Thinking about angular momentum

For many years, I have felt daunted by the quantum structure of nature. Don’t get me wrong – I studied it in lab class and I read a lot about it in my textbooks. It’s one thing to repeat an old experiment, or read a book; it’s quite another to be involved in a physical system where it’s do or die – get it, or don’t get it.

The bottomonium system seemed very mysterious at first. The upsilons, the etas, the h’s, the chis. So many Greek symbols, so many states, such a seeming mystery. I know I’m supposed to “just get” this stuff, and many of my colleagues do. But I also know who I am, and I am not afraid to accept that sometimes it takes a lot of time for a concept to puncture my cranium and sink comfortably into my brain.

Since I’ve been going around and giving all these seminars, I’ve had a lot of time to think about angular momentum. In nature, there are several kinds. The first is the most familiar, and it’s called “orbital” angular momentum. Find a friend, lock arms, and rotate yourselves around a common center until you puke. That’s orbital angular momentum – the rotating, I mean, not the puking. I guess the latter is linear momentum.

The other kind is internal angular momentum, commonly known as “spin”. It takes its name from the analogy that it’s a quantity that seems inherent to a particle, as if that particle were storing energy in its own rotation around an arbitrary axis. But the particle is not, in fact, spinning. That’s where the classical analogy breaks down. Do the calculation of how fast the electron, for instance, would need to be spinning and you quickly find that it violates a lot of what we know to be true about nature.

I’ve come to think of spin as a kind of vertigo. If grasping arms and physically rotating in space is orbital angular momentum, that sense of rotation you get in your head when no physical rotation is actually present – vertigo – is much like spin. It’s as real as actually spinning when you find you can’t walk straight as a result, and it plays a real role in defining how you interact with your environment.

Spin and orbital momentum, together, create a rich structure in even the simplest multi-particle quantum systems. It is the richness, evolving out of just a few particles, that defines the spectrum of states in a physical system. Let’s go back to bottomonium. Like the system from which bottomonium takes its name – positronium, an orbiting pair of an electron and a positron – this system has a ground state, and a multitude of states above that with unique combinations of orbital and spin angular momentum.

For example, positronium’s ground state is also called “para-positronium”, where the spins are oriented opposite one another in an antisymmetric configuration and the electron and positron have no orbital angular momentum. para-positronium is the state which, when formed, annihilates into a pair of gamma rays – this is the basis of a PET scan. If the spins align with one another, or are anti-parallel in a symmetric state, then we find ourselves in the ortho-positronium state. This lives much longer, because it has to annihilate into at least three gamma rays to conserve total momentum.

In the bottomonium system, the ground state is called the “eta” (instead of “para-bottomonium”, I guess) and borrows its nomenclature from its cousin quark, the charm quark, and her quantum state names. Because quarks are ruled mainly by the strong force, when the eta decays it does so into a pair of gluons instead of a pair of photons. The ortho-bottomonium state, called the “upsilon”, lives longer because it must decay into at least three gluons.

Nature is full of symmetry. The symmetry between positronium, its structure the result of the richness of electromagnetism, and bottomonium, its structure due to the strong force, strikingly unifies ones understanding of the two systems. There are key differences – coupling in electromagnetism is an order of magnitude weaker than coupling in the strong force – and this changes a lot of realities. But, fundamentally, if you can think about positronium, or the hydrogen atom, you are equipped to think about bottomonium.

The more I dig into the data of quantum mechanics, the less I think I do not understand it.

By steve

I am a husband, son, and a Professor of Physics at Southern Methodist University. Physics may be my favorite thing to do, but I like to do a little bit of everything: writing, running, biking, hiking, drumming, gardening, carpentry, computer programming, painting, drawing, eating and sleeping. I earned a Ph.D. in Physics in 2004 from the University of Wisconsin-Madison, I teach courses in physics and the scientific method at SMU, and I love to spend time with my family. All things written in here are my own, unless otherwise attributed; don't you go blaming my employer or my family for me.