Let us explain first, what is canonical quantization in general. Any system of which we have a classical description can be quantized following a process known as canonical quantization which was elaborated by Dirac in his thesis in 1925. The idea is the following. To give a description of a classical system, we use a set of dynamic variables whose values at time t, allow us to calculate at that time any quantity of the system, such as its energy, linear momentum, etc. Moreover, these dynamic variables obey first order differential equations of time, so that their value at initial time t_0 allows us to derive the future evolution after t_0. Now, among all the possible sets of dynamic variables, some will play a particular role. They come by pairs of canonically conjugate variables q_j, p_j that are called respectively the coordinate and its canonically conjugate momentum. The process of canonical quantization consists of replacing each pair of a coordinate with its conjugate momentum. By a pair of operators q_j_hat and p_j_hat, which do not commute. That commutator is equal to I hbar. As usual, hbar is the Planck constant divided by two Pi. If now, we consider two operators associated with different pairs of canonically and conjugate variables, they commute, their commutator is equal to zero. This is indicated by the second equation where the Kronecker symbol delta_jk is null if j is different from k and takes a value 1 if j equals k. As soon as we have replaced all pairs of canonically conjugate variables by operators whose commutator equal i hbar. We can describe the behavior of the quantized system using the quantized Hamiltonian associated with the classical Hamiltonian. In classical physics the Hamiltonian of a system is its energy expressed as a function of canonically conjugated variables. When you replace the classical conjugate variables by the corresponding quantum operators, you obtain the quantum Hamiltonian of the system, which is its quantum expression of the energy. You can now elaborate the quantum description of your system, since you remember from your course in quantum mechanics, that once you know the Hamiltonian of a system you can obtain all its quantum properties. [MUSIC] You know for instance, that the quantized energies E_n of the system are the eigenvalues of the Hamiltonian. And that the corresponding eigenstates phi_n constitute a convenient basis of the space of the states of the system. This equation is sometimes called the time independent Schrodinger equation. You also know that the time evolution of the system is given by a first order differential equation, also based on the quantized Hamiltonian. I hbar dpsi over dt equals H-hat applied to the ket psi that describes a state of the system at time t. This is the Schroedinger equation. It allows you to answer any question, provided that you can solve it. This is another story, not a simple one, but let us come back to our canonical quantization. At this point, a question immediately arises. How can we recognize pairs of canonically conjugate variables in a classical system? There is a general procedure for answering that question starting from a quantity named the Lagrangian of the system. Here, we will use a simpler and more pragmatic approach based on the classical Hamilton equations. which describe the dynamics of classical systems. The Hamilton equations are based on the classical Hamiltonian expression of the energy. They link the evolution of the two canonically conjugated variables of the same pair. They express the first order, time derivative of one variable as a function of a partial derivative of the Hamiltonian, relative to the other variable. They're almost the same except for the sign. I must admit that this is quite abstract especially if you see it for the first time. So, let us take a simple example to find how it works. Let us take the example of a material particle of mass m evolving in a potential U(x). We restrict ourselves to a one-dimensional problem. Let us try the hypothesis that the canonically conjugate variables are the position x and the momentum p = m dx over dt. The expression of the energy is then E(x,p) equals the sum of the potential energy plus the kinetic energy and we can take it as the Hamiltonian. We can then write the Hamilton equations. The first one gives p over m equals dx over dt. The second yields dp over dt equals minus dU over dx, that is a force. We recognize Newton equations of motion, that is to say, the correct dynamics of the system, and we can safely conclude that x and p are canonically conjugate variables. We thus quantize, by taking operators x_hat and p_hat that do not commute and write the quantum Hamiltonian as a sum of two terms associated with the potential and the kinetic energy. We can now write the Schroedinger equation and if we take the form relevant to the language of wave functions. We obtain the standard Schroedinger equation, that you have already used, I am sure, to study the behavior of a particle in a potential well. Or even to calculate the energy levels of the hydrogen atom. We could have made another conjecture. For instance, that the canonically conjugate variables are position and velocity. x and V equal dx over dt Would it work? To test that hypothesis we now assume that the Hamiltonian has the following form here, we consider V as the canonically conjugated of x. If now we write the Hamiltonian equation for that hypothetic Hamiltonian. We do not recover Newton equation, as you can check yourself calculating the partial derivatives. The dynamic variable x and V are not canonically conjugate variables. You may feel unsatisfied by this empiric way of recognizing canonical conjugate variables. And regret that I do not teach you the more general method starting from the Lagrangian. In my opinion, the latter is not more rigourous, since you must first guess the specific form for the Lagrangian, and then, you must verify that this expression leads to known dynamics of the system. For instance, Newtonian equations in the previous case. So the reasoning is basically the same as the one we do with Hamiltonian formalism. I do not mean to imply that there is no interest in using the Lagrangian formalism. But for this course, it is enough to start from the Hamiltonian formalism, where we guess which are the canonically conjugate variables and verify that they are indeed the canonically conjugate variables. We then use the following criterion, if the Hamilton equations associated to the energy of the system yield the known dynamics of the system, we can conclude that the energy was expressed as a function of canonically conjugate variables and proceed with the quantization. We will apply this criterion to the electromagnetic field to recognize the dynamic variables that will become operators obeying the canonical commutation relations. Before effecting the quantization of the electromagnetic field. I want first to show you how to do it in the particular case of a material harmonic oscillator. The reason is that the electromagnetic field behaves as a set of harmonic oscillators as we will learn soon. So if you want to learn quantum optics, you better know the formalism of the quantum harmonic oscillator in the form developed by Dirac. You have already learned it in your course of quantum mechanics, but you must anyway watch carefully the next sequence in order to refresh your memory. and check that you are comfortable with the formalism that will be used all along the course. Before starting that new sequence, you may want to search the following quiz. If you think that understanding canonical quantization is not your priority, you can skip the quiz.