Having trouble counting spin states

There is something funny about the fact that spin is supposedly "quantized" along the z axis. The thing is that it only really "works" because of a funny geometrical property of spheres. Let's say you have a spherical apple and you cut it into 1/4-inch slices. Everyone knows that you get a different amount of apple in each slice, but the funny thing is...you get the same amount of peel. What this says in quantum mechanics is that if you have a spin-11 system whose z-axis spin is randomly distributed among all values between plus and minus eleven, then those random spins are equally distributed among all spherical directions. It's a property of spheres that is intimately related to the fact that the surface area of a sphere is exactly four times the cross-sectional area.

I started trying to write about this...oh, maybe two months ago. I finally cobbled something together and posted it but I swept something under the rug that is very embarassing. It turns out I don't know how to count spin states. I thought I did, but I didn't. I'm going to try and explain the problem.

If you have one electron, there are two spin states. People say the spin must be either up or down, but that's nonsense. The spin can be in any direction, but those directions are expressible  as the superposition of an "up" state and a "down" state. The total spin is one-half.

If you have two electrons, there are now four spin states. Both up, both down, and one-up/one-down. Wait...that's only three spin states. Not exactly...it turns out that depending on the relative phase of the up-down combination, they can form a spin-zero state. They don't have to...they can form a spin-one state in a superposition of left- and right-spin. That's not spin-zero. There is a unique spin-zero state that has no spin anywhere, called the "singlet state". I've written about it here.

So for two electrons, you can describe the total spin by giving the complex amplitudes for the up- and down- spings of both electrons...that's four "basis" states. Or you can be more "elegant", as they say, and describe the total spin as the three amplitudes for the triplet states...that is, (+1, 0, or -1) along the z-axis, plus the amplitude for the singlet states. Either way, it's four basis states.

With three electrons, it's the same. You have now six amplitudes to describe the individual electrons. Or, you can group them in states...they can form either a spin-3/2 state with four z-axis levels, or a spin-1/2 state with two z-axis levels. Six states.

With four electrons...now bear with me...it's the same.; You now have eight amplitudes to describe the individual electrons. Or, you can group them into spin-2 states (five z-axis levels), spin-1 states (3 z-axis levels), and a spin-zero state. Eight states.

No...wait. There are nine states if you group them into total-spin states, and only eight if you just list the individual electron states.

And it gets worse. For five electrons, you have ten numbers to describe the individual spins. Or you can group them by total spin: spin-5/2 (six states), spin-3/2 (four states), and spin-1/2 (two states).

I don't think you can have more spin states than you get by listing the states of each individual electron. But at the same time, I don't see why there is any redundancy in describing a box of 21 electrons as having such-and-such amplitude to be l=11, m=6, such-and-such amplitude to be l=9, m=7, plus whatever. Because it works perfectly for a two-electron system. You have such-and-such amplitude to be in each of the triplet states, plus an amplitude to be in the singlet state. Why does this system break down when we add the fourth electron?

So there it is. I don't know how to count spin states. But at least I know that I don't know how to count them. That counts for something, doesn't it?

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.