The Road Goes Ever On

A bit more math. Has anyone ever heard of Clebsch-Gordan and Racah Coefficients? No-one outside quantum mechanics, I expect, as only quantum mechanics uses them. We certainly need them and they are very complex indeed.

There are no numbers, just symbols. Forgive me if I cut and paste from the 'net as there simply is not any other way to show them here.

These are the Clebsch-Gordan Coefficients.



These are the Racah Coefficients.

Before I lose the reader completely, let me tell you a story.

It begins with Newton's Laws, the first one to be precise. Harking back to my post on these laws, it states that "an object moving will keep on moving, or if at rest, will stay put unless moved"

We say a moving object has "momentum" and a stopped one, "inertia". The above math is all about momentum, and we measure momentum (p) most easily by saying it is " mass (m) x velocity (v) " . However the above is not about "linear" momentum, which is "mv", but "angular" momentum, which is defined similarly, thus: Angular momentum L is (angular inertia) I x (angular velocity) w

These are roughly the same as their linear counterparts. I is the equivalent of mass, and w is the equivalent of velocity.

I is a bit difficult to understand, it is the "moment of Inertia" of a body, and different bodies have different I's, some of them even have more than one. Here is a table to illustrate:


These objects have 2 distinct moments of inertia depending on which way you spin it.

Notice earlier how the words "velocity" and "angular velocity" were used. "Speed" and "spin" were not used because velocity is what we call a "vector" - it depends which way it points. Example, my car's speed is 30 kph no matter which way I turn it. However, if I drive north at 30 kph for an hour, then turn round and go south at the same speed for an hour, I will end up where I started. The speed was the same, but I used two different but equal (and opposing) vectors that cancelled each other out.

This, in essence is what these coefficients are all about. Example, if I take a vector east and travel for 12 km, and then south for 5 km, I won't finish 17 km away from the start, but 13km. (how do I know this? Try Pythagoras' Theorem). So when you add vectors, you have to notice which way they point and do the geometry.

Quantum mechanics does this, but at this level, momentum and angular momentum are quantised (only come in little bits at once), so special math are needed to do the geometry. These are the Clebsch-Gordan and Racah Coefficients.

Well, that is the background, angular momentum and vectors. Now for the rub.

The key to the idea is this. It is best to go back to my East-South car journey. How many ways are there to get from the start to the finish? I already gave one, but there are others, 5 km west, then 17 km east, and 12 km south is another. But the best way is the original one, 12 then 5 km.

Quantum mechanics, though, has to deal with all possibilities of the combinations of these vectors, and though none are equally likely, we need math to tell us how likely the car would would go West5, East17 (Brian, sit down!), South 12.

The Clebsch-Gordans tell us how many ways we can do it with two vectors (E - S , hence the j1 and j2), and the Racahs tell us how to do it with three (W - E - S, hence the j1, j2 and j3). The twiddly bits j12 and j23 just ask us which is first.

The "| >" , "< |" and "< | >" bits are actually indefinite integrals (yes, calculus) and the big mouths (the big bold symbols after the equals), are simply the adding up of all the bits we know about that follow after it.

That is the interpretation of the above. "Coefficients" is the name of a number we put in front of something we don't know to tell us how much to multiply it by.

The best way to put it is this: How do we get from A to B quantum mechanically? Answer: Add up all the probabilities of all the different ways we can do it to get "1" ( it is certain we will arrive at B). In this case, we have several quantum mechanical angular momenta, and we want to add them up. These formulae tell us how to do it and get an answer as to how probable it is that we will end up with a certain answer.

We don't know what it will be, but essentially we plug in the numbers for the J's and m's and M and we will get a big J. The j's are the individual angular momenta, and m's are the spin angular momenta, and the J and M are the probability of getting a certain total at the end from all the others.

Plug in the numbers, turn the handle, and out come the answers.

This is once more just an introduction, and the Road Goes Ever On.