Saturday, January 28, 2012

Guide to the Perplexed: Part IV

My series on Bell, entanglement, and the EPR paradox begins with a kind of restrospective essay on more or less how I got to where I am. It's not until my second post that I get into the very interesting history of how we actually got from Einstein to Bell. It's not until my third essay that I get to the crux of the matter: this whole business with the 22.5 degrees is highly overemphasized in the popular narrative. People don't realize that there are huge problems with causality even when the polarizers are aligned, "pre-Bell" so-to-speak. I explain why in this series of essays. Somewhere in the middle of all this I had another one of my Jewish digressions, this one on the fascinating history of double-dipping as originally discussed in the Talmud and later revived in a famous Seinfeld episode.  

My next article began with a discussion in where I guy posed the very interesting question: can you distinguish experimentally between a system where you have atoms in two different states, versus the same group of atoms except they are each in a superposition of those two states? It seems that the people who know how to do these things, using density matrices and such, conclude that there is no difference: and this has deep and far-reaching implications.

After this, I decided to talk about quantisation and the measurement postulate in the context of the
Stern-Gerlach experiment. It seems to me that people who should know better are awfully confused about where exactly the wave function supposedly collapses. Then, in following up on this article, I came across a fascinating Master's Thesis by a fellow from Utah named Jared Rees Stenson, who wants us to analyze the Stern Gerlach experiment in terms of a pure quadrupole field. Stenson does the very interesting analysis for the case of an unpolarized beam, but stops short of the polarized beam; so I set myself the challenge of doing this calculation. I spend the next three essays developing the necessary analytical machinery for my attack on this problem, and then, in my subsequent essay, I abandon all this machinery and simply guess the solution! It's more than a blind guess, of course: it has to satisfy some basic physical parameters, not least of which it has to duplicate Stenson's solution when applied to the unpolarized beam. But the real test would be whether my solution would meet the test of rotational symmetry. The quadrupole field has a four-fold symmetry which would be hard to duplicate unless my solution were just right. It would take some fancy spinor algebra but I should be able to test it against the special case of a 90-degree rotation.

I began girding my loins so to speak to tackle this problem when it slowly began to dawn on me: the quadropole version of the Stern-Gerlach experiment and the so-called "traditional" version were actually one and the same thing! What difference could the addition of a steady-state DC field have on the distorting effect of the quadrupole component? I realized that it was exactly like the way you calculate the tides: it wasn't the direct force of the moon's gravity that caused them, it was purely the distortional or quadrupole component of that field. Why should the Stern-Gerlach experiment be any different? If if that were the case, then the standard description you find everywhere of the beam splitting in two...had to be wrong, because the spatial symmetries of the quadrupole field demanded nothing less than a four-fold symmetry in the detection pattern!

Before going into my final calculation, I have one last brief digression on tides, where I consider the ocean as a driven oscillator, where there are three frequencies that need to be accounted for: the earth's rotation, the moon's period, and the natural frequency of the oceans. Leaving that problem for another day, I proceed to set up the final test of my solution for the polarized beam in the quadrupole field: can I take my solutions for the spin-up and spin-down cases, and add them together to get the correct solution for the spin-sideways case? The answer of course must be the original solution rotated by 90 degrees, and I show in this article that it does indeed work out.

Along the way I had a handful of random blog topics including a link to an awesome gospel harmony song by the Gaither Vocal Band, "There Is A River"; a promo for a physics retreat I held over Christmas at the Maskwa Wilderness Lodge; a link to where a guy from Jordon had been reading my blog in its Arabic translation; and a complaint about getting ripped off by my University of Winnipeg dental insurance plan

My next topic started with what I thought would be a simple calculation involving the reflectance of the moon, which got a little hairy when I realized that the seemingly flat appearance of the moon in the sky was inconsistent with the theoretical properties of the ideal diffuse or "Lambertian" scatterer. It turns out there are at least three moons which are interesting to calculate: the ideal Lambertian moon, the moon as a polished steel sphere, and the moon as a flat sheet of drywall tilted for maximum nighttime effectiveness. It turns out this last case has some eerie similarities with the mathematics of....the quadrupole Stern Gerlach effect! Check it out if you don't believe me.

Along the way I had a few more random posts: this one, a reprint of an old mail-out I did pointing out the arrogant and dismissive manner in which Israel had been presenting itself towards its neighbors; a topic which unfortunately has not lost its timeliness. Again with the Jews, I wrote up a historical analysis comparing the life of the Palestinians living under Israeli rule with the life of the Jews in the Czarist Pale of Settlement, which was later reprinted in the local Jewish weekly.   Then there was something messed up with my blog posts, and it turned out to be a technical problem which a guy from Finland helped me solve over at the Blogger Help Forum.

Most significantly, I was finally, after an acrimonious battle with my professors, expelled from the Teacher Certification program at the U of W. I started a separate blog to talk about that, which you will find if you follow the link.

And that pretty much brings us up to the present four-part series, A Guide to the Perplexed, which takes a retrospective look at two years of blogging to see what I've actually accomplished. On the one had, I look at it and see that I've actually done quite a lot of physics. On the other hand, the problem inspired the name of this blog is basically still with me: I still really don't know how to do quantum mechanics. The stumbling block is, and always was, how to handle a problem with two electrons in it. Oh, I know everyone says you just solve the Schroedinger equation in six-dimensional phase space; and I know there are some people who can actually do just that. I just believe that most of the people who talk about it really have no idea what they're talking about: the only difference with me is that I actually know that I don't know what I'm doing.

At any rate, that's how I felt when I started doing this retrospective series. The funny thing is that during the course of the week that it's taken me to get through all my old topics, an idea of a solution has started taking shape in my head. I'm thinking I might have an angle on the two-electron problem, and I wonder if it's for real. I'm going to leave off for today and come back to this question next time. 

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