Hello everyone, welcome back. We're now going to look a little bit more deeply into these mysterious gaps in the solar spectrum. So just to remind you when seen at high resolution, the light of the sun exhibits these dim spots which could be interpreted as absorption features. Indeed, that's what they are. There's a complimentary phenomenon with which you're probably familiar. You undoubtedly have seen these vivid gas signs, neon signs that are often called in the United States that exhibit very bright colors. And these are due to the very sharp emission lines that are observed in low density gases. Here are two pertinent examples, one is a spectra of atomic Hydrogen. It's a discharge lamp, and what you're seeing here is the increase in the wavelength going from left to right. And you see very distinct isolated lines. Red, a sort of a cyan, a blue and violet. Actually rather near our reference laser points this red line of Hydrogen 656 nanometers Balmer alpha, this is the Balmer beta and so on, so called. And down below the spectrum of neon, more complicated but as you can see, it consists of fairly well-resolved, quite sharp lines, rather than the sort of rainbow feature that we see for dispersed sunlight. Now as it happens, the emission lines that are observed in the low density gasses often correspond to the absorption features of the sun. In fact, this line at 656 nanometers is this large dark gap here. This line is down here I believe and the others have their counterparts in the relevant portions of the spectrum. So it became understood that the features seen in the absorption spectrum in the sun were due to the, the same transitions that were giving rise to emission in low density gases. But the presence of these sharp and distinct lines in atoms was a great puzzle because in classical mechanics, a system of interacting particles can have any set of positions and velocities, and there's nothing to single out a particular configuration for special attention. So, atomic theory at the end of the 19th century had many clever ideas deriving from things like vortices and fluids. Where there was some type of a regular structure that could be inferred. But none of these bore any fruit in terms of explaining the specific properties of atomic spectrum. I'll just mention in passing that the vortex knot idea is coming back into vogue in atomic physics due to the ability to produce vortex structures in gases, in those Einstein condensates. But, I mean these are not elementary atomic structures, they're rather a type of super fluid pattern that can be generated by appropriate excitation of the condensate. But the whole idea of the continuum models, as atoms as some kind of fluid, confronted a real difficulty in the discovery that atoms seemed to be made of particles interacting with electric charges, and thereby interacting by electric forces. Notable development was Rutherford's experiment in 1911 that showed that virtually all of the mass of the atom is concentrated in a minute fraction of its volume, one whatever that is, a quadrillionth. So it was as if this thing that had a pretty well-defined size is known from a variety of measurements, some as simple as looking at the thickness of an oil film on the surface of water. Taking the volume of the ordinary oil drop and seeing how far it would spread out gives a pretty good indication of the size of an atom. And most of it from Rutherford's experiment had to be empty space with all the mass concentrated at the center. So, this is actually a system somewhat like the planetary system. The sun is very massive, it sits near the center of the solar system, and the Earth has a much smaller mass and stands off from the sun and orbits it. We're going to see what classical mechanics has to say about the about the planetary, Atom. That is one consisting of two particles ordinarily well-separated in an orbit very much like the orbits of the planet the solar system. This year is the 100th anniversary of this great discovery by Niels Bohr, which is reported in the paper that's contained in the additional materials section for this course. And Bohr developed the postulate that there were certain planetary orbits, Of the electron about the proton in a hydrogen atom that would be stable and that would correspond to the stationary states of atoms. So he basically solved the equations of motion for the planetary system. There's nothing novel about that. Those solutions are well known. And than stated a postulate that certain of the orbits in an atom would provide the stationary states that were seen in optical spectra. And then that the radiation absorbed in atoms would be associated as in this case here with an atom falling from a higher energy state, larger orbit into a lower energy state and emitting a photon into the radiation field. So that the frequency of the light that is seen either in absorption or emission corresponds to the energy transfer between the atom and the radiation field. And in fact, the Balmer alpha line, the red line that we see in the absorption spectrum, is 656 nanometers, rather close to our reference laser. Okay, so let's see what classical mechanics tells about these hydrogen-like systems. So we have a single nucleus with a position denoted by r sub n, the mass M, the charge of +Ze. Z is the atomic number, so it'd be 1 for hydrogen, 2 for helium, 3 for lithium, and so on. And then e is the what we call the specific charge, that's the absolute value of the charge of the electron. Then there's the electron described by position vector r sub e, little mass little m and a negative charge of -e. We're going to keep the masses in play represented explicitly in this treatment because there's not such a big simplification from changing them to something else. And one of the exercises that we want to do is to go through the process that led to the discovery of deuterium. It's a very important event, both from the standpoint of scientific development of nuclear physics and also the development of nuclear energy and nuclear weapons. And it was a discovery that was made by looking at the Balmer series of hydrogen. Okay, now here are the Newton's equation. It's MA, mass times the second derivative, the acceleration of the nuclear coordinate. And then, this is Coulomb's Law. There's an equation of motion for the nuclear coordinate. The force on it is in the direction of the electron nuclear distance and it's inversely proportional to the square of that distance. You see there's a factor of r from the vector and a factor of r cubed below, so that's one over r squared force law. And then according to Newton's, what is it, Third Law, there's an equal but opposite force on the electron, so you have the same thing there. Now, I'm using this coordinate r here, which is the separation between the nucleus and the electron coordinates because that is in fact the relevant length that describes the electrical force law. What you can do now is you divide, okay? Take this equation and divide it by 1 over m, multiple by 1 over m. And you multiply this equation by 1 over little m. And you subtract them and so you get an equation of motion for the electron nucleus separation alone. So you see on this side you have the second derivative of that is just a function of itself. So we've now managed to get an equation that can be solved directly for the separation between the two particles. So from this equation, we go directly here, you see, we're just taking the mass, this term here is a dimension. This term here has the dimensions of 1 over mass, so we just invert it to get what's called the reduced mass and now we have a simple Newtonian equation for the motion of the relative distance between the electron and the nucleus. Now we define the momentum, p, associated with this coordinate. It's just the mass times the velocity. The usual definition, and this then allows us to rewrite the equations of motion as a pair of coupled first-order equations in time. So here's that equation we just made up. It's the r-dot is 1 over mu times p. And then p-dot is just taken from this equation, because p-dot is our double dot and so it's just minus Ze squared r over r cubed. Now, why do we reduce this to first order equations, you might ask? So I emphasized the development of equations that are first-order in time because the whole idea of solving the equations of motion for a system means that if you have a specification of the state at time T, you can find it's the state at some later time. For small time, that means that you need to have an equation of motion that is first-order in the time for that state. That's why Schrodinger's equation must be first-order in time. So the whole idea of quantum mechanics is that the wave function psi defines a state. And if we know psi at time t, and we know the Hamiltonian operator, then we can determine psi at some slightly later time. So this is something that seems rather different from some of the classical equations of motion with which you might be familiar. Here's an example to consider in an inline quiz. So I hope you appreciated from that previous example, that the familiar second-order wave equation that went off and deals with. It's a convenient way of solving the wave equation. But it's sort of a secondary equation that's based on primary first order equations. So, there's no real difference between quantum mechanics and classical mechanics in that respect. In classical mechanics, in order to solve the equations of motion, say for electrodynamics, to solve the equations of motion and time. You need to know the value of the electric field and its first derivatives in time. Let's say, as an initial value problem. Whereas in quantum mechanics, we just have this one wave function that provides all the information needed to solve the equations of motions once the Hamiltonian's known.