We will now apply the method of canonical quantization to the case of the material harmonic oscillator. The main reason is that the formalism we will develop will be used all along the course of quantum optics, since we will see that the quantized radiation behaves as a set of quantized harmonic oscillators. But it is also an excellent example of how the formalism allows one to obtain the Schroedinger equation associated to a system initially classical and then quantized. In order to solve the Schroedinger equation of a quantized material harmonic oscillator, we will use the method developed by Dirac. It has the advantage of being universal, that is to say that we can apply it to any vibrating system that can be modeled as an harmonic oscillator, in particular a mode of the electromagnetic field. Let us start then with a classical harmonic oscillator that is a material particle of mass m, which can move along the x axis and is attracted toward x=0 by an elastic force proportional to the distance. This force derives from a potential, proportional to x^2, which we can write 1/2 m ω^2 x^2. The total energy of the material particle is then the sum of that term of potential energy plus the term of kinetic energy. Note that we have used the momentum p to express the kinetic energy, so that the expression of the energy is put in the Hamiltonian form, we can thus write the Hamilton equations. We recognize the usual equations describing the dynamics of a material oscillator. More precisely, deriving the first equation with respect to time and using the second one, we obtain a second order differential equation for x. The solution is a function oscillating at angular frequency ω. It depends on two arbitrary constants: a and phi, which can be determined from the value of x and p at initial time t_0. The momentum is proportional to the derivative of x and oscillates in quadrature. As you know, it is convenient to introduce a complex amplitude A that we write with a so-called English or script font. The real oscillating quantity is proportional to twice the real part of the complex amplitude multiplied by a complex exponential evolving as -iωt. Note that there is a degree of arbitrariness in the definition. I could have, as well, used an exponential evolving as +iωt. Also, some authors would not use a factor 1/2 in the definition of the complex amplitude. In fact, there is no universal consensus and we have to make a choice. Our choice is clear on this transparency. It is the same as in our book. We make all efforts to be consistent with this choice in all the course. Let us proceed, now, with the quantization of the material harmonic oscillator. We have checked that x and p are canonically conjugate variables for the Hamiltonian H. So we can immediately write the quantum Hamiltonian replacing the classical quantities x and p by the quantum operators, indicated with a hat. As we have seen in the previous section, in the wave function description of the material particle, the operators x hat and p hat are respectively represented by x times, and ħ divided by i times d over dx. And the time independent Schroedinger equation writes: ... Solving that equation allows one to calculate the stationary wave function of the harmonic oscillator and the corresponding values of the energy. I will not comment much about that method to obtain the energy levels of the harmonic oscillator, since I want to insist on another method. Let me only say that when one imposes a condition that the wave function psi must be integrable over space, one finds a discrete set of solutions, parameterized by non-negative integers. More precisely, the only possible values for the energy are given by the well known expression E_n = n + 1/2 times ħω, with n a non negative integer. The associated wave functions are the product of a Gaussian that ensures an asymptotic null value, times a Hermite polynom of degree n. Do not confuse the notation H_n(x), which is a Hermite polynom of degree n, with the Hamiltonian H. I will not say more about that method, that you can find in standard textbooks on quantum mechanics, and we switch now to the Dirac method. Dirac introduced a completely different method to treat the eigenvalue problem of any harmonic oscillator. This method will give us the values of the quantized energy of the harmonic oscillator. We will also obtain fundamental relations for the corresponding states, that is the eigenstates of the Hamiltonian. We start again with a Hamiltonian of material harmonic oscillator. We are going to find the eigenvalues and eigenvectors of this Hamiltonian without using the wave-function representation. The only information that we are allowed to use is that form of the Hamiltonian, with the fact that operators x hat and p hat do not commute. More precisely, their commutator is equal to iħ. We thus look for the solution of the equation H phi = E phi, where phi is a vector in a Hilbert space describing the state of the system, and H the operator written above, with a commutation relation of x hat and p hat. Surprising as it is, Dirac found out this is enough to solve this equation, and found the possible values of the energies and fundamental relation between the corresponding states. Dirac was not only a genius as a theoretical physicist, he was also able to invent the new mathematical tools each time he needed one. The first step in Dirac's method consists of introducing dimensionless operators, capital X hat and capital P hat. Check that you can prove that square root of ħ over m times ω has indeed the dimension of a length, and similarly, that the square root of mħω has dimension of a momentum. The commutator of this dimensionless operator is of course dimensionless. Its value is i, as you can check by simple inspection. We have thus to find the eigenvalues and eigenvectors of the Hamiltonian H = 1/2 ħω times X^2 + P^2 with P and X hermitian operators, whose commutators equals i. At this point, Dirac introduces two new operators, which will play a major role in this course, and that we will use all along the course. They are a and its hermitian conjugate a dagger, which are combinations of X and P, called annihilation and creation operators for reasons that will be clear later. A simple calculation which appears now shows that a and a^† do not commute. More precisely, their commutator has a value of 1. This is a fundamental relation you should remember. We will use it all along the quantum optics course. Note that this a and a^† are not hermitian. Recall that an hermitian operator is equal to its hermitian conjugate. Here you see by simple inspection that a and a^†, which are hermitian conjugate of each other, are different. Since they are not hermitian, a and a^† do not represent physical observables, but all observables can be expressed with a and a^† and this will be done all along the course. We can for instance express X and P as a function of a and a^† and use this result to find a new expression of the Hamiltonian. At this point, I ask you to pause the video and do the calculation yourself. Do calculate X^2 and P^2 as a function of a and a^†, and write the Hamiltonian with a and a^† rather than X^2 and P^2. Remember that a and a^† do not commute. Did you find ħω times a^† a plus 1/2? If you did not get the 1/2 term, it means that you forgot that a and a^† do not commute when you squared X and P, so you have to redo the calculation. If you do not get these points right, you will not be able to follow calculations of quantum optics. So do not hesitate to do the calculation again until you get it right.