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PROFESSOR: OK, so we're good to consider, therefore, the two cases. So let's consider the first case, that this can be pretty important, this case one, when omega fi plus omega is nearly 0. In this case, what's happening? You have Ef minus Ei plus h bar omega is nearly 0, or Ef roughly equal to Ei minus h bar omega. So what has happened if you have a number-- so the question is, when this happens, which means that your omega-- that is, your perturbation-- is tailored to produce this, then Ef is Ei minus h omega. So you can think of the energy scale here. And Ef is lower than Ei. And the difference is h bar omega. So what is this process? This process is called stimulated emission. And why is that called stimulated emission? Because you're going from a state of energy Ei to a state of energy Ef that has lower energy. In that process, you're releasing energy h bar omega to the perturbation. So it's almost-- you would say it's stimulated because t...

PROFESSOR: Today, we have to discuss harmonic perturbations. So we've done Fermi's golden rule for constant transitions. We saw transitions from a discrete state to a continuum. And by integrating over the continuum, we found a nice rule, Fermi's golden rule, that govern the transition rate for this process. So the only thing we have to do different now is consider the case that the perturbation is not just a step that gets up and stays there, but it has a frequency dependence. So that will bring a couple of novel features. But at the end of the day, as we will see, our Fermi's golden rule is going to look pretty similar to the original Fermi's golden rule. A nice application of Fermi's golden rule is the calculation of the ionization rate for hydrogen, in which you take a hydrogen atom, you put it in an electric field or send a light wave, and then suddenly the electron and the hydrogen atom from the ground state ionizes. And we can compute alread...

We have seen two versions of the Bayes rule-- one involving two discrete random variables, and another that involves two continuous random variables. But there are many situations in real life when one has to deal simultaneously with discrete and continuous random variables. For example, you may want to recover a discrete digital signal that was sent to you, but the signal has been corrupted by continuous noise so that your observation is a continuous random variable. So suppose that we have a discrete random variable K, and another continuous random variable, Y. In order to get a variant of the Bayes rule that applies to this situation, we will proceed as in the more standard cases. We will use the multiplication rule twice to get two alternative expressions for the probability of two events happening. We will equate those expressions, and from these, derive a version of the Bayes rule. So we will look at the probability that the discrete random variable takes on a certa...

We will now go through an example that brings together all of the concepts that we have introduced. We have a stick of length l. And we break that stick at some random location, which corresponds to a random variable, X. And we assume that this random variable is uniform over the length of the stick. So its PDF has this particular shape. And for the PDF to integrate to 1, the height of this PDF must be equal to 1 over l. Then we take the piece of the stick that we are left with, which has length X, and we break it at a random location, which we call Y. And we assume that this location Y is uniformly distributed over the length of the stick that we were left with. What does this assumption mean? It means that if the first break was at some particular value, x, then the random variable Y has a conditional distribution, which is uniform over the interval from 0 to x. So the conditional PDF is uniform. A conditional PDF, like any other PDF, must integrate to 1. So the height ...

MARKUS KLUTE: Welcome back to 871. So in this section of our discussion of instrumentation, we talk about accelerators, and I'll do this in a little bit of an historic way showing you some of the developments over the last up to 100 years. We use electromagnetic fields in order to accelerate charged particles, and so the developments in electromagnetic and understanding electromagnetism led to then the technical developments or the technological development of accelerators and the availability of devices, which can be used in order to accelerate or modify particles. And so this goes back to Maxwell and Hertz discovering electromagnetic waves towards JJ Thomson, who was able to use cathode rays and the classical Lorentz force in order to understand the electromagnetic fields. If we study particle accelerators, we can see three different historic lines-- direct voltage accelerators, resonant accelerators, and transformer accelerators, and we go through those three diffe...

PROFESSOR: OK, Let's use the last 10 minutes to discuss an application. So that's our Fermi golden rule there. Let's leave it in the blackboard. So the example I will discuss qualitatively will not compute the rate for this example is auto ionization, or also called Auger transitions. So we imagine of-- I think the reason this example is interesting is that there is continuum states, sometimes in cases that you would not think about it, or you wouldn't have thought about them. So let's assume we have a helium atom. So you have two electrons, z equals 2, two protons and two electrons. And I will assume that we have a hydrogenic state. So you see the Hamiltonian of this whole , atom there's a P1 squared over 2n plus a P2 squared over 2n, roughly for each electron, plus or minus e squared over r1 minus e squared over r2, and then plus e squared over r1 minus r2, which is the Coulomb repulsion. So that's roughly the Hamiltonian you need to consider...

PROFESSOR: So what do we do? We are going to sum over final states the probability to go from i to final at time t0 to first order. Since the sum of our final states is really a continuum, this is represented by the integral of the f i t0 1, multiplied by the number of states at every little interval. So this will go rho of Ef dEf. So this is what we developed about the number of states. So I'm replacing this-- I have to sum but I basically decide to call this little dN, the little number of states in here, and then I'm going to integrate this probability, so the number of states over there, and therefore the dN is replaced by rho times dEf. So then this whole transition probability will be 4 integral, I'm writing now the integral, VfI squared, sine squared, omega f i t0 over 2 Ef minus Ei squared, rho of Ef, d of Ef. And you would say at this moment, OK, this is as far as you go, so that must be Fermi's golden rule, because we don't know rho of Ef, it...

MARKUS KLUTE: Welcome back to 8.701. So in this section, we talk about calorimetry. In contrast to the discussion of tracking detectors, here what we're trying to do is measure the energy of the particles. And we do this by basically destroying them. The underlying content is rather straightforward. We have a particle. And we put a piece of material in front of it, such that it slams into it, and the energy deposited by the particle is the energy, the measurement we try to undertake. So in nuclear and particle physics, that is exactly what we refer to as calorimetrics. So the detection of particles for measuring the properties through the total absorption in the block of matter. The common feature, or the central feature, is that the measurement is destructive. So again, in tracking detectors, we try to minimally disturb the particle and in calorimetrics, we try to destroy them. The exception to this might be a muon which might at high energies deposit only a small fr...

Good. So let's do, then, our transitions. So we do the constant perturbation. Constant perturbation. So as we said, delta H is equal to V, and it's time independent. It just begins at time 0. And we'll examine what's going on by time t0. So what are we going to do? We're going to examine a transition to go from some initial state i, initial state, to a final state f. So we don't have to say much about what the Hamiltonian is or anything. For us, V is going to have a constant. It's going to have some matrix elements that once we do an example you can calculate, but for the time being we need not know too much. So I'm going to use the key formula that was derived already about perturbation theory and how you get the transition amplitude. So we know that the coefficient c associated to the m state to first order in perturbation theory at that time t0 can be computed as a sum over all n integral from 0 to t0 e to the i omega mn t prime delta Hm...

MARKUS KLUTE: Welcome back to 8.701. So in this section, we'll look at tracking detectors. And before we look at tracking detector technologies, we want to remind ourselves how we measure the momentum of a charged particle. And this measurement is possible because charged particles are reflected in magnetic fields. We have already seen that in a homogeneous magnetic field, a particle follows a circle. And so from the measurement of the radius and the knowledge of the magnetic field, we can infer the transverse momentum of the particle. Typical particles also have a longitudinal momentum. And so therefore, the trajectory is actually of the form of a helix. And so we then can get back to the total momentum of the particle by knowing the angle or the component of the longitudinal and transverse momentum and then just calculate total momentum from there. So we have seen that particles, when they go through a piece of material, that the energy loss through ionization or br...

By now, we have introduced all sorts of PMFs for the discrete case. The joint PMF, the conditional PMF-- given an event-- and the conditional PMF of one random variable given another. And we're moving along with the program of defining analogous concepts for the continuous case. We have already discussed the joint PDF and the conditional PDF, given an event. The next item in our menu is to define a conditional PDF of one random variable, given another random variable. We proceed by first looking at the definition for the discrete case. A typical entry of the conditional PMF is just a conditional probability, but in different notation. And using the definition of conditional probabilities, this is equal to the ratio of the joint divided by the probability of the conditioning event. Unfortunately, in the continuous case, a definition of this form would be problematic, because the event that Y takes on a specific value is an event that has 0 probability. And we know that...

We will now use the Bayes rule in an important application that involves a discrete unknown random variable and a continuous measurement. Our discrete unknown random variable will be one that takes the values plus or minus 1 with equal probability. And the measurement will be another random variable, Y, which is equal to the discrete random variable, but corrupted by additive noise that we denote by W. So what we get to observe is the sum of K and W. This is a common situation in digital communications. We're trying to send one bit of information whether K is plus 1 or minus 1, but the observation that we're making is corrupted by a communication channel, by some noise that is present in the channel, and on the basis of the value of Y that we will observe, we will try to guess what was sent. The assumption that we will make about the noise is that it is a standard normal random variable. So suppose that we observed a specific value for the random variable Y. We wa...