How can I conceptually interpret spin precession?

Question

In Sakurai section 2.1, he works through an example of spin precession where we come to the conclusion that, when working in the z-basis and applying a magnetic field in the z-direction, the spin precesses in the xy-plane. I am wondering how I can conceptually interpret this precession?

Answer 1

It’s essentially the same as when rotating objects precess. For objects which are actually spinning, there is some axis about which they spin. This axis is likewise described as the direction of the angular momentum of that object where the sign denotes which direction they are spinning (right hand rule and all that jazz). Precessions are just when this axis is not stationary, but rather is also rotating. If you google “spinning top precessions” and look at the videos, there should be some good examples of what precession means physically in the context of objects actually spinning.

Now for elementary particles, their ‘spin’ is not quite the same as they are not actually physically spinning, but they still do have angular momentum which is what the spin is accounting for. Essentially by saying that a particle has spin, all we are saying is that it has some intrinsic angular momentum (but again, it isn’t actually spinning). As with before, this angular momentum is defined in some direction which classically would be the axis in which the thing is rotating about, but in the quantum world, we have other issues to worry about, mainly the Heisenberg Uncertainty Principle.

Because of Heisenberg, I am not allowed to know my particles momentum exactly, and this includes the angular momentum. As you likely saw in the calculations you are referring to, we can measure our angular momentum in one direction and have absolute precision in that measurement simply because angular momentum occurs in three dimensions, and with information from two directions still missing we are left not knowing the exact angular momentum of our particle and thus are safe. This is where the precession comes from. We know the magnitude of our angular momentum, and ‘how much of it’ is in the axis we choose to measure it in (the z axis in your case), but we don’t know what’s going on in the other two directions (x and y), so by Heisenberg, our particle is essentially allowed to distribute its “remaining” angular momentum anywhere it likes in the x and y directions respectively, thus it precesses about the z axis since the Heisenberg Uncertainty Principle tells us that we can’t know exactly how much angular momentum it has in those directions (note that I know the total angular momentum and the angular momentum in the z axis, so if I knew how much angular momentum was in either the x OR y direction, I would know the angular momentum completely since knowing x would give me y and vice versa, so I can’t know either).

The physical intuition for precessing is as I described it at first for classical spin, but for particles it’s more abstract since they aren’t actually spinning. Basically the precession comes from the fact that particles have intrinsic angular momentum (spin), but the Heisenberg Uncertainty Principle tells us that we can’t know exactly where that spin is, so if I measure it about some axis, it must precess about that axis in order to ensure that we don’t know where it is. In other words it just comes from our uncertainty about the spin, but this is an intrinsic uncertainty, not just an uncertainty so to bad measurements. Hopefully this long winded answer helps somewhat.