The discussion above shows that one must be careful not to confuse wave particle duality and complementarity. The most spectacular evidence that a photon is a wave, perfect interference fringes, is incompatible with the most spectacular evidence that the photon is a particle, a perfect which-path measurement. But the wave-like behavior is not incompatible with the fact that the photon can be detected only once. So, we can safely claim that the photon is both a wave and a particle not either a wave or a particle. This is wave-particle duality. The notion of wave-particle duality appears clearly in interference or diffraction experiments with massive particles, such as electrons, neutrons, atoms or even big molecules. One observes wave-like effects and nevertheless, we have no doubt that the objects are particles. In fact, the observation of a diffraction effect with electrons by Davisson and Germer as early as 1923 was a strong support to the de Broglie hypothesis that a wave with the wavelength lambda must be associated with a particle of momentum p, such that lambda times p equals the Planck constant. Why then, did it take so long to clarify that question in optics? The reason is that light produced by usual sources does not behave at all as a beam of particles, as we will see in future lessons, and in order to observe a particle-like behavior for light, it was necessary to develop one photon sources. You may think that it is a purely academic question without any practical consequence. In fact, in turns out that one photon sources are key components in the developing field of quantum technologies. I can testify that this was not anticipated by people who developed the first one photon source: we were only looking for a better understanding of nature, but it is the surprising character of such observations that prompted some imaginative people to find applications to the observed mysterious behavior. We will see some of these applications next week. For the time being, let us concentrate on trying to better understand nature. I suppose that many of you are puzzled by the concept of wave-particle duality. They probably find it impossible to develop a consistent image for something being both a wave that spreads in a broad region of space and a particle that follows a given trajectory. This difficulty, obsessed Einstein. Feynman named it a great quantum mystery. So, do not hope that there is a simple way to get comfortable with it. Several possibilities have been considered to comfort oneself. You may feel reinsured by the fact that the mathematical formalism raises no problem. As you have seen, it is the same formalism that has allowed us to describe both the wave-like and the particle-like behaviors. So, why bother? Why not "shut up and calculate" according to a famous expression? Well, many physicists and I am one of them are not satisfied with the consistent mathematical formalism only. They need also a description of the situation with images in our three dimensional world. Asher Peres, a sophisticated quantum optics theorist, once wrote: "experiments do not happen in a Hilbert space, they happen in a laboratory". This is why we need images in our ordinary three-dimensional world. Some physicists have tried to develop systematically such images. A well-known example, is the hidden-variable theory of David Bohm in which the particle trajectory is determined by a guiding potential. This theory, which is an elaboration of Louis de Broglie's double solution, is designed to yield exactly the same predictions as quantum mechanics. So, it cannot be submitted to experimental tests and a majority of physicists consider that it is a waste of time to enter into its complications. My personal solution is to think of a single particle as a wave packet propagating in our ordinary space, but I know that the wave packet will collapse instantaneously at the very moment when a detection happens. I am aware that this instantaneous collapse violates the rules that nothing can go faster than light. But on the other hand, I know that this instantaneous collapse cannot be used to transmit utilizable information faster than light. This is the simplest example of what is sometimes called the quantum non-locality, which we will encounter again in future lessons. Note that the Bohm hidden variable theory is also non-local. So, what is the conclusion you can keep in your mind? It is a difficult question to which you will progressively give your own answer. My personal answer is the following: there are situations that are really mysterious when we try to make images in the 3D world where we live. The mathematical formalism of quantum optics renders a consistent account of such situations. Recognizing a mysterious, typical quantum, situation may be the indication that there is a possibility to use it for a new application. Quantum technology is based on such ideas. So, thinking about quantum mysteries is not a waste of time. You will listen to me repeating this conclusion again and again in future lessons, but it is enough for today. See you next week for some applications of quantum mysteries in the world of quantum technologies. Bye, bye.