Teaching SPIN with First Year Physics

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Today we’re talking about SPIN. Quantized Spin. You know, spin in the sense of quantum mechanics. Let’s get to it!

From spin to SPIN

Spin is an easy concept to sketch, but a hard concept to explain. Quantum mechanics is hard, sure, but there’s also a lot of details that can be easily conflated and confused. Angular momentum and Magnetic dipole moments are the main ideas in today’s discussion. And the way we teach students about both just doesn’t line up with the reality of quantum mechanics. But it ALMOST does. It’s frustratingly close And that’s what’s so tricky about introducing the idea of SPIN.

So let’s start from the beginning, and add texture and nuance as we go.

Spinning Tops

All kinds of objects spin. Tops spin. The earth spins. Baseballs spin. Frisbees spin. Hurricanes Even ice skating athletes spin.

These classical, mechanical notions of spin are a rotation around an axis of symmetry of some object. The more complicated the object, the more nuanced the dynamics can be. Spheres are easy. They have a whole lot of symmetry, and relate better to point particles, so we’ll just stick with spheres.

Consider a spinning ball. What’s it’s angular momentum?

Well, angular momentum is a vector whose magnitude is proportional the ball’s rate of spin, and whose direction is along the axis of symmetry. You can find the axis of symmetry by drawing dots on the ball and watching them move. Most of them go around in circles. But exactly two such dots don’t: you might call them the north and south poles.

If you were to put an ACTUAL pole through the ball from south to north, you could still spin the ball and nothing would really change. That imaginary pole is directed along angular momentum vector. By convention the direction points from south to north.

So when we think about the angular momentum of a spinning ball, we think of an arrow sticking out of that ball.

Now. The angular momentum of that ball is KIND of like the angular momentum of an atom. Or a proton. It’s ALMOST the same, but not quite. There’s nuance.

We often think of particles - like atoms or protons - as tiny little spheres. They’re not, of course, spheres but from really far away particles can look like points. An by that we actually mean something really precise: particles often have spherical symmetry.

But a particle’s spin breaks that symmetry a little bit. A sphere that was completely still would certainly have that symmetry. But a spinning sphere has angular momentum, it has that imaginary pole. It’s got that VECTOR pointing in some direction. That vector is a preferred direction. It’s breaks the spherical symmetry. It’s a subtle point, but in a way, it IS a point that we teach to our students when we teach electromagnetism.

Dipoles

The second semester of a physics course is typically all about electromagnetism. Electric fields and magnetic fields. Now, full blown electromagnetism is actually quite hard - even for experts - so we usually only teach students about very symmetric, idealized systems. Points, lines and sheets of charge. Loops and lines of current. You know, those kinds of things.

The electric fields of point charges are VERY similar to the gravitational field of a planet, so of course we spend a lot of time talking about them. They have perfect, spherical symmetry. The electric field points radially - in all directions. But what happens when we give that charge a spin?

Well moving charges generate magnetic fields, but spinning points are a little awkward to discuss. So we can take the simple toy model of a loop of current. We definitely teach our students about current loops. Usually we ask them to compute the magnetic field from a loop of current in some computationally convenient place, like in the middle of the loop.

Taking the loop to be very, very small, we essentially get a model of a spinning charge, and the magnetic field points a definite direction: Up from the center of the loop - the axis of rotation in our model here. And at large distances, the magnetic field falls off like one by the distance cubed. In other words, it’s a dipole... magnetic field field.

Computing the full magnetic field from a tiny loop of current, even ignoring effects from the size of the loop, is pretty complicated. So we often tell students just to think of them as tiny little magnetic fields pointed along some axis of symmetry. We’ll often talk about the energy of magnetic fields and how those loops of current want to align themselves with other magnetic fields to lower the energy. Some might even ask their students to calculate the approximate energy of two loops of current right next to each other.

To do that, we sometimes think of the loops as having a basically constant magnetic field, B, and so the only variable is the direction of the magnetic field.

This really simplistic model turns out to be pretty powerful! Indeed as the Ising model for a ferromagnet, it performs pretty well! We can think of a bar magnet as being made up of a bunch of little current loops, all pointing together in the same direction to lower the energy of the system, which we then use to keep our graded exams on our refrigerators.

Amusingly, the collective behavior of all those tiny little loops of current - those tiny little magnets - forms one big magnetic that also can be approximated as a dipole field.

Linearity is such an intoxicatingly simple thing.

Quantum Spin

In quantum mechanics, Spin takes on a whole new meaning. It’s a property of particles, sure, but it’s more than that. It’s... somehow intrinsic to them. They’re always spinning. At the same rate. And that rate comes in half-integer units of the fundamental constant hbar. They might speed up or slow down, but there are rules governing how they can, and they can only jump between these discrete, fixed values. Their spin, in other words, is quantized. And that has LITTLE similarity to our classical understanding.

Of course, particles are also electrically charged. And they also have tiny magnetic fields. Those magnetic fields are basically dipoles. Are the magnetic fields related to how the particles are spinning? Well yes... but not in the way that you’d think. The classical models ALMOST get it right, but not quite.

Consider the neutron - one of the particles in the atomic nucleus. Th neutron is electrically neutral. Whence the name. Despite that fact, it has a fair sized magnetic field - almost as big as the protons!

Now that fact betrays some complicated inner structure of the neutron. But still. Classical electromagnetism is linear. In the classical theory, things add like you’d expect. Remember bar magnets made up of little magnets? Quantum electrodynamics - let alone particle physics - isn’t exactly linear.

Nevertheless, one thing still remains true. In a magnetic field, particles with magnetic moments preferentially change their spin to “align” with that field. Why? It lowers the energy, like you’d expect from elementary physics.

Practically speaking, you conflate the magnetic moment of a particle like an atom or electron or a neutron with it’s spin. While they’re precisely related for elementary particles. For composite things like atoms, it’s approximately true. In any case, the particle with it’s magnetic dipole field will certainly receive a KICK from a strong magnetic field.

To capture the essential physics in a discussion, it might be helpful to recount the Stern-Gerlach experiment, which aimed a beam of silver atoms - spin 1/2 particles - at a strong, sharp magnetic field.

The Stern-Gerlach Experiment

A beam of particles is just a random assortment of them moving through pace, one after another after another. They’re all in line sure - like a train of little spheres - but their dipole moments? Their spins? Those little arrows sticking out of the spheres? They could be pointing in any which ways.

If you aim a beam of particles at a strong, sharp magnetic field. Those particles would all get kicked individually. The AMOUNT and DIECTION of the kick depends on where their magnetic moment was pointing - that little arrow. Along the beam, those kicks would be just as random as the arrangement of those dipoles. So if a particle strongly aligned with the magnetic field, they’d get kicked a lot. Alternatively, if its spin is orthogonal to the magnetic field, they wouldn’t get kicked very much at all. The kick is proportional to cosine theta, but that angle theta are distributed randomly along the beam. At least this is what our little model of particles as little current loops tells us.

And if that model was right the beam of particles would fan out. Some getting kicked upwards and some kicked downwards, some barely getting kicked at all. The cross section of the beam - a little dot - would deformed into a little stripe.

Indeed, if you could launch a stream of really lightweight, refrigerator bar magnets at a REALLY REALLY STRONG magnetic field. That’s exactly what you’d see. That stream would smear out into a fan of magnets before smashing into whatever target you had behind the magnetic field.

But that little current loop model doesn’t work for particles subject to quantum mechanics. Small things like atoms or neutrons. Quantum mechanics says that particles can only have discrete values of spin. They can’t smear or fan out. They can only take on quantized values.

Otto Stern and Walter Gerlach did that experiment, with particles subject to quantum mechanics: silver atoms. Do you know what they saw? The beam of silver atoms didn’t fan out. It split. In two. Which is good because Silver’s outer most - and therefore most reactive - electron lives in the 5s1 state. Meaning it has zero orbital angular momentum, but net spin 1/2. Meaning that a beam of silver atoms really should split in two: corresponding to two distinct quantum states: plus or minus \(\hbar/2\).