 Welcome to Quantum Field Theory 6 – Interacting Fields. In this video, we bring together all our results so far to arrive at a quantum field theory in which the electron and photon quantum fields interact. This will allow us to describe the radiation and absorption of photons by atoms as well as other processes, such as a photon scattering from an electron. First, let's review our theory so far, starting with the photon field, which was the focus of video 3 of this series. We show that in quantum field theory, the classical electromagnetic field, described by vector potential A, becomes an operator, A hat. We express this as a summation over all possible modes, indexed by propagation vector k and polarization direction alpha of E k alpha over square root 2 omega k times A hat k alpha minus times E to the minus i omega k t times E to the i k dot x plus a conjugate term with plus and minus sign swapped. E k is a unit vector in the direction of photon propagation. E k 1 and E k 2 are two polarization directions, orthogonal to each other, and to E k. Spatial dependence is through factors E to the plus or minus i k dot x. This varies periodically along the E k direction, where the spatial period lambda is the wavelength of the radiation mode. The frequency omega k equals the magnitude of the k vector. The operators A hat k alpha plus or minus, create or destroy a photon in the mode k alpha. The field operator A hat contains creation and destruction operators for all possible photons. For compactness going forward, we will absorb the time factors into the creation and destruction operators. In video three, we develop this expression for the Hamiltonian of the radiation field. Here A hat k alpha plus times A hat k alpha minus is the number operator that extracts the number of photons in the k alpha radiation mode. In video two, we discussed how the one half term, summed over an infinite number of modes, gives an infinite so-called zero point energy. We will ignore this infinite constant as unobservable, a process we might call energy renormalization. The expectation value of the radiation Hamiltonian is then the sum over all modes of the mode frequency times the number of mode photons. Recall that we use natural units in which the speed of light C and Planck's constant over 2 pi, h bar, both equal one. So a photon energy, h bar omega, becomes simply omega. In video four, we develop the non-relativistic quantum field theory of the electron field and a hydrogen atom. The electron field becomes an operator, psi hat minus, equal to the sum over all electron modes of B hat k minus times e to the minus i omega k t times psi k of x. For the conjugate form, psi hat plus, we replace B hat k minus by B hat k plus and conjugate the other two factors. We will also absorb these time factors into the B hat operators. The psi k function gives the spatial dependence of the electron wave function for the k-th stationary state of the atom. These are the SPD and so on hydrogen orbitals. These are described in some detail in video six of the quantum mechanics series. The hydrogen atom Hamiltonian is 1 over 2 Me, p squared plus u of x, where Me is the electron mass, p is the momentum operator, and u is the potential energy due to the proton-electron electrostatic interaction. The stationary states satisfy h hat psi k equals ek psi k, where ek is the stationary state energy, h bar omega k, which reduces to just omega k in our natural units. The operators B hat k plus and minus create and destroy, respectively, an electron in the mode k. For a wave function psi of x, the expectation value of the Hamiltonian is the integral over all space of psi conjugate times the Hamiltonian applied to psi. Substituting the psi hat minus and plus operators for psi and psi conjugate, respectively, we obtain the quantum field form of the atomic Hamiltonian. This can be rewritten as the sum over m, sum over n of the integral over all space of psi m conjugate times en times psi n times B hat m plus times B hat n minus. The wave functions of the stationary atomic states are orthonormal. So this integral is 0 if m and n differ, and is em if m equals n. Therefore, the atomic Hamiltonian reduces to the sum over m of em times B hat m plus times B hat m minus. This product of creation and destruction operators is the electron number operator, and the expectation value of H hat a is the sum over all atomic modes of the mode energy times the number of mode electrons. For a single hydrogen atom, one mode will have a single electron while all the other modes will have zero. Now we come to the new material, the interaction Hamiltonian for the atomic and radiation fields. In video five, we saw that in classical mechanics, if a particle with charge q has momentum p, then replacing p with p minus q times a in the particle Hamiltonian will give the Hamiltonian for the particle in the presence of an electromagnetic field a. In the quantum case, we make the same replacement, but using the corresponding operators. In our case, the particle is an electron with mass m e and charge minus e. Therefore, the hydrogen Hamiltonian, 1 over 2 m e p hat squared plus u of x becomes 1 over 2 m e quantity p hat plus e times a hat squared plus u of x. We expanded this expression in video five. The result is the original hydrogen Hamiltonian plus two terms. Plugging these terms into the expectation formula gives us the quantum field interaction Hamiltonian. This expression is complicated, so let's work with one term at a time. We denote the part of the interaction Hamiltonian corresponding to the first term by H hat i prime, and the part corresponding to the second term by H hat i double prime. A hat x is the x component of the A hat operator. It's obtained by simply replacing the unit vector e k alpha with its projection along the x axis e k alpha dot e x, and likewise for a hat y and a hat c. It's convenient to break these into two parts, one with the destruction operators and the other with the creation operators. Since photons are destroyed when absorbed and created when emitted, these parts of the A hat operators correspond to absorption and emission processes. The momentum operators p hat x, p hat y, and p hat z are minus i times the derivative with respect to the corresponding x, y, or z coordinate. Let's first calculate H hat i prime with only the A hat x, p hat x term, and only the absorption term for A hat x. We obtain the expression shown here. This has the form sum over all electron states m, sum over all electron states n, sum over all photon states k alpha of a coefficient max prime times the product of the creation operator for electron state m, the destruction operator for electron state n, and the destruction operator for photon state k alpha. Much of the physics is buried in the rather intimidating expression for the coefficient. This is where the specific electron orbitals and photon propagation vector and polarization effects come into play. If we repeat this process for the y and z operator components and combine the results, we obtain H hat i a prime, the absorption part of the H hat i prime operator. Here the coefficient ma prime is the sum of the max prime, may prime, and maz prime coefficients. The same steps but using the emission part of a hat produces H hat i e prime, the emission part of the H hat i prime operator. Instead of photon destruction operators, this contains photon creation operators. The coefficient me prime is also the sum of x, y, and z components. These are the same as for the absorption case with e to the minus ik dot x in place of e to the ik dot x. The complete H hat i prime operator is the sum of the emission and absorption parts. This contains groups of operators that create a photon, destroy an electron, and create an electron. Or destroy a photon, destroy an electron, and create an electron. These are precisely the operations that in video four we claimed the interaction Hamiltonian must contain in order to account for photon absorption and emission processes. H hat i prime contains operators that can create or destroy any photon state and destroy and create any electron states. But this does not mean that all such quantum jumps are possible. The coefficients mb prime and ma prime determine the probabilities for these processes. If a coefficient is zero, then the process is impossible. The time dependent factors that we absorbed into the creation and destruction operators also play a role. In video four we saw that perturbation theory predicts a process will be significant only if the product of these time factors is constant through time. For emission this requires omega m plus omega k equals omega n which tells us that the energy of the created electron and photon must equal the energy of the destroyed electron. For absorption omega m equals omega n plus omega k. The energy of the created electron must equal the energy of the destroyed electron and photon. These requirements are simply statements of the conservation of energy. Let's now consider the a squared term of the interaction Hamiltonian. This is a constant times the sum of a hat x squared plus a hat y squared plus a hat z squared. Putting this term into our expectation expression produces h hat i double prime. A hat x squared is the product of a hat x with itself. This will have two summations over photon states. We use k alpha for the first summation and l beta for the second. Expanding the product of the corresponding photon creation and destruction factors will produce four types of terms. One will be a product of destruction operators. These will destroy two photons. One in state k alpha and the other in state l beta. Another product creates two photons. One in the k alpha state and the other in the l beta state. A third product destroys one photon in the k alpha state and simultaneously creates another in the l beta state. And the fourth product creates a photon in the k alpha state while destroying one in the l beta state. The third and fourth products are just different bookkeeping versions of the same process. When combined with electron destruction and creation operators, these describe two photon processes. These tend to be much weaker than one photon processes. The details require calculation of the process coefficients, which we do not do now. One situation where two photon processes can be important are when one photon transitions are not possible. Here are Feynman diagrams of these two photon processes. In a simultaneous photon destruction and creation process, a photon and electron interact. A quantum jump occurs in which they are both destroyed. Simultaneously, a photon and electron are created. For two photon absorption, an electron and two photons interact. They are all destroyed and an electron is created. For two photon emission, an electron is destroyed and an electron and two photons are created. One subtle aspect of two photon processes is that for given initial and final electron state energies, conservation of energy constrains only the difference or sum of the two photon energies. So there is typically a continuous range of possible photon states that can take part in the transition. And all those possibilities need to be taken into account when analyzing a two photon transition between given electron states.