Electron Spin for Toddlers

The subject of the “spin” of the electron comes up again and again, so as pointed out in a comment, I really ought to do a post explaining what it is and how it works. As a bonus, this gives me the opportunity to do the dorkiest thing anyone has ever done with a cute-toddler video, namely this one:

(That’s an early version of SteelyKid’s new favorite game. I’ll put a clip of the final version of the game at the end of this post.)

So, electron spin. Electrons, and all other fundamental particles, have a property known as “spin.” This is an intrinsic angular momentum associated with the particles, as if they were little spinning balls of charge. I say “as if” there, because they are not literally spinning balls of charge, because the spin angular momentum associated with fundamental particles has some properties that are very strange, and completely unlike the behavior of spinning basketballs, or gyroscopes, or figure skaters, or whatever your favorite example of a system with angular momentum is.

First, though, how can we say with certainty that these aren’t literally spinning balls of charge? Well, in addition to their angular momentum, the fundamental particles also have a magnetic moment associated with them, which is to say they behave like tiny little bar magnets with a north and south pole. This is exactly what you would expect for a spinning ball of charge, but if you go through the math, you find that it can’t possibly work.

If we want to say that the magnetic moment of the electron is due to the motion of a spinning ball of charge, then we can easily calculate what the spin rate should be, given what we know about the size of an electron. If you use the maximum size you could possibly associate with the electron, the “classical electron radius”, and calculate how fast a sphere of that size would need to be spinning to produce the observed magnetic moment, you find that a point on the surface would need to be moving at a speed several times the speed of light in vacuum, which is impossible. That’s also a gross overestimate of the size of an electron– as far as well can tell, the electron has no physical size. It’s a point particle, and thus doesn’t have a surface that can be physically rotating.

OK, then, maybe the magnetic moment is just one of those things, you know? Maybe the “spin” angular momentum isn’t really angular momentum at all. This is also false– spin angular momentum is real angular momentum. We know this because you can drive transitions from one spin state to another using polarized light, and we know from careful experiments done in 1936 that the angular momentum carried by light is real angular momentum. The angular momentum of a polarized photon can be used to make physical objects rotate, and it can also be used to make electron spins change states; this at least strongly suggests that spin angular momentum is real angular momentum comparable to that of spinning basketballs and all the rest.

The spin angular momentum of an electron does have some strange properties, though, that are very unlike those of ordinary rotating objects. For one thing, it has only two possible states, “spin up” and “spin down.”

To understand the meaning of this, we have to back up a moment and talk about classical angular momentum a bit. Angular momentum is a quantity associated with a spinning object, and it has a direction associated with it. The direction of the angular momentum of a spinning object is along the axis of rotation, and determined by the direction of spin. If you look down at a record on a turntable, and it’s spinning clockwise, the angular momentum points down, into the turntable. If it’s spinning counter-clockwise, the angular momentum points up, out of the turntable toward you.

For a classical object, this angular momentum can point in absolutely any direction, and can be moved around by exerting forces in the appropriate places. When we move to talking about quantum mechanics, which gets its name from the fact that the theory only allows certain discrete states, this picture needs to be “quantized,” but the transition from classical to quantum isn’t too bad. If you have a moving object in quantum mechanics, and ask what the allowed states of its angular momentum are, you find that they are integer multiples of Planck’s constant, up to some maximum value, and down to the same negative integer. That is, if the system you’re looking at can have up to 3 units of angular momentum, the states you can find it in are +3, +2, +1, 0, -1, -2, and -3 units of angular momentum (where the sign indicates the direction– +3 is up, -3 is down).*

Spin angular momentum has the same basic property– it moves from some negative value to some positive value in steps of one unit of Planck’s constant– but with an important difference: the allowed states have half-integer values of angular momentum. For an electron, the maximum possible spin is 1/2 Planck’s constant, so there are only two allowed states, +1/2 and -1/2. Since the sign indicates the direction, we tend to call these “spin-up” and “spin-down.” *

The first weird thing to jump out about this is that zero is not included. You can make a state that has zero spin angular momentum on average by adding together equal parts of spin-up and spin-down, but if you take one such system and ask “what is the angular momentum?” you will always get either +1/2 or -1/2, no matter what. There is no single measurement you can do on an electron spin that will ever give you an answer of zero. You might get an average value of zero after many measurements, but each individual measurement will give you either spin-up or spin-down.

This is a very strange idea, and very much unlike a classical system, or even quantized ordinary angular momentum. With an system having whole integer values of angular momentum, zero is always an option– if you think of it like a record player, you can imagine it as having settings for 33 1/3 rpm, 45 rpm, both forward and reverse, but the turntable also has a power button– you can always turn it off, and have angular momentum of zero. Electron spin is like a turntable with only forward and backward settings at a single speed, with the power cord wired directly into the mains so it can’t be shut off. It’s always spinning in one direction or the other.

That’s not the weirdest part, though. The weird part comes when you look at what happens when you flip the spin twice. In a classical system, or a system of ordinary quantized angular momentum, when you rotate the direction of angular momentum through 360 degrees, you get back to where you started. If you start out in a state with one unit of angular momentum, then do something to move it from +1 to 0, then 0 to -1, then -1 to 0, and finally 0 back to +1, at the end of all that state-shifting, you end up with a state that is indistinguishable from an identical system that just sat in +1 the whole time.

When you do this with half-integer spin– that is, take an electron in spin-up, flip it to spin-down, and then back to spin-up, you pick up a factor of -1. That is, the spin is pointing in the same direction it was at the start, but the overall wavefunction for that electron is multiplied by -1.

This is where the cute-toddler video comes in. If we imagine SteelyKid as a spinning electron, the climb-and-flip game in the video is like the rotation of an electron spin. At the start of the game, she’s in a spin-up state (that is, standing upright):

Halfway through the game, she’s in a spin-down state (upside down):

And at the end of the game, she’s coming back to spin-up:

If you look closely at the first and last pictures, though, you can see that her state isn’t identical– in the final picture, her arms are twisted around (I picked a frame where she’s not quite back to vertical so this would be clearer). She’s made a 360 degree rotation of her body, but her overall state is different than it was at the start.

Of course, this analogy isn’t perfect– if you rotate a spin by 360 degrees, and then by 360 degrees again (in the same direction), you get back to exactly the state you started with. If I had SteelyKid do two consecutive climb-and-flips without letting her shift her grip in between, we’d end up in the emergency room with her shoulders dislocated, and Kate would divorce me.

To get a more complete analogy, all you need is a coffee cup, or other handled container. Hold the cup in your right hand, with the handle facing away from you. You can rotate the handle so that it faces you by curling your right arm toward yourself, and if you continue that motion, you can bring the handle back to pointing away from you, but your arm will be twisted into a slightly awkward position. If you continue the rotation through another 360 degrees, thought, you get back to the original situation, with your cup held comfortably pointing away from you.

That’s a more complete analogy, but nowhere near as cute. Coffee cups don’t giggle, or narrate their rotation. (“I’m climbing up!”)

This strange rotation property is characteristic of half-integer spin angular momentum, and is one of the strangest properties of quantum physics. It’s also one of the most important properties.

What does it mean for the electron wavefunction to be multiplied by -1? Not much, if you only have a single electron. Multiplying the overall wavefunction by -1 is the same as shifting the phase of a wave by 180 degrees, which does not by itself produce any interesting effects. A light wave that is delayed by half a wavelength before reaching you looks just the same as one with no delay, and an electron spin with a factor of -1 in front of the wavefunction behaves just the same as one where the wavefunction is multiplied by +1. (Note that the factor of -1 is on the wavefunction, not the spin– it reflects a change in the external context, not a change from +1/2 to -1/2. In the video, the factor of -1 refers to the state of SteelyKid’s arms, while the spin-up/spin-down information refers to the orientation of her body.)

This factor is critically important, though, when you talk about comparing spins to each other. In the same way that delaying a beam of light by half a wavelength has no effect on the intensity of that single beam, but takes you from constructive to destructive interference when you add it to a second beam of light, the effects of spin rotation have dramatic consequences when you are dealing with two or more electrons. In particular, this behavior under rotation is related to the Pauli exclusion principle that students of chemistry know and love.

Pauli exclusion says, in the form that we usually talk about, that no two spin-1/2 particles can ever be found in exactly the same state. This is a result of a deeper requirement, which says that the state of a collection of identical spin-1/2 particles can’t depend on labelling individual particles– if you switch the positions of electron 1 and electron 2, the physics has to be the same, up to an overall factor of +1 or -1 in front of the wavefunction.

Switching the state of two particles, though, is mathematically related to rotating the two spins by 360 degrees. And as we just saw, when you rotate a spin-1/2 particle by 360 degrees, it picks up a factor of -1. Thus, when you are dealing with spin-1/2 particles, swapping any two of them has to give you the same wavefunction you started with, multiplied by -1. When you work out the consequences of this, that means that there is no way for two spin-1/2 particles to be in exactly the same state at the same time (I talked about how to understand this a couple of years ago, copying an explanation from Feynman).

This also gives you features like the cancellation described in the topological insulator post, where there’s an interference between the part of an electron’s wavefunction that sees its spin rotate clockwise by 180 degrees, and the part that sees its spin rotate counter-clockwise by 180 degrees. If you think about it a little, you can see that getting from one of those states to the other involves a 360 degree rotation, which means that one is just a factor of -1 from the other. When you add those two pieces of the wavefunction together, then, they exactly cancel, which is why the surface states in topological insulators are insensitive to impurities, which is a big part of why they’re considered so cool.

So, that’s what you need to know about electron spin. To summarize, in bullet-point form:

• Electrons are not literally spinning balls of charge, but they do have intrinsic angular momentum.
• Spin angular momentum is real angular momentum.
• The angular momentum of a spin-1/2 particle like an electron is never zero.
• Rotating the spin of a spin-1/2 particle by 360 degrees doesn’t get you exactly the state you started with, in the same way that the climb-and-flip game doesn’t get your toddler back exactly the state she started in.
• This spin rotation property is one of the weirdest but most important features of the quantum theory of fundamental particles.
• SteelyKid is the cutest toddler in the universe.

OK, strictly speaking, that last one isn’t about electrons. But it’s true, and important. As additional proof, and as a reward for reading all the way to the end (you did read all the way to the end, didn’t you?), here’s the more advanced version of the climb-and-flip game, developed a little later that evening.

She’s the cutest electron-analogue ever.

(* As pointed out in comments, I’m glossing over one aspect of this– the quantum states I mention in this post are the angular momentum along some axis in space. The choice of axis is arbitrary– “up” and “down” are names given as a matter of convention, and “parallel” and “antiparallel” might be more appropriate. Once you’ve made your choice of axis, you will only ever measure the angular momentum to be one of a limited number of possible values, as described in the text.)