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.