probability distribution: it is wholly defined by its mean, whereas other distributions (e.g. the Gaussian) require more parameters to specify them. Likewise, the one photon state is wholly defined by these means as a function of position in space and time - entangled multiphoton states are much harder to describe because they require much more than the means of observables to define. If these means propagate precisely following Maxwell's equations, then they have all the full blown "wave" properties we study. They undergo diffraction and interference. And, for destructive measurements (where the photon is absorbed) the intensity $|\vec{E}(\vec{r},\,t)|^2 + c^2 |\vec{B}(\vec{r},\,t)|^2$ becomes the probability density to absorb the photon at position $\vec{r}$ and at time $t$ when the relevant Maxwell equation solution is properly normalised.

I say more about these ideas in my answer to the question "How can we interpret polarization and frequency when we are dealing with one single photon?".

I would also recommend reading the Wikipedia article on Quantization of the Electromagnetic Field as a gentle introduction.