LightPhoton said:
I was reading Optics by Hecht
I don't have this textbook, but from what I can gather it appears to be a textbook mainly on classical optics. However, the statement you refer to is a statement about quantum optics, i.e., based on quantum electrodynamics, QED. And, as I commented in post #3 just now, this thread is in the QM forum, so presumably QED is to be the basis for discussion.
For a good layman's discussion of how QED treats this case, including discussion of experiments in which QED's predictions about things like this are confirmed, you might try Feynman's book
QED: The Strange Theory of Light and Matter.
With all that said:
LightPhoton said:
the law of reflection is only valid statistically
It depends on what you mean by "the law of reflection". See below.
LightPhoton said:
some photons might reach the observation point (P in image, S being the source) by following different paths, that is, the paths for which the angle of reflection incidence is not equal to the angle of reflection.
Some photons might--at least if we are using the path integral formulation of QED (which is what Feynman uses in the book I referred to above) and being somewhat hand-waving in our description.
A more accurate description is that the
probability of a photon going from S to P by bouncing off the mirror in between depends on the specific point on the mirror at which the photon bounces. The highest probability is for the classical path, the one where the angle of incidence (note my correction in the quote above) equals the angle of reflection. As you move away from that point on the mirror, the probability decreases, but it is nonzero for a fairly wide range of points on the mirror, so the total probability for a photon to go from S to P by bouncing off the mirror will include a fairly wide range of points on the mirror; it cannot be accounted for solely by photons traveling on the classical path.
I should also correct a common misconception: in the above, when I talk about photons following different paths, I am
not talking about
different photons--i.e., I am
not saying that some photons follow one path and some another, and adding all the photons together gives the observed intensity of light going from S to P by reflecting off the mirror in between. QED's model does not say that. What QED's model says, in the path integral formulation, is that
every single photon travels on all possible paths, and each path has a probability amplitude associated with it, and the final probability for a single photon to go from S to P by bouncing off the mirror in between is obtained by adding together all the amplitudes.
I don't think "the law of reflection is only valid statistically and sometimes fails" is a good way to describe the above. Again, I don't have Hecht's textbook so I don't know the context in which the statement you refer to was made. (I also don't know how much your paraphrase of the statement distorts Hecht's intended meaning.)
LightPhoton said:
Have there been such experiments performed where the experiments deduced the path followed by photon and showing that sometimes the law fails?
No, because, as above, a photon does not have just one path, and "sometimes the law fails" is not a good description of what QED actually says about this scenario. But experiments have certainly confirmed the predictions of QED in this regime many, many times.