Course Report: Atoms and Molecules in Three Classes

I got way behind on my reports from my Modern Physics class-- the last one was over month ago, and the class has since ended. There's enough material left to be really awkward as a single post, though, so I'm going to take my cue from Brandon Sanderson and split it into three parts.

The remaining material is from the sprint-to-the-end "Applications of Quantum Mechanics" portion of the class, and breaks into three roughly equal chunks. The first of these is dealing with atomic and molecular physics.

Class 22 presents the full quantum model of Hydrogen, starting from the Schrödinger equation in spherical coordinates. One of the first times I taught this class, I wrote that up on the board, and the students just goggled at me. "This is a joke, right?," one of them asked.

It's not a joke, though it is a horrible-looking mess. I put it down mostly for cultural purposes, so they've seen it at least once, but we don't really do much with it. (Beyond explaining that the physics convention for naming the polar angles is the reverse of the math convention-- theta is phi, and phi, theta). I sketch out the idea of separation of variables, which breaks it into more tractable pieces, and then give the solutions in terms of spherical harmonics and named polynomials. This introduces the n, l, m quantum numbers for the energy levels, and I talk a bit about how to identify those.

Class 23 picks up from the very brief discussion of angular momentum at the end of the previous class, and introduces the Zeeman Effect as proof of the existence of the degenerate sublevels created by all those angular momentum states. This is done entirely through semi-classical arguments about orbiting electrons acting like current loops, and so on.

I then say that this explains almost all of the observed transitions in hydrogen, with the exception of the tremendously important 21-cm line. I explain how this can't be explained as a transition between closely separated Rydberg levels, because the transition is seen in cold clouds of interstellar gas, but it must be some low-lying state resulting from new physics.

This is a lead-in to the Stern-Gerlach experiment, and the notion of electron spin. I then explain the hyperfine splitting is very vague and semi-classical terms (the spin of the proton in the nucleus creates a weak magnetic field that interacts with the spin of the electron to produce two closely spaced energy levels), and say that the addition of spin allows quantum mechanics to explain all of the observed energy levels in hydrogen (which isn't quite right, because of the Lamb shift, but it's close enough for government work).

Class 24 moves on to multi-electron atoms, and a whirlwind tour of degeneracy-breaking effects-- electrostatic shifts, spin-orbit coupling, the hyperfine interaction. These are pretty much just named, with a tiny sketch of how they work. I also talk about Pauli Exclusion, and how it leads to chemistry.

Showing my own biases, I then talked for a little bit about Rydberg atoms, and how they can look very much like the original Bohr model states. I throw in a few things about cool applications of Rydbergs, from terahertz radiation detectors to cold plasmas.

Finally, I talk a bit about interactions between atoms, and how they can be understood in terms of little dipoles that are either aligned or opposed, giving you either attractive or repulsive interactions. This lets you qualitatively understand the sort of molecular potentials you draw in the Born-Oppenheimer approximation, and sets the stage for solid state.

These three classes are ordered in a way that I think provides a reasonable narrative flow through all of these topics (it borrows heavily from the Six Ideas books in places). This is not, however, the way that the textbook we were actually using does it, leading to a lot of section-skipping and jumping around. Which is a little inconvenient, but not nearly as bad as it gets later.

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Hmmmm. When I do a bit with QM (for grins to fill the last few days of the semester), I make a point of writing H*psi = E*psi, and H = T+V. Putting it in spherical coordinates is just math, given on the inside back cover of "div grad curl and all that" and my intro mechanics book.

Why make it look impossible when it is so logical?

*Grin* I used to show the full H-atom Schrodinger equation in spherical coordinates when I was teaching Freshman Chemistry. I didn't expect the students to understand the equation (and I told them as much), but I wanted them to know that there _was_ an equation that led to all of the pictures of orbitals that they'd been seeing throughout their education.

By Grant Goodyear (not verified) on 31 Mar 2009 #permalink

When I do a bit with QM (for grins to fill the last few days of the semester), I make a point of writing H*psi = E*psi, and H = T+V. Putting it in spherical coordinates is just math, given on the inside back cover of "div grad curl and all that" and my intro mechanics book.

Why make it look impossible when it is so logical?

They've already seen the H=T+V explanation of the S-equation, and have been working with the 1-D time-independent version for a couple of weeks. I show them the full 3-d version here because they're going to be working with it next term, in the intermediate class, and they should have some idea what they're in for.

This class wasn't as horrified by it as some of the other classes I've had. I did have to go in and write it on the board before class, though, because if I try to write it on a chalkboard at lecture speed, it's completely illegible.

it is helpful to explain where the ideas came from. Rather than introducing the final polished theory as if it came from some superhuman intellect, it is less intimidating to see the progression of models that started fairly simple and worked at some level (a sign they were on right track) but unaccounted stuff turned up so they were modified, etc.

That's great, Chad, but I'd suggest putting it up as grad squared, then writing it all out, then separating out the part that is "just" the angular momentum operator (since the radial and angular parts separate so cleanly into distinct eigenvalue problems). I think that clarifies the structure of the problem, which could help the less mathematical ones survive in a QM class.

If you thumb through the books they use for intermediate mechanics or math physics at Union, you ought to be able to find the expression of grad squared in different coordinate systems (usually rectangular, cylindrical, and spherical). The old edition of Marion had it all in one place in the appendix, as do several other books. That way they will know how to find it themselves in the future.

By CCPhysicist (not verified) on 01 Apr 2009 #permalink

The Schrodinger equation for an atom labeled psi has a new science solution of immediate results, perfect for brief classroom presentations, named the CRQT (Clough Relative Quantum Topological) GT integral atomic wavefunction. This model solves the Schrodinger equation by separation, simplification, rearrangement, and series differential expansion within [ Gravity <---> Time ] boundaries. Since psi's space is bonded to it by gravity, the psifunc has spacetime limits.
The CRQT approach builds a topological data point image of one atom by integration of the relativistic transforms for time, mass, and energy into the quantum functions for frequency and wavelength. The combined waveparticle equation generates the picoyoctometric 3D animated interactive image of a psi (Z) as having a nucleus radiating force particles of four types by [ e = m(c^2) ]{ chronons of timefield, probablons of probability fields, varietons of magnesymmetry fields, gravitalons of gravitational field }. The atom pulsates in alternate cycles of nuclear radiation and absorption by the frequency [ nhu = e/h ].
Progress of the nuclear force fields outward leads to their condensation by field bonding effect to form energy intermedon particles, due to radial dilution. The topological functions of each of the 26 sizes of intermedons predicted may be solved by writing the psi's internal momentum function, rearranging it to the photon gain rule, and integrating that to give the symmetopol functions for { positrons, workons, thermons, electromagnetons, magnemedons }. That all works by the way the GT integral is made up with quantized variables for: time, space, magnetism, gravity, symmetry, heat, probability, electric +/- charge, workons.
The resultant 26 energy particles fit quantitatively as the 5/2 kT J heat capacity energy, and intersect the values for the fundamental physical constants: h, h-bar, delta, nuclear magneton, beta magneton. A full discussion of RQT physics with essays, graphics, and software build projects for atomic imaging may be found online at: | |. The images of the h-bar magneparticle of ~ 175 picoyoctometers are on display there.
CRQT physics conforms to the unopposed motion of disclosure in U.S. District (NM) Court of 04/02/01 titled The Solution to the Equation of Schrodinger, and U.S. copyright TXu1-266-788, The Crystalon Door.