To complete this first lesson, I want to say a few things about the lowest number state associated with zero photon that we call vacuum. Normally, we think of the vacuum as the absence of everything, in particular, the absence of radiation. This is very true in classical physics. But in quantum physics, the state with zero photon has a remarkable property that has physical consequences, the electromagnetic field is not null, it has fluctuations. Let us first calculate the average of the electric field operator E in vacuum state 0. Using the expression of E, we find that the average is null because a applied to |0_l> gives 0 and similarly for a_dagger applied to <0_l| on the left. But a null average does not mean necessarily that the quantity is null. It can be spread around 0. To check this possibility, we calculate the average of the squared electric field in the vacuum. When one squares the expression of E and calculates the average in the vacuum, all terms are 0 because either of a on the right or a_dagger on the left, except for one term that you see here. Using the commutator of a and a_dagger, we find one for the average of a a_dagger and thus, E1 squared for the square of the field. So the electric field has a dispersion. One also speaks of fluctuations. The root mean square value of the fluctuations has an amplitude E1. That is to say, the one photon electric field. In fact, as in the case of the material harmonic oscillator, this fluctuation can be linked to the fluctuation of the conjugate canonical observables, Q_l and P_l. As shown here, the electric field can be expressed as a function of Q_l and P_l by expanding the exponentials, i k r , and using the expression of Q_l and P_l. Let us now calculate the fluctuation of Q_l and P_l, whose averages are null in the vacuum. A calculation similar to the previous ones gives their dispersions, root of hbar/2. The product, delta P delta Q = hbar/2. That is to say the minimum value permitted by the Heisenberg relation. The vacuum is thus a minimum dispersion state. That is to say, the state in which the fluctuations have the minimum value compatible with the Heisenberg relation. The fact that the electromagnetic field has fluctuations in the vacuum has remarkable consequences emblematic of quantum physics. First, the vacuum fluctuations are responsible of the fact that an atom in a level that is not the ground state will eventually decay by spontaneous emission. Even in the vacuum, the atom is not isolated. It is coupled to the quantized field and its fluctuations. We will come back to this point later. Vacuum fluctuations have also an effect on the position of the energy levels of the atom, which are slightly modified compared to what is calculated in the absence of the coupling to the quantized electromagnetic field. This slight modification is called the Lamb shift, from the name of Willis Lamb, who measured that shift in the hydrogen atom with his collaborator Retherford. This measurement was a masterpiece of experimental physics, just after World War II. The result agreed remarkably well with the calculation based on quantum electrodynamics theory just developed by Feynman, Schwinger, and Tomonaga. Nowadays, more and more precise measurement of the Lamb shift of various level of the hydrogen atom are still one of the best benchmark of QED, the quantum electrodynamics theory. I could cite several other effects due to vacuum fluctuations. For instance, the Casimir effect, that is to say, the attraction between two metallic plates close to each other. Or a small anomaly in the magnetic moment of the electron linked to the spin 1/2 of this particle. So believe me, vacuum fluctuations are not a fantasy of theorists. They have consequences, and we will see some of these consequences in this course.