# Course Report: Historical Quantum Mechanics

In the last report from my modern physics course, we wrapped up Relativity, and started into quantum mechanics, talking about black-body radiation and Planck's quantum hypothesis. The next few classes continue the historical theme

Class 10: I make a point of noting that Planck himself never liked the idea of quantization of light, and in fact never applied the idea to light directly. His quantum model for black-body radiation was based on the idea of having "oscillators" in the object emitting the radiation. Einstein was the first to apply the idea of quantization to light directly, and take the whole thing seriously.

The next step in the story is the photoelectric effect, and I discuss this in the style of the Six Ideas That Shaped Physics text, imagining ideal experiments that can be done with ideal ammeters and voltmeters. If you take two metal plates connected by an ideal ammeter, and illuminate one of them, the liberated electrons will travel to the other plate and back to where they started through the ammeter, giving you a measure of the number of electrons produced. If you replace the ammeter with an ideal voltmeter, the electrons collect on the other plate, and build up until the repulsive force from the extra negative charge is strong enough to stop the next electron from reaching the plate. At that point, the potential difference between the plates is the "stopping potential," which is equal to the maximum kinetic energy of an emitted electron.

I talk about the predictions the wave model of light makes for the photoelectric effect, and how they utterly fail to match the experimental observations. Then I talk about Einstein's quantum model, in which a single photon supplies all the energy, and how that fits the data better. I also mention Millikan's experiment and his grudging agreement that the Einstein model works.

That takes a bit more than half a class, including one example problem (because the photoelectric effect lends itself to two-equations-two-unknowns problems, and the more practice they get with those, the better), so the second half of the class is devoted to the other great particle model of light experiment, the Compton effect. I talk about how relativity says that a photon should have momentum that is inversely proportional to the wavelength, and that an X-ray scattering off an electron should lose some momentum, and end up at a longer wavelength. I don't go through the derivation in detail, saving that particular carnival of algebra for a homework problem.

This is probably the single lecture from this course that I've given most often. We used to do the photoelectric effect in the first-year E & M course as well, and it's only a little longer than the spiel I gave for the physical constants workshop back in December. I could more or less give this one in my sleep.

Class 11: Having dealt with light as a particle, we turn to electrons as waves, starting with the Bohr model of hydrogen. I talk about the empirical success of the Rydberg formula, and how a classical atom-as-solar-system model can't possibly work. Then I explain how Bohr showed that you can explain the spectrum of hydrogen if you make a couple of really odd ad hoc assumptions about the behavior of electrons in atoms.

I spend about half of the class showing how these assumptions let you reproduce the observed spectrum of hydrogen and hydrogen-like ions. I make a point of stressing that the Bohr model is still wrong-- it's a miserable failure for atoms more complicated than hydrogen, and is totally lacking in physical justification for its weird assumptions.

At the very end of the class, I squeeze in Louis de Broglie, and the idea that electrons should behave as waves. I show how assuming that electrons are waves lets you justify the Bohr quantization condition by saying that the stationary states of hydrogen are states for which the electron orbit is a standing wave. Which is really weird, but might be crazy enough to be true.

I got blanker looks than usual when I mentioned standing waves. I didn't figure out why until Monday, when I realized that we have once again changed the curriculum for the courses before this one, and standing waves aren't covered in those courses any more. We really need to stop changing the goddamn curriculum, because this is about the third time I've taught this class without knowing the exact background of the students taking it.

Class 12: Having introduced the idea of de Broglie waves, we move on to direct proof of the wave nature of the electron, in the form of the Davisson-Germer experiment. Davisson and Germer were studying the scattering of electrons off nickel, and seeing nothing too surprising until they broke their apparatus. In the process of repairing things, they inadvertently melted their nickel target, which settled into a single crystal when it cooled back down. After the repairs, they saw a huge number of electrons coming out at one particular angle, which was a signature of diffraction.

I do a two-minute recap of Bragg diffraction of x-rays, then explain how the Davisson-Germer case is similar but not identical to the Bragg situation. I work through how to find the diffracted peaks, and show that when you use the de Broglie formula for the electron wavelength, it fits the Davisson-Germer result very nicely.

From this, I segue into a slightly hand-wavy discussion of wave packets, and how something like a gaussian wave packet is about the best you can do for describing something like a photon or an electron that has both wave and particle properties. This is more or less the same as Chapter 2 of the book-in-progress, and here as there, I use the wave packet stuff to get the idea of the uncertainty principle. This also leads nicely into:

Class 13: I spent this period having the class look at traveling wave solutions in Mathematica, as a way to get a sense of how to use the program. For homework, I ask them to work through another Mathematica notebook on superposition of waves and Fourier series, which lets them see semi-quantitatively that producing a sharp change in a wave pattern requires many more Fourier components than a broader distribution. This sets up Friday's lecture, in which I try to be a little more quantitative about uncertainty.

The Mathematica experiment was marred slightly by two things: 1) two students missed Monday's class, and thus didn't get the basic tutorial on Mathematica, which means I can expect some panicky email at about 11pm Thursday night (homework is due Friday) asking how to do the homework, and 2) Mathematica has gone through a version upgrade since the last time I did this, meaning that some of the steps in the notebook (which I cribbed from somebody else) have been rendered obsolete by newer commands. I'll have to re-write them before the next time I teach this.

Today's lecture will also be a technical review. I found out a few years ago that contrary to the expectation of basically everybody in physics, the math department does not discuss complex exponentials in the calculus sequence. Which means that students don't see Euler's theorem in any of the math classes that are required for the physics major, which makes solving the Schrödinger Equation a little rocky when they first hit it in my class.

Thus, Class 14 is The Swashbuckling Physicist's Guide to Complex Numbers, in which I demonstrate the important properties of complex numbers through plausibility arguments and hand waving, rather than formal proofs. Mathematicians would undoubtedly be appalled, but if they want students to see something more mathematically rigorous, they should teach it themselves.

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I'm a mathematician and I'm not too appalled by the lack of formal proofs about the complex numbers. The nice thing about them is that plausibility arguments DO hold up--the complex numbers are very well-behaved. You could get really rigorous and construct the complex numbers from the reals, which are constructed from the rationals, etc., and go on about algebraically closed fields, but it wouldn't be any more enlightening to the students.

The best argument I've seen for complex numbers is their requirement for solving cubic equations with real coefficients and real solutions-- the intermediate steps require them, even if they cancel in the solution. It is interesting to remember that back then (16th century), negative numbers were still sometimes called "fictitious"

Physics majors aren't required to take a course on complex numbers at your school?

It is interesting to remember that back then (16th century), negative numbers were still sometimes called "fictitious"

In a way, they still are. "Negative" comes from a Latin word meaning "to deny". In this case, what they were denying was the reality of numbers less than zero. Because until the invention of credit, you couldn't have less than nothing.

By Eric Lund (not verified) on 04 Feb 2009 #permalink

I have found that students can relate to the idea of complex numbers as "rotational numbers", that is, they are a convenient mathematical device for doing rotational algebra. That is, if I want to add two angles u and v, I can look at

cos(u+v) = cos(u)cos(v) - sin(u)sin(v)

and

sin(u+v) = sin(u)cos(v) + cos(u)sin(v).

Then things work out dandy if I define

cos(u+v) + i sin(u+v)
=(cos(u) + i sin(u))(cos(v) + i sin(v))

where i^2 = -1. So I can sell the utility of the complex plane in terms of the utility of keeping track of angles. Which is not too far from the truth.

BBB