Quantum Mechanics and the Area of a Sphere

I told you two weeks ago that I thought there was some kind of cosmic connection between quantum mechanics and the formula for the surface area of the sphere. I got half-way through my description of a hypothetical spin-11 system when I decided to take a little break. Now it's two weeks later and I'm still pretty convinced there's some kind of connection there, but I can't quite nail it down.

Here's the thing. Quantum mechanics tells us you can fully describe the spin state of a spin-11 system by specifiying exactly 23 numbers, corresponding to all possible z-axis spins between plus and minus eleven.  Be careful: I'm talking about a system where we know for a fact that the total spin is eleven. We just don't know the z-component. More specifically, we don't know the distribution of the z-components. There's nothing in quantum mechanics that says the system has to be in exactly one of those z-spin states.  It just says that the complete spin state is fully described by listing those twenty-three numbers. (Complex numbers, actually, but that's besides the point.)





Notice that this "total spin state" is something much more complicated that what we call the classical angular momentum. This state we've created certainly has an angular momentum in the classical sense, but there's nothing that says it has to equal 11 Planck units (the dimensions of Planck's constant are the same as angular momentum), and there's nothing to say that it has to point in the z direction. This quantum system still has a "classical" angular momentum; if we know the 23 spin parameters we can easily calculate the classical spin;  and it can be anything from zero to eleven, and it can be oriented along any axis.

Now I'm going to do something which I suspect is experimentally impossible but I really don't know: I want to restrict myself to cases where the total classical spin is 11 units, or at least very nearly so. This is a tiny fraction of all possible spin-11 systems. And then I want to pick a random axis in 3-dimensional space and call it my z-axis. And then I want to list the 23 complex numbers describing the spin state. Actually, at this point I want to convert from amplitudes to probabilities, which means squaring out the complex quantities so I have a list of real numbers between zero and 1.

Now I want to do the same thing for different random choices of the z-axis. So I have a hundred different lists of 23 numbers, numbered from +11 to -11. And now I want to ask: what are the average values? Is the average of spin-7 higher than the average of spin-4...or all the all simply equal to 1/23?
Remember, they can be interpreted as probabilities, so they have to add up to 1.

And here at last is the connection. If...and only if....the surface area of the sphere is equal to exactly four times the cross-sectional area...then all the probabilities must equal 1/23. I think I'm correct in this, but I'm not about to argue the steps. Qualitiatively, I'm trying to say that all z-values of spin are equally likely; but it turns out its very difficult to say precisely what you mean by a statement like that. I don't want to make things more complicated than they have to be, but this is honestly the best I've been able to do so far. I think it's some kind of cosmic connection but I'm still not quite sure...

Jewish Lightning

I write a biweekly column in the local Jewish newspaper, mostly about Yiddish-related topics. Bernie, the editor, gives me quite a lot of slack, but even so I was pleasantly surprised when he let me run this article. I thought I'd repost it here for your enjoyment, as a brief diversion from the physics.
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Jewish Lightning

They say the Eskimo has forty-three different words for “snow”. This may be an exaggeration, but it illustrates the point that a language will evolve to reflect the things that matter to a particular society. In this light, it is fitting that the Jews should have a special word to denote one who has lost his posessions in a fire: acordingly, from the Hebrew, we have in Yiddish the word nisraph.

A Nisraph is the title of a humorous piece by Sholom Aleichem. I have translated here a short excerpt:

 “I come from the village of Boslov. A nice little place. The kind of place where you show up with your pockets full and leave with your pockets empty. You know how they send people to Siberia when they want to punish them? Better they should save the trainfare and send them to us in Boslov instead. We'll know how to treat him. First we'll set him up in a little shop, then  we'll give him a line of credit so he can fill it up with merchandise, and then, when his shop burns down leaving him with nothing but the shirt on his back...we'll jump up and down and point our fingers at him, and shout: "Jewish Lightning! Jewish Lightning!"

Now, at some time in our lives, most of us have heard it said that Jews burn down their stores to collect the insurance money. We rightly consider this accusation to be just another vicious anti-Semitic slander. But if you’re like me, you probably thought that it was a New World invention; a sort of milder 20^th-century adaptiation of the classic Blood Libel, a fable which might have played in Kiev or Odessa but would have been a little too medieval-sounding to attract much credibility in Chicago or New York. Nevertheless, it’s clear from the above passage that we carried this stigma with us even in the Old World.

So how does such an anti-semitic slur come to be the topic of  a satirical piece by Sholom Aleichem? To understand this, we must delve into the original Yiddish text. Now, "Jewish Lightning" is admittedly a very picturesque expression; but of course, that's not what Sholem Aleichem uses in the original. The expression he uses is so Jewish and so quintessentially Yiddish that it deserves a full explanation.

"Borei me-orei ha-eish" means, literally, "blessed be the kindling of the fire". It is from the prayer recited on Saturday night for the lighting of the Havdallah candles, marking the end of the Holy Sabbath and the return of the Gray Week. Now, the Bible is often praised for its poetry, but the fact is in the original Hebrew, the poetry consists almost entirely of the use of imagery and metaphor. Actual rhyming poetry, and especially rhymes combined with metrical rhythm, is so rare that one has to consider its occurence to be almost accidental. And yet those instances of accidental rhyme and rhythm are some of the most compelling lines from the Bible and from the prayer liturgy. "The mighty hand and outstreched arm: yad khazaka u-vizroa netuya." "Borei me-orei ha-eish" is certainly another such instance.

Furthermore, one can readily see how the magnificent roaring flame of the  triple-wicked Havdallah candle, so unlike the steady, modest glow of the ordinary Friday-night Sabbath candles, would have inspired in the imagination of the Jewish Merchant of Old Russia nothing so much as the image of a warehouse, chock full of merchandise and insured to the hilt, going up in flames. We are, after all, a poetic race if nothing else.

Which brings me to my final point: if we are allowed to think that as a race, we Jews are smarter than everyone else (don't deny it! you know we do!)...then aren't we ALSO allowed to admit the possibility of other, less praiseworthy tendencies? It's nothing to hide or be ashamed of...it's just one more aspect of the complicated, intricate enigma that is who we are. 

How to Caluculate the Temperature of the Sun

I told you last week that I thought there was some kind of cosmic connection between the laws of physics and the suface area of a sphere. In particular, I thought that there was something special about the exact ratio of 4:1 between the cross-section of the earth's disc and the total surface area of the planet. It turns out that I was misled by some rather surprising numerical coincidences. Let's recall how that worked out.

It all the fact that the angular diameter of the sun is very close to one hundredth of a radian. That's a nice round number. If we take it as being exact, it has the interesting consequence that the sun occupies a fraction of the total sky amounting to one part in 80,000; or, if you count the "total" sky as being both the day sky and the night sky, one part in one hundred sixty thousand. The fact that it comes to a nice round number goes back to that exact ration of 4:1 between the disc and the sphere.

Then we notice that 160,000 is a perfect fourth power...namely, 20 to the fourth. It happens to be a law of thermodynamics that a black body radiates heat according to the fourth power of absolute temperature. And as I showed last week, that means that, assuming the sun is a black body, it ought to be exactly 20 times hotter than the earth. Actually, it doesn't even have to be a black body...a "gray" body does just as well. Either way, it comes out pretty close...if the average temperature of the earth is 300 degrees K, that gives us 6000K for the sun, which is pretty close. (The notorious "greenhouse effect" throws things off a little, but not enough for us to worry about here.)

These are very cool calculations...but when you look them over, they really don't depend in any critical way on the 4:1 ratio. Except that having a nice integer ratio makes the numbers come out more nicely. Other than that, the physics of the calculation must hold true whatever the geometric ratio between 2-d and 3-d area. You really can't calculate the area of a sphere by taking careful measurements of the temperature of the earth and the sun.

And yet...I still can't get it out of my head that there is some kind of cosmic connection between the physics of the universe and that 4:1 ratio. If not in the realm of thermodynamics, then what about quantum mechanics...specifically, the nature of angular momentum?

We've all heard about how angular momentum is quantized: that an electron can have either "spin up" or "spin down" but nothing in between. I should put in a disclaimer here: even though "everyone knows about that", the fact is it's not true. The spin of an electron can be aligned in any possible direction. What quantum mechanics actually tells us is that no matter what direction the spin is aligned, we can treat the electron as being in a superposition of two states: x amount of "spin up", and y amount of "spin down".

We are further told that in quantum mechanics,any system has a certain property called "total spin", and that quantity must be an integer or half integer. For an electron, the total spin is 1/2. For a random collection of 22 electrons, the "total spin" must then be....eleven?

Not necessarily. The total spin of such an ensemble can take on any integer value from zero to eleven. So they tell us...


But once again, this isn't true. It's not even true that a system of two electrons can have a total spin of either zero or one. What is true is that the physical state of the two-electron system can be fully described as a superposition of two systems, one of which has spin-zero, and the other of which has spin-one.  There is nothing in the physics that restricts the relative proportions of those two states.

Furthermore, the spin-one state is itself not fully described simply by the fact of its total spin. A full description of the spin-one state requires in addition, three more parameters to completely specify it.

You have considerable freedom in choosing which parameters you want to use. But one interesting  choice is to choose z-axis spin, where z is an arbitrary axis chosen in one specific direction. A spin-one combination of electrons is totally described, in terms of its spin state, by specifing:

1. it's total spin (spin one in this case)
2. its z-spin=1 component;
3. its z-spin=0 component; and,
4. its z-spin=-1 component.

Similarly, a box containing 22 electrons has a spin state which is a combination of total-spin 0, 1, 2, 3... all the way up to 11. I don't know of any way of preparing a box of 22 electrons so its total spin is 11. I would guess that it may be impossible. But in any case it is at least theoretically possible that such a box of electrons might randomly find itself in a state where its total spin was eleven. The probability of such a coincidence must be vanishingly small, but there it is. And in that unusual circumstance, you would then still need 23 more numbers to describe the actual spin-state of the system:

1. z-spin = 11
2. z-spin = 10
3. z-spin =  9
*
*
*
23. z-spin = -11

And there it is. It is a fact of quantum mechanics that any system whose total spin is exactly eleven can be completely described, as far as its spin state, by listing the 23 numbers (complex numbers, by the way) corresponding to z-spin of 11, 10, 9.... all the way down to minus eleven.

And in my opinion, there would be very serious problems with such a description were it not for the mathematical fact that the ratio of the surface to the disc happens to be exactly 4:1 for a perfect sphere.

Let's talk about that when we return.

Cosmic Connections

There is a funny thing about the area of a sphere. If you look at the moon, you see a certain cross-sectional area. But the actual area of the moon is exactly four times what you are looking at. It's a funny thing, that factor of four.

If you know the area of a sphere, the volume is "trivial", as the mathematicians like to say. From the formula for the volume of a cone, 1/3(base)x(height), you get the volume of the sphere. The surface of the sphere is simply the "base" of a generalized "cone". Archimedes famously determined the volume of a sphere by a very ingenious and very different argument involving a cone inscribed in a cylinder. I don't know if he knew about the ratio of 4:1, but that's another pathway to the volume formula.

Of course it works both ways: if you know the volume of a sphere, as Archimedes did, you can back it up to get the formula for area. Again, I don't know if that's what Archimedes did. It would be nice to find out.

But without solving first for volume, is there an obvious way you can derive that special 4:1 ratio? It turns out you can indeed argue it from geometry without too much difficulty. It starts off looking a lot like Archimedes' argument: first, you incribe a sphere in a cylinder. And then you take a disc-shaped slice perpendicular to the axis of the cylinder. This is just like Archimedes so far. But then, to calculate volume,  Archimedes inscribes a double-cone inside the cylinder, and it gets pretty intricate. For the surface area, you don't need the cone. You just compare the areas of the cyldrical section and the spherical section, and it's easy to see they are equal. From this, you immediately get the surface of the sphere.

For a long time, I've had the idea that this ratio had some kind of cosmic connection with physics. I don't remember where I got this idea, but think it might have been a pure accident. See, there's a funny thing about the apparent size of the sun. The apparent diameter of the sun, as viewed from the earth, is pretty close to one hundredth of a radian. That means if you hold your hand out 50 centimeters in front of your eye, and stick out your pinkie finger, the sun will span about half a centimeter as measured across your fingernail. One percent.

You have to be careful not to compare apples to oranges, or in this case radii to diameters. Your arm is a radius and the sun is a diameter, so you the true ratio is of course 1:200. Since area goes as the square of the linear dimension, that means the sun occupies one part in forty thousand as compared to...the area of the sky? No, because we don't yet know the area of the dome. We're really comparing flat circles, which would be the equivalent flat area of the sky. The beauty of the 4:1 business is that we can immediately see that the area of the dome is exactly twice the area of the corresponding disk; so it follows that the ratio of the sun's area to the total area of the sky is 1:80,000 which is a pretty cool result.

Here's where it gets cosmically weird. The dome we see is exactly one half of the actual "sky", because there is just as much sky on the other half of the world. So the area of the sun to the total sky becomes 1:160,000 or exactly half of the visible ratio.

The funny thing is that number 160,000 happens to be a perfect fourth power: specifically, it is 20^4. You know it's not that unusual to hit a perfect square. Perfect cubes are not that common. Perfect fourth powers are pretty rare...obviously, there are only 19 of them smaller than 160,000. But so what?

Fourth powers are not only a bit rare among natural numbers, but even more rare in physical laws. Most laws of physics have squares in them, but there is a law of thermodyanamics that says a black body radiates heat according to the fourth power of absolute temperature. So if you make something twice as hot, it radiates not twice as much heat, not four times as much heat (square law), but sixteen times as much.

Now we're going to put it all together. The temperature of the earth is around 300 degrees Kelvin (absolute scale). The earth is therefore radiating heat into the vastness of space according to the Laws of Thermodynamics. Unless that heat is being replaced by an equivalent source, the earth must therefore be cooling down. The fact that we aren't cooling down tells us how much heat we are aborbing from outer space.

If outer space consisted of a gigantic dark sphere at a temperature of 300 degrees Kelvin, we would obviously be in thermal equilibrium. We would be radiating heat out to the giant sphere, and it would be radiating heat back to us. Both bodies would be radiating heat at the same rate. But in fact there is no giant sphere out there, just the endless vacuum. So we are getting nothing back.

Except for this tiny patch of the sky occupied by the sun. Since it is only one part in 160,000 of the total sky, and it is obviously doing the whole job that our hypothetical giant sphere was otherwise doing, it must be giving off power at a rate 160,000 times as great as the black body of 300 degrees. But in that case, we know how hot the sun is! Knowing that a black body radiates heat according to the fourth power of absolute temperature, we take the fourth root of 160,000 and find that the sun is exactly 20 times as hot as the earth, or pretty close to 6000 degrees Kelvin.

And that's damn close to the actual temperature of the sun.

Dimmer Circuit Puzzle

Have you ever had your kitchen dimmer switch turned down really low, and then the refrigerator cuts in and the lights go right out? The funny thing is that when the compressor turns off, the lights don't even come back on again. You've got to turn the dimmer up quite a bit to bring them back.

It works best in older houses where the fridge plug is on the same circuit as the overhead light, but you can often see it quite clearly even when they are on different circuits. I was at a friend's house the other day and I was showing them how this works, when something very unexpected happened. The fridge cut in and the lights got brighter. I thought we were imagining it but we watched for several cycles, and it kept happening. Then we tried turning on the microwave, and the lights got dimmer. But the fridge definitely made it get brighter.

I think I figured out what is going on, and the answer is kind of interesting. It's really a two-part question: part 1 is just why is a dimmer switch turned down low so very sensitive to small changes in line voltage; and part 2, why the anomalous result of the lights getting brighter?

I'm including a simplified schematic of the dimmer circuit for your edification: I'll give you my answer when we return.

(I found the schematic on this very impressive website.)

Curriculum? What curriculum?

This is the final installment of my story of how I got fired from UCN back in 2006. My inexplicable refusal to follow the curriculum was the reason they gave for letting me go. But there's an ironic twist to that story....

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That just about covers it except for one small item you might still be wondering about: the mysterious eighth paragraph. I have said already that in seven paragraphs out of eight I am accused of not following the curriculum. What about the eighth paragraph?

In this paragraph Henning describes my very first day of work, going over how I was shown filing cabinets full of material including worksheets, old tests, blueprints, etc. It happens to be the longest paragraph of the entire letter, and yet it doesn’t include any particular complaints against me.  So what is it doing in my letter of rejection?

The eighth paragraph purports to show that the College provided me with all the resources I needed to do my job: which is to say, they basically shoved me in front of a filing cabinet and said “Knock yourself out”. But oddly enough, in the long itenerary of resource materials listed by Henning, there is one item conspicuously absent: a copy of the apprenticeship curriculum! I became aware of this deficiency in November and immediately wrote my supervisor requesting that one be provided. Selwin ignored my request; or to be more precise, he first said that he would get it for me, later said there were complications (what happened is that Murray actually freaked out when he heard I was asking after the curriculum!),  and finally never followed up at all, which was typical behavior on his part. (Everyone who has ever worked with Selwin knows this is true.) The ultimate irony is that the very curriculum which I am accused of not following is a curriculum which the College wouldn’t give me. 

In Which I Call A Spade a Spade

For the last two weeks I have been re-telling the story about how I was fired as an instructor from the University College of the North. Today's installment is about an incident that did me untold damage in terms of my reputation, but which ultimately was not even mentioned in the letter which listed the reasons for my dismissal. The reason it didn't make the list is because the College knew it was a malicious slander from the get-go. The report by independent consultant Joyce Oddidison was subsequently destroyed, but the damage had already been done. Here then is the continuing story...

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I suppose in the end it’s all very well for me to write a self-serving letter which makes me out to be the hero and everyone else to be the villains. But what does it really prove? Ultimately, you can say that I was probably fired because I was a bad teacher. The point is, that’s not what the College chose to say. They said that I wasn’t following the curriculum, and that I was repeatedly warned and given a chance to comply. That is a lie and a slander which Katherine McNaughton, Selwin Peter, and Murray Oman conspired to establish and maintain, and to which Denise Henning has signed her name. The truth of my claim can be clearly seen in the patently absurd examples which the College has cited as so-called evidence of my inappropriate teaching.

I wasn’t fired because of the smokestack or the water heater or even the slope question. It’s almost as though I was fired simply because something about me inspires a profound, inexpressible loathing among certain people; that in an institution which claims to value “diversity” as its highest ideal, I was just too different. And the best excuse they can come up with is the smokestack issue. In fact, over the course of the year there was a never-ending stream of malicious gossip about me, including at least two claims of sexual harrassment! All of these were investigated by management in a desparate attempt to throw dirt on my reputation and  none of them were found to have any validity. That’s why they’re not brought up in my letter of rejection. But at least one of these incidents deserves further mention, because everyone on staff at the College heard about it at the time and I would like to set the record straight.

Elvis Balfour was a student of mine who lost his temper in class one day for no sensible reason, jumped over the desks and began shaking me by the throat. I remained calm and put up no physical resistance. When it was over, I told him to stop acting like a baby and get back in his seat. Subsequently through no action on my part, by the end of the day Elvis had been expelled from the course by Murray Oman.  

What happened next is confusing: Murray has been an instructor for many years and has routinely expelled students in the past without controversy (including the very same Elvis Balfour on a previous occasion). For unknown reasons, this time the College chose to disregard its own zero-tolerance policy on violence, refused to ratify the expulsion and instead relocated Elvis to The Pas where he was allowed to finish the course. An outside consultant by the name of Joyce Oddidison was hired by the College to investigate the affair, and she ultimately produced a report that blamed me for provoking the incident.

Oddly, Ms. Henning chooses not to mention the Elvis Balfour incident in her letter. Why, when she is clearly trying to make the strongest possible case for my dismissal, does she not invoke this glaring example of  misconduct? The answer is simple: there never was any misconduct on my part and Henning knows it. The report by Oddidison is a pack of lies, which she was hired by Selwin Peter to produce in order to justify his own idiotic decision to keep Balfour in the course.

Most notably, the argument Oddidison uses to condemn my behavior is based on a shameless doctoring of my own written submission. In particular, she places my words, “Stop acting like a big baby...” ....BEFORE the violent outburst rather than after, turning it into a provocation. Perhaps she thought I was lying. If so, she should have said so, and introduced testimony from other witnesses to back up her conclusion. That’s not what she chose to do: instead, she falsely manipulated my own words to make me guilty by explicit admission!

The College was an active participant in this slander and they know it to be false. The proof is simple: if they believed their own report, they would have included its conclusion in the letter which gives the reasons for my dismissal. Instead, they say I was fired for measuring the height of a smokestack. What jackasses they are.