 We currently have quantum field descriptions of atomic electron states and the radiation field, but we lack an interaction mechanism that can account for emission and absorption of radiation by the atom. This should take the form of an interaction term in the Hamiltonian. We will develop the interaction term and examine its effects in the next two videos. For now we can get an idea of the type of terms that must appear in the interaction Hamiltonian with the graphical aid of Feynman diagrams. The total energy is expressed as a sum of three terms, atom, radiation, and interaction. The atom Hamiltonian has terms such as e k, b hat k plus, b hat k minus. The energy times the number operator for the k-th orbital. Suppose the m-th orbital contains an electron. We represent this on a space-time diagram by an arrow labeled with the corresponding quantum state. The radiation Hamiltonian has terms such as h bar omega k, a hat k alpha plus, a hat k alpha minus. The energy of a photon in the mode with propagation constant k and polarization alpha times the corresponding number operator. Suppose there is a photon in one of these modes. We represent this on the Feynman diagram by a squiggly arrow labeled with the corresponding quantum state. The electron and photon are shown as colliding. If there was no interaction mechanism between them, they would pass through each other and continue on unperturbed. If there is an interaction mechanism specified by the interaction Hamiltonian, then there can be an interaction leading to a quantum jump. We'll represent a quantum jump by a small circle at the vertex where the electron and photon symbols come together. Suppose this quantum jump consists of the destruction of the electron and photon and the creation of a new electron in the n-th orbital. Our Feynman diagram shows the original electron and photon converging on the vertex where they are destroyed, and the created electron emerging from the vertex. The quantum jump is described by the product b hat n plus, b hat m minus, a hat k alpha minus. This destroys the photon, destroys the n-th orbital electron, and creates the n-th orbital electron. We conclude that the interaction Hamiltonian must contain terms like this to describe the absorption of a photon by the atom. Another possible process is that we start with an electron in the n-th orbital. A quantum jump occurs, which destroys this electron, and simultaneously creates a photon in the k alpha mode and an electron in the n-th orbital. To describe this emission process, the interaction Hamiltonian needs terms like a hat k alpha plus, b hat m plus, b hat n minus. This destroys the n-th orbital electron, creates the n-th orbital electron, and creates the photon. Of course, these processes can only occur under certain constraints, such as conservation of energy. The energy of the n-th orbital electron must equal the energy of the photon plus the energy of the n-th orbital electron. Presumably these interaction terms will include coefficients that specify the probability of the corresponding interaction. We will work out the interaction mechanism in the next video, and the interaction Hamiltonian in the video after that. Now we turn to the related topic of perturbation theory. This is the technique we will use to actually solve complex problems in quantum electrodynamics. Here we will see the intimate connection between Feynman diagrams in perturbation theory. Suppose for a system that satisfies the Schrodinger equation with Hamiltonian h hat 0, we can find explicit stationary states with spatial part phi n and time dependence e to the minus i omega n t. The spatial parts satisfy the time-independent Schrodinger equation with energy h bar omega n. But we are interested in solutions of a modified system with Hamiltonian h hat 0 plus h hat interaction. Where this form of the Schrodinger equation is too difficult to be explicitly solvable due to the additional interaction term in the Hamiltonian. If this added term is in some sense small compared to the original h hat 0, we can think of it as a small perturbation of the system, and try to develop a solution in terms of the solutions of the original unperturbed system. For an rotational simplicity going forward, we assume units in which h bar equals 1. We express the solution as a linear combination of the stationary states of the unperturbed system. The coefficients cn of t can vary in time so the perturbed system can evolve from one state to another. For the left side of the Schrodinger equation, i, time derivative of the quantum state, the terms in our sum have two time dependent factors, so the derivatives have two terms, derivative of the first times the second plus the first times the derivative of the second. This gives us i times sum over n of c dot n of t, the time derivative of the nth coefficient times e to the minus i omega n t times phi n, plus i times sum over n of cn of t times minus i omega n times e to the minus i omega n t phi n. The right side of the equation we break into a sum of h hat 0 and h hat interaction terms. The h hat 0 operator simply adds a factor omega n to each term in its sum. The h hat interaction operator operates on the stationary states phi n in its sum. Since i times minus i is just one, the two box expressions are equal and they cancel. This leaves i times sum over n cn dot of t e to the minus i omega n t phi n equals sum over n cn of t e to the minus i omega n t h hat interaction phi n. To isolate one of the c dot terms we project both sides onto a state phi f. Since the projection of state n onto state f is zero unless n equals f, in which case it is one, the left side is reduced to the single term i c dot f of t e to the minus i omega f t. On the right, each sum has a factor h hat interaction operating on phi n projected onto phi f. We call this the fn matrix element of the interaction Hamiltonian. Solving for c dot f of t this is equal to minus i sum over n cn of t e to the i omega f minus omega n t times the fn matrix element. This gives us a differential equation that determines how the cf of t coefficient of our solution evolves with time. Unfortunately there are an infinite number of these equations, and each equation depends on the infinite number of coefficients cn of t. There are no practical techniques for solving this infinite set of coupled differential equations in closed form. Instead, we will use perturbation theory to generate a solution by making successively more accurate approximations.