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.