As usual in quantum mechanics, having the Hamiltonian of the system we want to know its eigenvalues, that give us the quantized energies of the system, and the eigenvectors whose total set forms a complete basis of the space of states. This will allow us to encounter the notion of photon, the most emblematic concept of quantum optics. This is the quantized Hamiltonian describing the electromagnetic field in the mode L. We want to solve its eigenvalue problem, that is to say, obtain its eigenvalues, the quantized energies, and the corresponding eigenstates. We already know the solution of this problem. Since we have the Hamiltonian of an harmonic oscillator here. I repeat the solution presented about the material harmonic oscillator now applied to the mode l. Let us write explicitly the eigenequation. We have kept the index l to remember that we are in the mode l and we use the index n_l to characterize each eigensolution. In speaking, I may sometimes forget to mention l and only speak of the energy E_n, and the state phi_n, but remember that in fact we consider only one mode, the mode l. We know, thanks to Dirac, that the eigenvalues of H are E_n = hbar omega_l times n_l+1/2 with n_l, a non negative integer. We also know that the eigenstates obey these remarkable relations which connect the various eigenstates to each other. Moreover, we know that there is a state playing a particular role It is a state associated with n_l equal zero. That is the state with the lowest energy. It is determined by the equation: a |phi_0>=0 involving the destruction operator, a_hat. From that state we can generate all the other states by applying N times the creation operator and normalizing. We thus obtain a complete basis of the state space. Remember also that the eigenstate of the Hamiltonian are eigenstates of the number operator N_l. They are often noted as state |n_l>, and they are called number states. They are also known as Fock states from the name of Vladimir Fock, a Russian theorist who developed this representation. These states are well suited to introduce the notion of photon as we see now. The number state |n_l> has a well defined energy which is larger than the lowest possible energy by n hbar omega_l. The classical electromagnetic field has also momentum, which can be considered responsible for radiation pressure when a beam is absorbed or reflected. As for the energy, it depends on the volume of quantization and is given by the modulus squared of alpha_l times hbar k_l. The corresponding quantized observable associated with the momentum in the volume of quantization is p equals hbar k times a_dagger a. We have again a_dagger a, and thus the numbers state n_l is also an eigenstate of the momentum operator, with eigenvalue n_l hbar k_l. Finally, the state |n_l> has well defined energy n_l hbar omega, and a well defined momentum, n_l hbar k_l. It has thus the same energy and momentum as n_l particles, each with energy hbar omega_l and momentum hbar k_l. We call these particles photons. I have two comments about momentum. First you must be aware that the formula we have used is valid for running waves only. If you have quantized on other types of modes, for instance, standing waves, as you will see in the homework, this formula does not hold. Second, if we calculate the rest mass by using the well known relativistic formula, relating it to energy squared minus momentum squared times c squared, we obtain 0. In relativity a particle of 0 mass travels at the velocity of light. It should not come as a surprise but we see that everything is consistent. Einstein noticed that point as soon as he introduced the notion of quantum of light in 1905, the same year when he published the founding paper of relativity. Who else could have noticed that remarkable fact? Note that Einstein was speaking of quantum of light, Lightquanten. It is the American chemist, Gilbert Lewis who coined the name photon almost two decades later. By the way, if you've never tried to read papers by Einstein, you should. They are accessible to anybody with a reasonable undergraduate culture in physics, with the exception of the papers on general relativity, that demand to master the formalism of tensors. The papers of Einstein are wonderful. You can find them on the web translated in many different languages English, French, many others. The 1905 paper on the photoelectric effect where he introduces the quantum of light is a true pleasure to read.