We have just seen that the complex amplitude E_l totally describes the state of the field since it allows us to express E as well as B. Its dynamics is given by a first order differential equation. As a complex number, E is determined by two real variables, its modulus and its phase or its real, and imaginary parts. We are going to see that its real and imaginary parts are canonically conjugated variables within a multipliying factor. Let us first introduce a so called normal variable alpha_L, defined by writing the complex amplitude E_l as i times E1_L times alpha_L. The constant E1_L will be the determined later and will have a clear meaning. It has the dimension of an electric field so that alpha_L is dimensionless. The i factor is arbitrary here, but it is a choice made by most authors in quantum optics. So we better use that choice to facilitate reading books or articles in quantum optics. The evolution of alpha is obviously the same as the one of the complex amplitude E_l. This equation fully determines the dynamics of the electromagnetic field in the mode l. We now introduce the real and imaginary parts of alpha within a factor that will make future formulae simple. Let us now write the energy of the electromagnetic field in the mode, knowing the structure of the mode, and the amplitude E_l. It involves two terms, E squared and c B squared, but for a running wave, you can use the expression of B to find that the second term is equal to the first term, hence, the second expression of the energy. Then, a standard calculation allows us to express the integral of E squared as the volume of integration multiplied by two times the square of the modulus of the complex amplitude. The factor of two stems from our definition of the complex amplitude. In fact the volume of integration is no different from the quantization volume that we will define soon. We now make the following choice of the constant E1_l. It is the amplitude of a field that has an energy hbar omega_l, in the volume of quantization V_l, as you can check, using the expression just established. Why introduce hbar omega in a classical calculation? Because we intend to quantize and this choice will make formula simpler, as I told you. We take hbar omega as a characteristic value of energy in the problem, because we are not naive. We have heard that this is the energy of a photon, right? I'm sure you already know that the energy of a photon is hbar omega, or maybe you know h nu. The constant, E1_l, is called the one photon amplitude of the mode l. We will remember that it is proportional to the square root of the inverse of the quantization volume. With this notation, we have a remarkable expression for the energy. Do you recognize a form we have seen for any harmonic oscillator? Then you are ready for effecting the canonical quantization of the field in the mode L. I must first tell you some secrets about the volume of quantization. What is the meaning of this volume of quantization that I have introduced? It is a volume where I consider there is radiation. If I have a cavity as in the homework of this lesson, it is a true volume, a real volume. The volume of the cavity delimitated for instance by mirrors. The mirrors impose boundary conditions which in turn impose conditions on the k vector. This is why I have indicated an index L on the volume, to emphasize that the volume and the mode are not independent of each other. But, what happens when I have a traveling wave extending to the infinity? Well, it is quite simple. I define a fictitious volume larger than the volume in which I am interested and if I proceed correctly, the results will not depend on the size of the volume. Let be more specific. We choose a cubic box of size capital L and decide that it is our volume of quantization. So the volume of quantization is L cubed, but I have a free wave, extending to the infinity. So in fact, I want to be able to prolongate the field smoothly beyond the boundaries of the cube. There is an easy solution to that demand, based on the fact that such a wave is periodic in all directions of space. This condition is known as the periodic boundary conditions. We impose that L be, an integer number of spatial period of the wave along each axis. It writes, k_x times L, equals an integer number multiplied by two pi, and similarly for the other components of k_l. You can check, for instance, that the field at X=0 Y=0 and Z=L is equal to the field at X=0 Y=0 and X=0 and similarly for other directions. Note that the integer numbers n_x and n_y and n_z, can be negative in order to span all the possible directions of propagation. If we represent the extremity of vector k in the k-space, it cannot lie anywhere. It has to be on a three dimensional cubic grid with an elementary period of 2 pi over L along each axis. Here, I'm representing the case of nx equals 2 and y equals 2 and z equals 3. To conclude this section, let me repeat that the periodic boundary conditions are well adapted to quantization of free waves. But there are other situations where other choices would be more appropriate. We are now armed to proceed with the quantization of the electromagnetic field. We have seen that the state of the field of the single mode L, is characterized by two dynamic variables, Q_l and P_l, and we have written the energy as a function of these two particular variables. We are going to see that they are canonical conjugate variables, so quantization will be immediate. Recall that we have written the complex amplitude of the field as a function of a dimensionless variable alpha_l, using the so called one photon amplitude of the field E1_l. Introducing the real and imaginary parts of alpha_l within a factor involving hbar, we have obtained a remarkable expression for the energy. It is immediate to check that the Hamilton equations for Q_l and P_l yield an evolution equation for alpha_l. That is the one we have obtained from Maxwell's equations applied to the field in the mode. We can thus conclude, that Q_l and P_l are canonically conjugate variables, and proceed with a canonical quantization introducing the operators Q_hat and P_hat with commutator i hbar. And writing the Hamiltonian as omega_L over 2 times Q_l_hat squared plus P_l_hat squared, we recognize the quantum harmonic oscillator, Hamiltonian that we have already studied. In order to use Dirac formalism we introduce a_hat and a_hat dagger, which can be defined as functions of Q_hat and P_hat and which correspond also to alpha and alpha conjugate. The commutator equals 1 and they yield, the remarkable expression for the Hamiltonian H = hbar omega_l times a_dagger a + one-half.