The Wave Function of Helium

When I started this blog two years ago, I said I didn't know how to handle two electrons at the same time. And then, almost right away, I got into some very cool calculations about the helium atom. Which is, of course, all about two electrons at the same time. The catch is that I cheated: I optimized the wave function of the helium atom assuming the two electrons were in what's called a "product state", where, to use the Copenhagen terminology which I abhor, the probability of finding electron A is independent of the position of electron B.

It's cheating, but it's the same cheat that everyone else uses. (Except that I happened to do an especially nice job of streamlining the math by using some very cool scaling arguments!) You get a pretty good approximation to the ground state energy level, and you can't really improve on it unless you go the whole hog and calculate the 6-dimensional wave function of both electrons.

Then I did something that was very very cool. I generalized the solution method to apply to the whole isoelectronic series of Helium: that is, every possible ion consisting of a nucleus with two electrons. That would be H(-) (yes, there is a stable species of Hydrogen with an extra electron), He(neutral), Li(+) and Be(2+) etc. The funny thing about the series is that the higher up you go, the more accurate it gets. I actually worked out a table of values which I promised to post, where I compared the calculated energies to the experimentally determined values. Then I got distracted and I never did post my table. Well, here it is now:

The calculated values come from a formula I derived for the series. The formula has two parameters, k and Z. Z is just the nuclear charge, and k is what I call a relaxation parameter whereby the wave function spreads itself out to accomodate the mutual repulstion of the two electrons. It works like this: You put the first electron into the atom, and it naturally goes into the ground state which is just a scaled version of the Hydrogen ground state. Then you put in the second electron and assume it goes into exactly the same state. But it turns out there's an advantage to be gained, in terms of lowering the energy of the system, if the wave function spreads out just a little bit. Here is the formula which we have to work with:

You can see there are three terms in the formula: the first is the potential energy, the second is the kinetic energy, and the third is the interaction energy of the two electrons, due to their mutual repulsion. The k factor tells you how much the wave function spreads out to accomodate this repulsion, and you calculate it just by using basic first-year calculus to minimize the energy. For helium you get a relaxation factor of 27/32. (Other people get 27/16 but they are defining their terms a little differently from me.) It turns out the relaxation factor approaches one as z approaches infinity. This means that the interaction energy becomes less significant as the atoms get bigger. For U(90+) the electrons just basically share the scaled-down hydrogen ground-state orbital. (That's uranium stripped down to its last two electrons, in case you didn't figure that out.)

My formula is very accurate at the high end, but starts to diverge at the low end. In fact, for the hydrogen negative ion, it does something quite bad. My calculated value is -12.86 eV, and the actual energy is  -14.35 eV, which is more than 10% off. That might not seem so bad, but in fact it makes a huge difference in the physics. The binding energy of ordinary hydrogen is 13.6 eV, or as it is known, one Rydberg. My calculated binding energy is just under one Rydberg, so it is actually preferable for the hydrogen to eject the extra unwanted electron. In fact, the true binding energy is just slightly greater than one Rydberg, and this makes the configuration stable with two bound electrons. In fact, H(-) is a stable species, which you wouldn't have known from my calculations.

Why doesn't my formula work so well for the lightest atoms? Because the electrons have a more intricate way of minimizing their energy, which goes beyond what I can account for with my simple model. You have to treat the system in 6-dimensional phase space in order to do the optimisation. Not about to happen on this blog. (Not without some tricks anyhow).

But that's not exactly where I wanted to go with this. There's something very funny about the binding energies, and I'm not sure anyone has actually solved this problem. As the atoms get lighter, the binding energy gets less and less. What is the lightest atom which is stable with two electrons? Obviously, hydrogen. But what if you could shave a bit of charge off the proton? Wouldn't it still be stable, up to a point? I'm interested in that very special atom consisting of something like 0.92 protons and two electrons, where the binding energy is...exactly zero. Has anyone calculated this atom? Because I think I know how to write down the wave function for it, and I mean the real wave function, not just an approximation.

The Spreading of the Wave Packet

One of the big problems for the wave interpretation of quantum mechanics is the spreading of the wave packet. Shroedinger did some very clever stuff with his equation to construct cases where a little packet of waves maintained its spatial integrity, most notably in the case of the harmonic oscillator. However, once the electron was propagating in free space, there was nothing he could do to keep his wave packets from spreading out. How was one to reconcile this with the obvious fact that an electron produced at point A was always observed to travel intact to point B?

This is a very puzzling question indeed, but I must ask: just exactly which experiments do we have where an electron is produced at point A and detected at point B? Just as I argued in an earlier essay that we have no pea shooters for photons, I don't believe we really have any pea shooters for electrons either. Oh, there are obviously devices which produce a constant stream of electrons; and the rate of production of these electrons can be slowed down, probably to any arbitrary extent. But isn't the appearance of single electrons still unpredictable? I do not believe we have any reliable method for producing a single electron at point A and detecting it at point B. So what is the problem with the spreading of the wave packet?

People say that you can actually see individual electrons in a cloud chamber. The traces are undoubtedly very compelling: yet they may not be what they seem. In 1927 (or 1929? could it have been that soon after the Schroedinger equation)  Nigel Mott published an analysis which showed that for a spherically propagating wave, the most probable observed ionizations would be those lying on a straight line: in other words, the straight-line rays of the cloud chambers were in fact just what you should expect for spherically-propagating waves generated according to Schroedinger's equation.

The common belief that we can actually see individual particles, whether photons or electrons, is with us at every turn. Even Feynmann is guilty of it when he talks about the photomultiplier tube: he says you can actually see individual photons, and they are indicated by the click of the detector. There is a video of him kicking around YouTube where he makes this point, and it strikes me that he is uncharacteristically agitated, for want of a better word, when he makes this argument. Is he showing his frustration because deep down he knows his reasoning is flawed?

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 StackExchange.com 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. 
  

Guide to the Perplexed Part III

It had been five months since my last post when I returned to blogging with an article on the collapse of the wave function. One of the most famous examples of wave function collapse is the appearance of flecks of metallic silver on a photgraphic plate exposed to the light of a distant star. Any reasonable calculation of the energy density of an electromagnetic wave shows that it is impossible to gather enough energy to drive the chemical transition AgBr => Ag + 1/2Br2, and this circumstance is taken as proof that the energy of the light must be concentrated into particles called photons. In this article, I outline the thermodynamic argument which shows that when we treat the whole silver bromide crystal as a solid solution, the reaction actually becomes spontaneous at very low concentrations. In other words, the energy necessary to detect the light is already present in the crystal, and you don't need to invent a "photon" to supply it.

I don't know if it was before or after I wrote that article that I discovered something that would change my life: Google Blogger tracks your statistics! I discovered that people were actually reading my articles, and suddenly everything was different. I don't know if you'd call it an awful lot of hits, but I was getting two or three hundred clicks a month. This changed everything.

The first thing I posted was an old article about the Mid-East conflict that I had first circulated by email back in 2006. Then I got right back to physics. My next article was about something that I figured out over twenty years ago. I had been trying to calculate how much power you could absorb from an AM radio station with a well-designed crystal radio, and I discovered that the theoretical power was independent of the length of your antenna. This seems like an absurd result, but there is a formula for it in the books, and it's true. The catch is that you can't easily build a perfect antenna because of imperfections in real materials, mostly due to the resistivity of copper. But in theory the result is true and it is mathematically derived. What is different in my approach is that I show how you can understand the result pictorially, and working from simple pictures you can get a pretty good ballpark of the exact theoretical result.

What the whole world seems to have missed about this calculation is its enormous implications for quantum mechanics. In Schroedinger's picture, a hydrogen atom is nothing more or less than a tiny crystal radio antenna, and everything that a hydrogen atom does, in terms of its interaction with the electromagnetic field, can be understand in terms of its properties as a classical antenna. In particular, this new perspective makes a mockery of those old textbook calculations where you evaluate the photo-electric effect by looking at the cross-sectional area of an atom. The effective electrical cross-section of an antenna has nothing to do with its physical cross-section, and this is a purely classical effect that you don't need to explain with "photons".

Meanwhile, now that I was posting again my hit count had taken a sudden upswing. Google doesn't just give you the hits, it tells you the country of origin and the search engine keywords; so I was pretty excited one day to notice my first visitor from the Palestinian Territories, so I couldn't resist giving a friendly shout-out to whoever he was. It was only aftertwards I realized that he might have just as easily been an Israeli "settler"; but either way, I'm glad to see him.

Another surprise from the Blogger statistics was the number hits I got from people who googled "perturbation theory" and "ladder operators". So I wrote a follow-up to my earlier musings on the subject. It's a more mathematical topic than I normally ought to bite off, but I still think my pictorial perspective adds something to the big picture.

In my many internet discussions over the years about the photo-electric effect, I had often heard of the semi-classical school of Jaynes and Scully. I had always assumed that my approach was more or less in line with theirs, and when people ridiculed me for my ideas, I would sometimes invoke Jaynes for moral support. I was pretty shocked only last year to learn that I was wrong: Jaynes and Scully take a classical field and apply it to the quantum atom, but then instead of allowing the atom to evolve through time evolution from the excited state to the ground state (which is what I do), they still calculate the quantum leap transition probabilities.  In other words, if my approach is "semi-classical", then Jaynes and Scully should actually be considered hemi-semi-classical. Or whatever. You know what I mean. I explain it all in the linked blogpost.

Coming up with a semi-classical ("no-photon") explanation for the photo-electric effect was a defining moment in my life, and for ten years I would go on the internet and try to argue it. The most common way people would shoot me down was to say "maybe you can explain the photo-electric effect, but you can't explain the Compton effect." And they were right: I couldn't. Until one day I did. This was a paradigm breaker! I thought for sure I would win the Nobel Prize for this. Sadly, it was not to be. It turns out my explanation was identical to the explanation that Schroedinger had already published in 1927. It's true that in 1919 Compton had "proved" that you couldn't explain the effect according to the wave theory, but that was because he treated the electron as a little charged ping-pong ball. The wave theory explanation is a totally natural outcome of the Schroedinger equation of 1926, but by then the photon paradigm had been so firmly established that even Schroedinger was ignored and marginalized when he argued against it.

In the meantime I had gotten into a discussion on StackExchange.com about transmission line impedances, and so to get myself back in the game, I re-did some old calculations about a parallel-plate waveguide. You get some very interesting results if you just assume that for a freely propagating wave between two plates, there must be no net attraction or repulsion between the plates. This discussion led to some very cool calculations of transmission-line impedances, which I calculate as usual with very pictorial methods. Next I take a pretty big leap and apply these methods to calculate the radiation resistance of a half-wave dipole. Everybody knows this is supposed to come out to 73 ohms, but for most people that number is pretty mysterious. I don't get it exactly right, but I definitely justify it to within a reasonable accuracy, and all by very graphic methods.

In my next post, I find myself again dragged back to the Middle-East conflict. Here I resurrect an old proposal of mine to adopt the usage of Arabic Script to write Hebrew. It's a fantastic idea on all kinds of levels, and it would do huge things to bring Arabs and Jews together, but nobody in Israel is listening to me. I'll keep trying.

As I mentioned, Google Blogger gives me statistics on country of origin, and naturally the U.S. and Canada lead the list, followed by Germany, Russia, and the United Kingdom. Surprisingly, the next spot on the list is up for grabs, and is hotly contended by such countries as India, the Netherlands, South Korea, and...tiny Slovenia, which for a brief moment edged out the other contenders for sole possession of sixth place. I acknowledge them in this post.

My next series of articles was motivated by a question from my nephew, who asks "why is energy e=mc^2, and not m-c-cubed  or whatever? Although this question can be easily answered with dimensional analysis, the actual reason is harder to justify than you might think, and I was led pretty far into uncharted territory (for me anyway) when I tried to justify it via relativity.

My next series of articles deals with perhaps the most baffling and certainly the most talked-about paradox in all of quantum mechanics:  the question of entanglement, with Alice and Bob and the crossed polarizers and all that stuff. It gets pretty involved and I think we'll continue with this topic when I return.  







   

What if the Moon were Made of Golfballs?

I've been talking about the reflectance of the moon in recent posts, and the issue came up of why the appearance of the Moon is so out of whack with what should be called for by the theory of diffuse reflection. The classic Lambertian reflector has the singular property that it looks just as bright whatever your viewing angle: but this property does not extend to the angle of illumination! In other words, it most certainly does not look just as bright no matter what angle you illuminate it from. So why does the moon look uniformly bright all the way across its disk? Why does it look like a flat cut-out?

It occurs to me to ask the question: what if the moon were made of golf-balls? Each golf-ball would be a mini-moon: when the moon was at half-phase, each of the little golf-balls would look like half moons to us here on earth. Except they'd be too small to see. So we'd just see the average of all of them, and the local average would be the same wherever we looked. So maybe the moon would look uniformly bright everywhere.

Except the golf-balls low on the horizon would also be partly in the shadow of other golf-balls, so the fringes should still look darker. In other words, I'm not completely buying my own explanation. But it's a thought.

Meanwhile, I tried to apply some calculus to this question, and I'm not too sure of my technique, but I got an answer and I wonder if anyone out there would like to double-check it. I compared the real moon...that is, the ideal, bright-white Lambertian "real" moon...to a flat cut-out sheet of drywall. According to my calculation, if you compare these two models at midnight on a full moon, the flat drywall cut-out provides 50% more night-time illumination than my "ideal/real" Lambertian moon. I'm not going to try and repeat my calcuations here, but I'm just wondering if anyone out there wants to see if they get the same answer as me.

Guide to the Perplexed: Part II

In my last post, I catalogued all my blog articles for 2010, up to August of that year. What followed was a long period of inactivity, which followed a visit to Winnipeg from my grad school buddy Richard Epp. I remember it was a really hot summer day, and started telling him about Quantum Siphoning. Richard objected to the interpretation of the wave function as charge density. He pointed out something I already knew: if the charge is distributed, then the calculation for electrostatic energy is messed up. How could I answer that objection? I had no answer, and every time I tried to work on physics, that was all I could think about.

Then in February I stumbled across a website called The Foundational Questions Institute which was apparently sponsoring an essay contest on the question "Is Reality Analog or Digital?". Incredibly, the deadline for submissions was that very day, so I immediately wrote up an article which I gave a title that says it all: There are No Pea-Shooters for Photons.  The targets of the essay are the three pillars of the photon theory, as universally recognized in the popular narrative: the Blackbody Spectrum, the Photo-electric effect, and the Compton Effect. In my essay, I show how each of these has an intuitively reasonable wave theory explanation, which could not have been understood by Planck, Einstein, or Compton in their time because the wave theory of the electron was not revealed by Schroedinger until 1926, long after the particle paradigm had taken firm hold. The title of the article refers to the well-known post-modern justification for photons based on experiments where you supposedly fire one photon at a time and track where it goes.  I point out, as I did in an earlier blogpost called The Clicking Detectors, that none of these experiments are really quite what they claim to be, for the plain and simple reason that there really are no pea-shooters for photons. 

When I submitted my essay, I was in a rush to meet the deadline, and I didn't feel that I had dealt as well as I might have with the question of the blackbody spectrum, so I went back to my blog and that became the subject of my next article, which soon mushroomed into a nine-part series. The basic idea was that instead of attacking the ultraviolet catastrophe at the electromagnetic level, you take it on at the mechanical level. All electromagnetic radiation has to have its source in the mechanical vibration of charges, and if there is no mechanism to set those vibrations in motion, you don't have a problem at the electromagnetic level. The one nagging problem with my argument is that thermodynamics doesn't depend on specific mechanism:  there's something called the Equipartition Theorem which supposedly rules no matter what, and I had to somehow explain it away. I struggled with this through my next four blogposts until I made a huge breakthrough.

There's something people do in physics which drives me crazy and that is putting all their arguments in the most abstract, mathematical form. I need to see real examples and real mechanisms, and in particular I wanted to figure out just how an equilibrium is established between a mechanical oscillator and the electromagnetic field. This doesn't look like it should be an insurmountalbe problem: we know how to do the driven harmonic oscillator with damping: just apply that to an atomic oscillator with the electromagnetic field as the driving force. How hard could it be?

The problem is that we don't just have a driving field of known intensity: we have a random, distributed field. There is no simple number we can pick out of the blue and say "we are driving the atom with an oscillating field of so-and-so-many volts-per-meter: the field strength is expressed in volts-per-meter-per-hertz, and how the hell do you interpret that. What happened is after years and years of not knowning how to handle this question, all of a sudden I figured it out! It's a beautiful, very pictorial explanation that starts off by considering that familiar old chestnut of statistical theory, The Drunkard's Walk . Analyzing the driving force on the oscillator as a special case of the Drunkard's Walk, I show that you are allowed to truncate your frequency distribution at any arbitrary limits and you still get the same oscillation regardless. I then carefully count up the cavity modes of the electromagnetic field, and then run a numerical example of a special case to come up with the amazing conclusion, which is the basis of the Rayleigh-Jeans derivation of the ultraviolet catastrophe: the energy per mode of the electromagnetic field is equal to the energy per mode of the mechanical oscillators!

What this means is if you can show that the high-frequency oscillations are suppressed at the mechanical level, then they are automatically suppressed at the electrical level. You don't need to throw out Maxwell's equations to avoid the ultraviolet catastrophe.

My next post was four weeks later. A year previously, I had been very excited to work out the solution for the problem of two electrons sharing the same potential well. It came as a huge surprise to learn that in contrast to the well-known single-electron case, with two electrons the shape of the solution depended on the size of the box. I had a huge argument with a guy named SpectraCat in physicsforums.com over this: he said that the shape of the solution depended on the strength of the interaction, and I said that was exactly the same as saying it depended on the size of the box. Of course I was right: the very suprising thing about it was that the case of the very small box corresponded the case of independent particles, and vice versa: the strong interaction corresponded to the case of the very large box! It may seem counterintutitive, but in my analysis of the iso-electronic series of helium, I show that's exactly how it works. A "helium-like atom" is any atom stripped down to its last two electrons. The series actually begins with hydrogen: it turns out that the negative hydrogen ion, consisting of a proton and two electrons, is marginally stable. As you add more protons to the nucleus, the electrons get more tightly bound: helium, lithium, beryllium, etc: I actually found binding energies for all those atoms and you can clearly see from the values for the energy, that as you go to higher atomic numbers, the electron configuration approaches the simple product state of hydrogen-like orbitals. In other words, as the box gets smaller, the interaction of the electrons becomes insignificant.

What brought me back to this question was I realized I had made a big mistake: I had botched the symmetrization! The wave function was actually quite a bit more complicated than I had drawn it, because my answer did not preserve the correct symmetry for fermions: the function must reverse polarity when you switch particles. You can always do this by taking appropriate sums and differences of whatever function you already had, and that's what I do in this article.  It turns out this is the exact same symmetrization method that needs to be applied to the case of two isolated hydrogen atoms. You are not allowed to simply say that the electron here is spin-up, and the electron there is spin-down: try writing it down that way, and then interchange electron A with electron B. You know that according to theory, you must get back the same function with a negative sign, and you'll see that you don't. The wave function isn't right until you symmetrize it the way I've shown in my article, and it turns out that this leads to some very distrubing and surprising consequences.

In the meantime, I wrote a couple of other articles in April about Fourier Transforms, Ladder Operators, and Pertubation theory. My son's friend was taking a course in Mathematical Physics, and I had been helping him with assignments. It drives me crazy the way they suck all the physics out of these things and just give you math question that amount to manipulation of symbols according to a set of rules. That's not physics to me, and it's not even math. For me, it's all about the interpretation, and I got into some very cool stuff in this article about solving differential equations with Fourier transforms. The very last problem on the homework assignment was a weird-looking differential equation that I vaguely recognized as having something to do with the quantum harmonic oscillator: I couldn't quite put my finger on it but finally figured out that it had to do with ladder operators. One thing led to another and I started writing some very cool stuff about Perturbation Theory. This is something that's taught, as usual, as a set of rules for symbolic manipulation of functions, but here I make it into something pictorial, relating it to ladder operators and Taylor expansions. It's a bit half-baked, but it's still good.

And then it stops. I don't know exactly what happened, except I remember I was working all summer on a construction survey crew, and we had a lot of fifty-hour weeks. My next article wasn't until almost six months later, and that's where we'll continue when I come back again.

A Guide to the Perplexed:Part I

I started this blog almost two years ago with a rant about how I didn't feel like I knew what I was doing any more, and quantum mechanics was to blame. Since then, I've posted eighty-something articles, including some pretty cool stuff, and maybe you're wondering: do I feel any better today? That's a tough question. Let's start by going through my old posts and listing the topics I've covered. That will be a useful thing in itself.

One of the main reasons I started this blog was because I felt I had all kinds of original ideas on how to do physics problems, and I wanted to start recording them. How original were these ideas? I don't know, you can judge for yourself. I just thought I did physics differently from other people, and I wanted to stake out my territory. So it's a bit ironic that my first physics post after the inaugural rant was entitled Something I just Figured Out Yesterday . This is what happened. I complained in my first post that I didn't know how to do quantum mechanics. I actually was quite good at doing most of the undergrad level calculations, like the hydrogen atom. More importantly, I felt I wasn't just good at doing the calculations, but I could actually interpret the results with some insight. The big problem was when I came up against two electrons at the same time. I just couldn't begin to handle multi-electron problems.

So I was thinking about this, and it occurred to me that there was one two-electron problem that I had  to be able to solve: the problem of two isolated hydrogen atoms. There are two protons and two electrons. Since I already knew how to solve the single hydrogen atom, how could it be any harder if there were two of them. They key was the shift of perspective: I was going to solve it as a two-electron problem.

I came up with an apparent solution that raised more questions than it answered: it took me quite a while to find  my mistake . In the meantime I put up a very important post called The Clicking Detectors . This one wasn't brand new, it was stuff I'd figured out a long time ago but never written up. People have the idea that you can split a beam of photons with a half-silvered mirror, and the statistics of the detector clicks show that each photon must have gone either one way or another. This is supposed to be a proof of the particle theory of light. In "The Clicking Detectors", I show that any reasonable wave theory of light would give the exact same detector statistics.

I didn't post again for about a month. What had happened in the meantime is that I'd got into a very intense discussion thread at physicsforums.com on the nature of wave function collapse, and in the course of that discussion I actually figured out a mechanism whereby you can explain all kinds of collapse phenomena via the ordinary time-evolution of the Schroedinger wave function. I called the mechanism Quantum Siphoning . In my article on the Clicking Detectors, I said that any "reasonable" wave theory would give the right statistics: the catch was, I didn't actually have a reasonable wave theory at that time. Quantum Siphoning turns out to be that theory. If I ever win the Nobel Prize, it will be for Quantum Siphoning.   


My next article was a reposting of something I'd actually written about a year earlier in private correspondence with my physics buddy, Richard Epp, who was in grad school with me and now works at the Perimeter Institute. Ramanujan and the Casimir Effect shows how you can use some of the weird seemingly divergent series studied by Ramanujan to calculate the force between parallel plates due to the Casimir Effect.

In the meantime, I was still engaged in the quest for a two-electron problem that I could actually solve, and I was now working on  The Double-electron Potential Well , which became a series of two articles. Not the third-year physics problem with fictional "non-interacting electrons", but the real thing. I had some preliminary sketches of two-dimensional wave functions that I had posted on physicsforums.com and I had written that I was having trouble getting a handle on it when a very smart guy named Peter Atcam suggested I "turn off" the interaction and then turn it back on again very gradually. This was a hugely productive suggestion that not only clarified the situation with the two-electron well, but led to a whole new topic, The Iso-electronic Series of Helium .You can find solutions for the helium atom on the internet, but no one does it with quite as much finesse as I do in this four-part series of articles. In the meantime, my article on the two-electron well holds the distinction of being my most widely-read posting
of all time, according to Google Blogger statistics; which is  a bit ironic since it contains an error of omission which I didn't find and correct until over a year later.

I was just finishing my series on helium when the tin-pot dictators of physicsforums.com led by the omniprescent ZapperZ kicked me off the forum for the third and final time. I talk about the circumstances in my article Banned for Life .  All I can say now is I've been banned from better places than physicsforums.com, including a couple that I mention in my next article, A Tale of Two Strikes . But back to physics. My next article came in response to a question posted in physicsforums wherein I show how the anomalous specific heats of diatomic gasses at very low temperatures can be intuitively understood as a consequence of the wave nature of matter .

When I talked about Quantum Siphoning, I said it represents my best shot at getting a Nobel Prize. I wasn't joking. You might think it's not much of a shot, and maybe it isn't but if it's one in two hundred, it's still a better shot than anyone else who's every read any of my blog posts. It's a shot because if it's right (and it is right) then it's a paradigm breaker just like relativity or the uncertainty principle: it's not just a different calculation but a whole new way of looking at physical reality. What makes me think I'm capable of making a breakthrough of this magnitude when people smarter than me have tried and failed?

I'm a bit of a student of the history of quantum mechanics, and from my readings I know that there was no one who would have been more sympathetic to my viewpoint that Schroedinger, who was himself ridiculed by the "mean physicists" (Born, Heisenberg, Lorentz etc.) for trying to find real-time, causal mechanisms for quantum phenomena: and make no mistake, these would be wave-theory mechanisms, not particles. The particle concept is the source of all evil in physics: the quantum leap, the collapse of the wave function, the Alice-and-Bob correlation of entangled particles. No one hated particles more than Schroedinger: so if quantum siphoning is the answer to these paradoxes, why didn't Schroedinger invent quantum siphoning? 

The pathway to the truth is fraught with obstacles; and what makes me different is I have always done things my own way. Over a lifetime of tinkering with ideas, I have put together my own unique bag of tricks. These are mostly things that other people already knew; people much smarter than me, including Schroedinger. But no one single person, not even Schroedinger, ever knew exactly all the little things I knew. It is the strange uniqueness of my own personal bag of tricks that has placed me in the fortuituous position of being the paradigm breaker.

I'm not going to list all of my tricks right now: after all, that's why I started this blog in the first place. But my next series of articles, spread over three months and beginning with the post Karma and Carbon Monoxide , tells the story of one of the most unlikely links in the chain which led to Quantum Siphoning. If you read through the previously-mentioned physicsforums thread where I ended up getting banned for life , you will see that at one point in the argument I was dealt a near-death blow relating to the activation energy needed to drive the chemistry of the photographic process. Incredibly, I recovered from this blow with an ingenious thermodynamic argument where I showed that despite the unfavorable enthalpy, that at the very low concentrations of metallic silver in a developable exposure, we were entitled to treat the silver halide crystal as a solid solution, and taking into account the Gibbs Free Energy of the total system, it could be shown that the thermodynamics of the transition was actually borderline favorable! This argument led directly to Quantum Siphoning, and I never would have been able to make it if not for the story told in this series of posts about a project I was given as a junior engineer in my very first job out of university at a government research station in Pinawa, Manitoba: hence, Karma and Carbon Monoxide.

This brings us up to August 2011, except for one article from the summer of 2011 which I haven't yet commented on: Why do Solids Absorb Light . This turns out to be one of my most-often googled posts, and not without good reason. It is a question which is asked all the time, and people never give a decent answer. The standard answer about electronic transition levels is fine for explaining why light would be scattered, but where is the mechanism for absorption? I have to admit my answer is a bit sketchy, but I'm pretty sure it is a semi-classical mechanism very similar to the one Schroedinger demonstrated for the Compton effect. That in itself is another very important paradigm which is buried in obscurity in the conventional group-think, but more on that later.

After my Carbon Monoxide series ended, I stopped posting for about six months, not to resume again until 2011. I think we'll stop this recap as well for now, and continue next time from where we leave off today.











     

Explorer Bad, Firefox Good

I mentioned in some earlier posts that some of my articles were screwed up: not that I had physics mistakes, but that the files seemed to be corrupted. When I clicked on the link, they would show up for a second, and then go blank. I had to take down the articles and repost them.

Yesterday I decided I ought to go through my whole blog, all 84 posts, and make sure they were all working. I was appalled to find about ten of them crashed the site, including some of my best posts with really good search engine exposure. This was very disappointing.

I prepared to delete the bad posts by making backup copies. and then it occured to me to check out discussion groups for possible help. A quick search showed that Google Blogger maintains its own help forum. I posted my problem and within an hour a user named "mspotilas" replied:

Blogger rolled out threaded/two level commenting, and that does not work on Internet Explorer. With Firefox and Chrome you should be able to open the posts with comments.

To make them work on Internet Explorer, too, while waiting for Blogger to fix the bug, you can change in blog's settings your comment form from embedded form to full page or popup. That disables the threaded commenting, and also Internet Explorer will work again.

You can see the exchange here at theBlogger Help Forum . Could it be the browser? I could hardly believe it so I tried the links in Firefox. They all worked! The lesson: Explorer bad, Firefox good.

But there was one puzzling thing: My problem had nothing to do with comment fields. It was just random throughout my posts. Or was it??? I looked through my list of posts again. Son of a gun, if the bad posts weren't all the ones with comments. How could I have missed that? It was 4:00 am when I went through my files, but still...

POSTSCRIPT: I hope Potilas doesn't mind me posting his user profile here. For your information:
(Oh, and he's also got a YouTube channel mostly about classical piano ).

Introduction

Sairastan MS-tautia ja käytän Bedrocan-lääkekannabista taudin kurissa pitämiseen.

I have multiple sclerosis and I use medical cannabis (Bedrocan) to control the disease and symptoms.

Gender

Male

Other names

MS-potilas

Is Israel an Apartheid State?

I originally posted this article two months ago but something funny has started happening to some of my links, whereby they seem to show up for about a second and then go blank when you click on them. So I'm reposting it here.

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These days you read about things happening in Israel that would have been unthinkable in the idealistic days of the first few decades of the beleaguered state. Even then, with its back to the sea and deadly enemies surrounding it on all sides, Israel was widely vilified as a second South Africa; even more outrageously, it was often compared to the Nazis. Sadly, with the passage of time, some of the old slanders have begun to take on elements of truth as the rights of Arabs under Israeli rule have come under increasing attack. Not that Arab life is Israel is remotely comparable to the worst excesses of South African apartheid, let alone the grotesque comparisons with Nazi persecution. Ironically, however, the critics and defenders of Israel alike have missed the most obvious and glaring comparison : there is an excellent case to be made for equating Arab life under Jewish rule with Jewish life in Imperial Russia!

We Jews are supposed to place great importance in knowing our history, but our collective memories of life in Russia are oddly skewed. I happen to have some expertise in these things because I am one of the few people in my generation who is able to read the literature of our people in its original language, and I have done so extensively. So I know something about Jewish life in Russia. I also know that if you ask modern Jews to give a single word that most succintly expresses the nature of that life, they will almost unanimously say "pogroms".

Unfortunately, it seems we Jews, like all other peoples, remember only what it suits us to remember. We choose to remember life in Russia as an unmitigated series of horrors not least because it helps to justify our Zionist mythology. (By the way, because I call it a "mythology" does not mean I don't personally buy into it. I am a proud Zionist, but I am not proud of everything Israel does!) In fact the pogroms were a horrible episode in our history, but they were far from a dominant feature of Jewish life in old Russia. The actual picture is much more rich and nuanced. It is true that there were many episodes of persecution and injustice, but it these were interspersed with periods of great freedom and opportunity. And as we review the history it is surprising how many parallels we will find with lot of the Arabs in Israel.

We can begin with how the Jews came to be Russian subjects. Medieval Poland was a place of refuge for Jews fleeing the persecution of the Crusaders, and once established we fluorished there. Poland was in those days a huge kingdom covering much of present-day Eastern Europe; over the centuries, it was gradually picked apart by the surrounding powers of Austria, Germany, and Russia, until with the final Partition of Poland in 1793 the Russian Tsar awoke one day to find himself the proud ruler of close to a million Jews. Thus the Jews became unwanted subjects of Russia much the same way as the Palestinians became unwanted subjects of the Jews: through military conquest.

These Jews were not the educated intellectuals of North America who typify the modern Jewish stereotype: they were mostly primitive, black-frocked and bearded religious fanatics. The Tsars mistrusted the Jews for their alien beliefs and their close ties to their co-religionists living in hostile states across the border. Sound familiar? Keep reading. Over the course of the nineteenth century the pendulum swung from one extreme to another, the government sometimes trying to integrate the Jews into the modern economy as productive citizens, and sometimes trying to contain them by harsh discrimination. A great concern was the Jewish birthrate, with early marriages and up to a dozen children being the norm.

There is much nonsense written about the actual facts of daily life. People say that Jews weren't allowed to own land. They certainly were: however, there were restrictions on where they were allowed to buy land. Sound familiar? And there were certainly cases where the Jews were cheated out of their lawful property rights by the government. But at the same time we were entitled to go to court and contest such expropriations, and occasionaly we would win these cases. Mendel Bailiss was famously acquitted by a Russian judge and jury in the infamous blood-libel trial of 1912. But on the whole there is little doubt the courts were stacked against us. Sound familiar?

One of our greatest grievances against the Tsar was the "Pale of Settlement". Jews were forbidden to take up residence outside the areas which basically constituted the original Polish kingdom: in other words, they weren't allowed to leave the Occupied Territories to live inside the Green Line where there were greater economic opportunities. Oops, it wasn't called the Green Line...that's what we have in Israel.

Now let's remember a thing or two about the pogroms themselves. There were three significant waves of pogroms. The first was in the 1880's in the aftermath of the assasination of Tsar Alexander. Although the news of these outrages terrified the Jewish community throughout Russia, the total number of fatalities in this period was in fact less less than one hundred. A more serious outbreak began with the Kishinev pogrom in 1903, and over the next few years perhaps a thousand Jews died in the unrest.

There is a lot of nonsense about Cossacks and Russian police officers leading these outrages. In fact, while high officials in the Government undoubtedly knew and approved of what was going on, the fact remains that Russia was a country of law and justice and it was unthinkable for the police to allow these things to go on with their knowledge, let alone to participate in them.

The catch was: if they didn't know, they couldn't very well do anything about it! It's called plausible deniabilty and it's the oldest trick in the book. As long as they could pretend they didn't know, they would let it go on; but after a day or two they would invariably show up and restore order. A few ringleaders might be slapped on the wrist, but that would be the extent of it.

How similar is this to the pogrom which we allowed the Christian Phalangists to carry out under our noses in the Sabra and Chatilla refugee camps in 1982? It is true that Ariel Sharon was eventually found accountable and demoted from his cabinet post, but that didn't stop him from later becoming Prime Minister. How do you think our Arab citizens should have felt about that?

There were some fifty deaths in the Kishinev pogrom, and the world was shocked. It was a time when people believed that freedom and human dignity were marching forward, and the backwardness of Russia was a huge embarrassment both inside and outside the Empire. The plight of the Jews attracted worldwide attention, and an upsurge in the Zionist movement was one of the immediate consequences.

This is not quite the end of the story. In 1914 war broke out and within four years the old order of kings and emperors simply ceased to exist. It was replaced by a new harsh world of nationalisms and ideologies. Civil war raged in Russia and in the Ukraine, a nationalist government took over that virtually declared war on the Jews. A hundred thousand died in the pogroms of 1919-20, and the world scarcely took the time to yawn. Even the Jews hardly remember these martyrs, as their suffering was eclipsed by the much greater disaster of the Holocaust twenty years later. But we ought to at the very least not blame those pogroms on the Tsar, who had already been killed by his Bolshevik captors.

History is a funny thing. We choose to remember whatever suits our purpose, and it suits our purpose to demonize the Tsar and everything he stood for. Yet when the Palestinians do the same to us (and they do), we feel aggrieved. "Why don't they appreciate that living under the Jews, they are far better off than their bretheren living under brutal dictatorships elsewhere in the Middle East?" We ought to remember that their attitude toward us is simply human nature, and it is not so different from our attitude towards the Russian Empire. Perhaps we have more in common than we like to admit.

Again with the Jews...

A couple of months ago I posted on this blogsite a letter that I had first circulated five years ago, "Ten Things We Jews Believe About the Middle East" . I have always been a staunch supporter of Israel, and until 1992 I supported every military strike and every new settlement. Then Arafat signed the Oslo Agreement, and the conflict was over. We had won! or at least, we had established our right to exist within recognized borders. There was no more need to fight.

But in the aftermath of this great event, nothing seemed to change! For the next eight years it was business as usual. The occupation went on, and the settlements grew and multiplied. Why were we surprised in the year 2000 when the intifada broke out? Did we look at our own behavior? No! We said that the Arabs were finally showing their true colors; that the intifada proved that they had never given up their goal of driving us into the sea. That's when I sent out the letter I referred to above.

As you can guess, the letter was not well received in the Jewish world. The negative reaction was not by any means unanimous, but it was nonetheless...well, it was negative. Among non-Jews, on the other hand, my letter was very well received. I got so used to this pattern that one day, when an acquaintance I thought to be non-Jewish wrote critically, I answered back (totally unselfconsciously, I swear!): "I didn't know you were Jewish, Ralph." Ralph took offense at that comeback: what right did I have to assume he was Jewish? It was only very recently that I learned the rest of the story: Ralph isn't Jewish, but his wife is.

Anyhow, what brings all this to mind is that I was browsing through some old correspondence and I stumbled across a follow-up to my original letter. The facts cited are a bit out of date, but the general picture has hardly changed. Here then is what I wrote five years ago. God help us if we don't change our attitudes.

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ISRAEL SHOWS ITS CONTEMPT FOR THE ARABS.

Six weeks ago, I sent out my ten-point plan for attitude change along with my two-point action plan for peace in the Middle East (see below). Since then, the news from Israel has only strengthened my conviction of the dire need to change our attitudes towards our Arab neighbors before we bring down disaster on ourselves. Consider these five examples of Israeli behavior:

1. Syrian President Basher Assad proposes peace talks on the Golan issue and we brush it off with contempt.

2. Arab League Secretary Amr Moussa calls for a renewed initiative based on the 2002 Beirut peace plan and we ignore him.

3. Israeli Prime Minister Ehud Olmert stands on a platform in Moscow beside Vladimir Putin and warns Iran to "be very, very afraid" of what Israel will do unless Iran backs off its declared plans for peaceful nuclear development. (And just days ago another cabinet minister mused publicly about a "first strike"!)

4. Foreign Minister Tzipi Livni accepts and then rejects an invitation from Qatar to take part in a UN conference in the Gulf States where she would have an opportunity to present Israel's best diplomatic face to its neighbors. Reason? She refuses to attend the same event as representatives of the elected government of the Palestinian people.

5. Prime Minister Olmert appoints ultra-right-wing-nationalist Avigdor Lieberman to a senior cabinet post. Remember the fuss we made when Joerg Haider became a member of the Austrian government? Our man Lieberman makes Haider look like a boy scout. And now he's our point man on the Iran issue!

I thought we Jews were supposed to be smart. Maybe we are when it comes to inventing the theory of relativity or finding the cure for polio, but it seems like we've got a few things to learn about civilised relations between nations of different backgrounds. If it's not already too late.

I get to re-use a graph from another problem

Last time I said it was interesting to calculate how effective the moon would be if it was a simple hunk of drywall, just a big round cut-out. Actually the interesting calculation is if we hang the drywall up in the sky, and then optimize it by tilting its angle so as it revolves around the earth, it always delivers the maximum amount of illumination.

It's  not hard to calculate that the effective illumination is maximized if the angle of tilt simply bisects the angle between the sun and the earth, as measured from the drywall. It's a consequence of Lambertian diffuse reflection and plain old geometry. So, for example, when the moon, sun, and earth form a right angle, the effective illumination is exactly 50% of what it was in "full moon" conditions: you tilt the drywall at 45 degrees so it intercepts 71% of the sunlight, and because of the viewing angle it's apparent size is only 71% of the full disk. Compound these two effects and you get 50% power. We can sketch the function of effective illumination power as the moon revolves around the earth. We imagine the sun is at the bottom of the picture, and the graph of radiance looks like this:

Look familiar? It's the same graph I plotted last month for the atom deposition pattern of a Stern Gerlach beam in a quadrupole field ! Interesting.

But that's nothing. The sheet of drywall does something very cool as it moves around the circle, always tilting itself so that it delivers the maximum power to the surface of the planet. It starts off at top dead center, square to the sun. As it moves around the ring clockwise, it tilts 45 degrees when it is at 3:00 position; then, in the 6:00 position (between the sun and the earth) it tilts at 90 degrees, parallel to the sun's rays, so it intercepts nothing; moving on, at 9:00 it is tilted 135 degrees to its original orientation, until finally when it returns to home, it is....flipped by 180 degrees! You have to make two full revolutions before it is restored to its original orientation. Now, where in all of physics has anyone ever heard of a situation where you need to make two full revolutions to get back to where you started? (HINT: That's a trick question!)

You better paint both sides of the sheet of drywall.

The Polished Steel Moon

Not everyone knows this, but twenty years ago I used to be the math guy on community access TV here Winnipeg. “Math With Marty” started in 1989 and ran for three years; it quickly attracted what everyone calls a cult following. Just recently I started putting up old video clips on YouTube. I guess it was one of those old shows that got me started on the topic of the moon.

I originally posed the problem in terms of billiard ball collisions: If the moon is a giant billiard ball, and you shower it with regular billiard balls at high speed, how would you describe the distribution of billiard balls bouncing off the moon? You can watch me solve this problem here Actually, in the opening clip, it’s not me solving the problem, it’s my friend Neil, who was co-host and lead guitar player on the show with me. I pick it up toward the end of the clip, and…well, you can see for yourself.

Although I solve the problem in terms of billiard ball collision, it’s obviously exactly the same problem if you consider it as light reflecting off a polished sphere. And the answer is the same: the scattered light is uniformly distributed through space. All angles are equally illuminated. If you can’t see the source beam, you have no way of knowing even what direction it came from based on the scattered light, because the scattered light goes equally in all directions.

Despite its deceptive simplicity, this is not a trivial result. In fact, it is only true in three dimensions, as we see from Neil’s attempt to solve the two-dimensional equivalent. In my way of thinking, when things come out this way it seems to illustrate some kind of cosmic property of the universe. In other words, I don’t know what it means, but it must mean something.

What would the moon look like if it were a polished steel ball? Evidently it would appear equally bright from whatever angle we viewed it, regardless of the relative angle of illumination. In fact, all we would se would be a bright glint of the sun, and we wouldn’t even know whereabouts on the face of the moon it came from. (except that the silhouette off the moon was blocking whatever stars would have been behind it.)

The interesting question is: would this polished ball be a better or worse source of illumination, on average, than our ordinary every-day moon? Bu ordinary and everyday, I mean of course the theoretical moon which ought to be a Lambertian scatterer, an ideal bright-white sheet of paper. In other words, all the light that comes in must go out, just like the polished ball, except now the scattering is diffuse. Cosine-law and all that.

Which one would provide more illumination to the earth, on average? It’s a funny question, and for the longest time I was drawn to the tantalizaing prospect that either moon would be equally good. Actually, in practical terms, the real moon is more useful because it scatters preferentially in the backwards direction, so it is a better night-light than a daytime light. This is in contrast to the polished ball, which scatters in all directions equally, so much of its utility is wasted in brightening the day by an infinitesimal amount. The point is, what goes in must go out, and since over the course of a whole month the moons are on average located at all possible angles with repect to the sun, the earth must receive the same total illumination from either one of them.

Except it’s not quite right. The moon does a circular orbit in the place of the sun, but it is never found, for example, above the north pole. This screws up everything. It’s actually a case where the two-dimensional case has a much tidier solution. For cylindrical earth-moon systems, the average illumination of polished versus difffuse is of course equal. Not for the three dimensional case.

Because of the cosine-law for the scattering angle, the diffuse scatterer keeps more of its illumination in the equatorial plane, which is of course where the earth is. The polished sphere wastes more of its scattered light outside the plane. So if the purpose of the moon is to illuminate the night, the actual moon is actually a better moon than the polished sphere after all. Partly because it keeps more of its light in the plane of the solar system, and more importantly because it is a more effective back-scatterer, so it wastes less power on the daylit skies. Did God maybe know what he was doing when he put it up there?

The unfortunate thing about the “real” moon, the diffuse scatterer, is that I haven’t found any neat and tidy way to do the calculation. I wanted to use its equivalence “on average” with the polished moon to draw some nifty conclusions, but I still don’t know how. The polished moon, at worst case, reduces to a Grade 11 science problem in focal lengths, so I think I can do it. I just don’t know exactly what  I’ll do with it.

A couple of funny things about the real moon. First, in terms of total reflected radiance as an illuminator of the nighttime sky, it ought to have an equivalent polished version. Not the polished sphere: that is clearly different. I’m saying that for some distorted, squashed-down version of the sphere, we should be able to generate a polished surface that has the same illuminating effect as the real, diffuse moon. That would be an interesting thing to calculate. I’m going to guess that it might be a cycloid of revolution, but that’s just a wild guess.

The other point I still wanted to come back to was the one I talked about last time, the departure from Lambertian diffusion. As Wikipedia points out, if the moon were a Lambertian scatterer, it should look darker around the edges and brighter in the center: in other words, it should looks more spherical. One of the most obvious facts of the moon is that it just doesn’t: it looks more like a flat dish than a round ball. Wikipedia explains this as a departure from Lambertian scattering: since the outer edges are just as bright as the inside, the scattered power must be greater at lower scattering angles.

I said last time that I didn’t buy it, and now I have a good reason to back this up. The Wikipedia theory would seem to explain the flat-dish appearance of the full moon, but then it’s a total contradiction with the half-moon, which also appears uniformly bright. If the flatness of the full moon is indeed due to enhanced low-angle scattering around the outside, the the half moon should show even more drastic darkening toward the diametral line, which is the zone of very oblique illumination. In other words, if the scattering is enhanced at the low angles, it must be depleted at the high angles. But the half-moon looks just as uniform as the full moon. You can’t have it both ways.

I still say it’s a psycho-visual effect having to do with the saturation of the eye receptors, the rods or the cones or whatever they are. Some of those photo-shots of the full moon look pretty dramatically spherical. I guess the effect shows up when you balance your light levels properly.

There was one more type of moon configuration which turns out to interesting to analyze, and that’s the big flat drywall cutout moon. After all, when we look at the moon, it looks flat…so why now analyze how it would behave if it were just a big round hunk of drywall stuck up there in the sky. I figured out some cool stuff about it, but I think we’ll leave that for out next post.

Teacher Certification: Let's try this again

I posted this article a few days ago and something really weird happened to the page. Whenever I click on a link to this page, it shows up for about a second, and then goes blank. I'm going to try reposting it here to see if it happens again.
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For those of you who don’t know me, I recently went back to university after many years to certify as a physics teacher. For years I had considered doing this, but was reluctant for two reasons: first, I always thought that I should be involved in things at the university level, because it was so important what was going on there. I just didn’t think high shool mattered that much. Second, the idea of going back and doing two years of university to “learn how to teach” seemed like something of an idignity to me.
Actually, I said there were two reasons, but there’s probably a third: the idea of choosing one single career seems to close the door on all kinds of other possibilities of where life might lead. Maybe I got to a point in life where it seemed like there weren’t all that many doors left regardless, so I bit the bullet and put in my application. I was accepted last year at the University of Winnipeg.

It was the best decision I ever made! The great thing about the U of W post-degree program is that they put you into the schools right away, one day a week in practicum. I loved working with the kids, and my co-op teachers were great about letting me go up to the board and do random topics. But the biggest impression I got from being in the practicum was that I had something different to offer that the kids were desperately hungry for but just weren’t getting from the system. They needed me, and I wanted to be there. All of a sudden, it was important.

The university courses were a bit annoying, but I still enjoyed them. The profs were not very smart, and they really had nothing to tell me about teaching that I didn’t already know. But it was actually fun doing assignments, and even educational sometime. At the U of W they give you a lot of assignments to write up lesson plans, so you have to go through the Provincial Curriculum and see what’s required. I really got to learn the science curriculum inside and out. At the other university across town, they say they had a lot of essays to write about the so-called great philosophers of education: what did Dewey say about this, and what did Piaget say about that? That would have been unbearable for me.

But the profs couldn’t stand me. I was older than all the other students, and I obviously knew a lot more about my subject areas (math and physics) than anyone on the faculty. It seems they all took it into their heads to cut me down to size, to prove to me that I wasn’t as smart as I thought I was. One thing led to another, and the end result was, as of yesterday, I am kicked out of the program.

How did it happen? It’s funny how a negative vibe gets picked up on and amplified. One by one people started complaining about little things that happened in class, until one day the dean called me up and told me I had to report for a meeting. “What is the subject of the meeting?” I was told that there had been numerous complaints about my conduct in class. I replied with what was to become my mantra over the next eight weeks:

“Please put the allegations in writing and I will respond to them”.

Just what were those allegations? That’s a long story, and I’m going to save it for another day. For my former fellow classmates who might be wondering why I didn’t show up for the test this morning, if you happen to be reading this, now you know.

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I think I might want to stick to physics on this blog, so I've set up another blogsite to talk about my situation with the university. Check it out at Due Process, Natural Justice, and the University of Winnipeg

The moon was a ghostly galleon

I started working on this problem the other day and I thought it would be a piece of cake, but now it’s driving me crazy. The question is: how bright would the Moon be if it was a flat cardboard cutout instead of a round ball? Would the full moon look just the same?

It’s a funny question because to me, the moon always did look like a cardboard cutout. I never get a strong impression of roundness, or sphericity. When I look at the moon, it just looks like a flat disk to me. In my mind, I had already assumed that this was some general property of reflectivity, emissivity, and angles: for a uniformly illuminated object, the angle of viewing does not affect the apparent brightness. That was the general principle, roughly speaking, and I pretty much assumed you could go to the bank on that. Was I right?

Here’s a thought experiment. Let’s say you take a big sheet of painted drywall and hang it in outer space directly overhead. When it gets dark at night, have someone (!?) slowly rotate the drywall this way and that way. What can you conclude about the drywall? All you see is a white rectangular shape. No, that’s not even true; it won’t even be necessarilly rectangular. A parallelogram at best. The question is: what can you tell me about the drywall? Can you tell how far away it is? Can you tell how big it is? Can you tell what angle it is tilted at? Can you tell its true shape? Or is it just a featureless white quadrilateral occupying a measurable angular fraction of the sky?

Like I said already, I had this general principle in my mind whereby you’d see exaclty the same white color no matter which way you tilted the drywall, so it wouldn’t make any difference how far away or what angle. That’s why the disk of the moon is uniformly white. It doesn’t matter what angle the sun hits it or how you view it, the sand which covers the moon is all illuminated to the idential brightness, and that’s all you can see from the earth.

That’s what I thought, and I had a little math problem I wanted to solve (I’ll tell you what the problem was eventually) so I tried to apply this principle. To my great annoyance, things just wouldn’t add up. The total available radiant power surely depended on the angle at which the sunlight hit the drywall, so obviously an obliquely oriented sheet couldn’t capture as much sunlight as one tilted straight on. So therefore the difference must be compensated for by the angle of viewing: the sheet which intersects less radiant energy must at the same time present a larger profile to the viewer, and vice-versa. So everything balances out.

But this was crazy! There are two angles in the problem and they are completely independent: the angle of illumination and the angle of viewing. There is something called “Lambertian emission”, or ideal scattering, and it is indeed the principle that something in uniform illumination looks the same no matter what angle you view it from. It works like this: an illuminated sheet puts out, let’s say, 100 watts per square meter per solid radian (figure it out!) when viewed head on, but only 71 watts per square meter per solid radian when viewed from  an angle of 45 degrees. The result is that you see exactly the same brightness from whichever angle you view it: you don’t need to get 100 watts when you view it at a 45 degree angle, because the apparent size of the sheet is is only 71% of it’s true size. On account of the cosine of 45 degrees. It all adds up.

(I’m going to come back later to that business of the watts per square meter per solid radian. It might be important…)

So the illuminated sheet looks the same from all angles. But not if you tilt the sheet! You can move around all you want, and you’ll see the same apparent brightness from any viewing angle. But if you tilt the sheet with respect to the sun, everything changes! You’re not intercepting the same amount of radiant energy, so you can’t expect to have the same brightness. That’s the difference. I got those two things mixed up.

But if that’s how it goes, why does the moon look uniformly bright? On a full moon, the center is illuminated directly by the sun, and the fringes only obliquely. A given square meter of lunar surface near the edges is intercepting less illumination than a square meter in the middle of the disk, so how can it look just as bright? Something doesn’t add up.

I checked Wikipedia, and they actually comment on this very question, and I quote:

“…if the moon were a Lambertian scatterer, one would expect to see its scattered brightness appreciably diminish towards the terminator due to the increased angle at which sunlight hit the surface. The fact that it does not diminish illustrates that the moon is not a Lambertian scatterer, and in fact tends to scatter more light into the oblique angles than would a Lambertian scatterer.”

So Wikipedia agrees with my perception of uniform brightness, and they agree with me that this conflicts with Lambertian scattering. But is their explanation correct? Now, I think Wikipedia is a phemonenal resource, and the quality of science and math is generally first-rate. But this explanation is a little to easy. There is a natural way for objects to randomly scatter light, and it is called Lambertian. The moon, according to Wikipedia, deviates from this natural scattering pattern in some random arbitrary way, and as a result, appears…randomly blotchy?...no!...it appears perfectly uniform! How can this be? It’s too perfect.

Lambertian scattering predicts a global effect whereby the object appears uniformly bright no matter the viewing angle. What then is the concise princible whereby an object might appear uniformly bright regardless of illumination angle??? It’s totally farfetched to think you might get such a neat tidy result simply on account of “deviation from Lambertian scattering”, without some bigger principle at work. In fact, I just don’t buy it.

So I googled images of the moon, and found some beautiful shots. Check out this article buy a guy named Kash Farooq at The Thought Stash . Now look at the moon. It looks round! Yes, we all know it’s round, but I mean it looks spherical! The camera doesn’t lie. Could the flatness we are all used to be the result of psycho-visual effects rather than pure physics?

I think what’s going on is that in the usual viewing conditions, the brightness of the moon saturates the eye’s receptors. Once you hit white, it’s white, and it doesn’t get brighter by adding more white. I have to admit I’m not completely comfortable with this theory, but that’s the best I can come up with.

In Grade Seven we had a poem on the curriculum called “The Highwayman” that begins with the very memorable line:

“The moon was a ghostly galleon tossed on stomy skies…”

I’m sorry if my moon turns out to be just a sheet of drywall, but in physics if you want to get to the bottom of things you sometimes have to reduce things to the lowest common denominator. We’ve got some interesting calculations coming up, and I’m not yet sure if they’re going to work out. I told you there was a problem that got me going on this topic, and it has to do with the surface area of a sphere. The question is: is there anything in the physics of emissivity that relates to the fact that the true area of an illuminated spherical surface just happens to be exactly twice the cross-sectional area? We’ll return to this quesiton on another day.