What is spin anyway?
What is spin anyway?It's a legitimate question to ask not only the first time someone mentions it to you, but often still after a few years of university, simply because people don't often spend much time talking about the foundations of quantum mechanics, preferring to get on and calculate something.
Let's try and come up with a few approaches to answering this question, starting from a non-technical point of view, and then starting to introduce some maths.
Let's start off with something we basically understand - classical mechanics. Here, there is a simple notion of what it means for something to be spinning around, and what's more there is a standard way of measuring it called angular momentum. Just as there's the more familiar (linear) momentum which means that 'on average everything keeps moving in the same direction at the same speed', angular momentum 'keeps everything spinning in the same direction at the same rate'.
What do we mean by this? One of the classic examples is to spin around on a swivel chair with your legs stuck out, then bring them in close to the centre of the chair. You speed up, because the angular momentum of things going round in smaller circles is smaller, so you have to speed up to compensate.
Similarly, just as when two things hit each other their individual speeds can change but their total momentum stays the same, every time two spinning objects touch each other, they might change their spinning speeds, but the total angular momentum will stay the same. For example, every time you start turning the wheels of your car, you spin the earth the opposite way.