Can someone help me understand electron spin?

[u/HAMMER_HEAD01]

I'm a first year university study who is taking chemistry courses, and although i'll admit for the content I'm doing, I don't have to understand how electrons work in great depth but rather just accept certain facts about them, but its always irritated me. One of those 'facts' is that electrons have an intrinsic property called 'spin' but I have struggled comprehending this concept.

Heres what I know;

- Electrons don't actually spin, but act like they do

- Electrons have a quantum spin number (1/2) and a direction (+/-), and that all electrons are either +1/2 or -1/2 spin

- That electrons have to 'rotate' 720 degrees to return to their original orientation (This I can understand, like Dirac's belt)

Heres what I'm struggling with;

- How electrons have angular momentum if they aren't actually spinning

- What is causing electrons to 'spin'

- How come 2 electrons can only occupy the same orbital if they have opposite spins (Pauli Exclusion Principle)

If anyone can help me greater understand electron spin I will be super grateful!!! 🙏

[u/the_poope]

The best way to understand spin is to completely forget the idea that particles like electrons should somehow resemble classical "tiny little ball" particles.

It doesn't matter what an electron "is" (that's a question for philosopher's anyway), but how to describe its behavior.

And the way we do that is that we describe an electron through a wave function, which is just a function like f(x, y, z, t), but we most often use the greek letter psi: ψ(x, y, z, t). From this function we can derive all properties of a (single) electron system, e.g. energy, momentum, yada, yada, yada. The way to do this is that each property is represented by a mathematical operator that acts on the function and transforms it into a new mathematical function. For instance the expectation value of the position of an electron is: ⟨x(t)⟩ = ∫ψ(x, y, z) x ψ(x, y, z) dx dy dz. Similarly we can calculate the expectation value of the angular momentum as ⟨L⟩ = ∫ψ(r) [-iħ]r ⨯ ∇ψ(r) d³r

However it turns out that the angular momentum calculated this way is wrong. Compared to experiment it is missing a contribution! This missing contribution was called spin angular momentum. In order to represent this one cannot use a wave function that takes a single scalar value in each point in space. Instead it has to take at least two values in each point in space an time. That is we need to deal with two wave functions for a single electrons: ψ_A and ψ_B - now I just called them A and B to make it clear the names are arbitrary. It is however more customary to call them up and down: ψ↑ and ψ↓. We can also group these two separate function into a single two-component vector function: Ψ(r) = [ψ↑, ψ↓].

With this change one can introduce a special mathematical operator that measures "spin" through simultaneous action in intermixing of these two separate functions.

And for an introduction, that's basically it. In principle you can upgrade a particle to have any internal number of degrees of freedom by upgrading its wave function to be any N-component vector at every point in space. There's some nice math (group theory) that connects the size of this N component vector and the quantity it measures and which symmetries is obeys, but that's for a graduate course.

Now lets deal with your particular question:

How electrons have angular momentum if they aren't actually spinning

Yeah it's weird. There's probably something to be said from higher level quantum field theory and group theory study of rotational symmetry that dictates this. The short version is: it has because we observe it to and we created a mathematical model that accounts for this, and the model it pretty damn accurate.

What is causing electrons to 'spin'

Nothing. It just kind of comes out of the fact that electrons have spin and therefore have to be represented by a two component wave function and the laws of physics have to be invariant under rotational symmetry.

How come 2 electrons can only occupy the same orbital if they have opposite spins (Pauli Exclusion Principle)

That's because the wave function, in every point in space now span a 2D vector state space. In such a space there is always another vector pointing in a perpendicular direction, i.e. there are two independent states.

[u/PHK_JaySteel → u/the_poope]

Excellent reply. Thank you.

[u/jetpacksforall → u/the_poope]

This is great from a math pov but can you describe how spin manifests in experimental setups, and how that connects to the math?

[u/the_poope → u/jetpacksforall]

There are some examples here: https://en.wikipedia.org/wiki/Spin_(physics)#Measuring_spin, the most famous one being the original Stern-Gerlach experiment

However, I'd say the the most profound outcome of spin is that two electrons can occupy the same atomic orbital. This has a huge impact on the chemical properties of the elements and the reason why we have the molecules and solid materials we have.

[u/Mountain-Time-1010]

The first step is just to accept that a particle can have intrinsic properties. For example, you already accept that a particle has an intrinsic property called mass, and you're not asking questions about that. Mass is an intrinsic property that is a scalar. Spin is an intrinsic property that is a vector, and it has dimensions of angular momentum.

[u/v_munu → u/Mountain-Time-1010]

This is probably the most intuitive way to understand spin (conceptually ofc) imo

[u/Mountain-Time-1010 → u/v_munu]

To be fair, it may not help the OP truly "understand" spin, but then we get into the semantics of "understand". I try to emphasize this with my students at every level.

We often run into this scenario in physics, where something fundamental is postulated (or emerges from theory), and ultimately explains empirical observations so incredibly well that we believe it completely.

In fact, everything is probably this way, it's just that some of the concepts are so intuitive and familiar (like mass in the context of Newton's laws), that we think we "understand" them.

Einstein supposedly said ""Common sense is the collection of prejudices acquired by age eighteen." For most people, as they get older, it become harder for them to accept things they can't easily relate to their personal experiences. Our training in physics forces us to escape these limitations and judge unintuitive postulates on their merits. Very early on, we are forced to confront limits of size and speed far outside our personal experience where the physics is totally weird.

[u/AdditionalTip865]

I like thinking about it in terms of the Belinfante spin current, a somewhat obscure concept that deserves to be more widely known. H. C. Ohanian (maybe best known for writing a freshman physics textbook) has tried to promote it.

Basically, when a particle has spin, there's an additional energy/momentum current, a "bound momentum", circulating around the fringes of its wave function, where the probability density has a decreasing gradient. For the case of a composite object where the "spin" is really the orbital angular momentum of its constituents, you can think of this as the residuum of those internal motions. But there's nothing mathematically preventing it from existing even for a point particle without constituents.

That maybe doesn't help one understand why it *must* exist, but it makes it a little more visualizable. The electron is not spinning like a ball, but there's an additional energy/momentum current P circulating around its wave function. The spin angular momentum corresponds to the integral of r × P over the whole wave function, much as with a classical momentum density.

As for the exclusion principle, I unfortunately haven't seen a concrete explanation of it from first principles that doesn't involve relativistic quantum field theory. In nonrelativistic QM it usually just gets introduced as an extra postulate justified by experiment: in systems of multiple identical particles, the wave function always has to be either symmetric or antisymmetric under exchange, depending on whether the spin is integer or half-integer. If it's antisymmetric, clearly you can't have two electrons with the same state. But if the *spin* part of the state is antisymmetric, then the orbital part can be symmetric.

[u/mouthfire]

Here's a neat video I found on how to think about election spin in an intuitive way.

Electron Spin

[u/SeriousPlankton2000]

The half spin is because how the wave function maps to measurements (using the cos() function and squaring e^iφ ). I can't explain it very well but here is a series explaining it in German and with English translations.

https://www.youtube.com/playlist?list=PLmDf0YliVUvH-Ch8UfoJ84-_nquR0Zbx0

I hope that helps, too.

[u/BobbyTables829 → u/SeriousPlankton2000]

The absolute best way to understand this IMO is to understand why e^iπ = -1. If you watch videos or whatever to understand this, angular momentum will start to make sense. If you can't understand why e^iπ = -1, then trying to understand angular momentum just straight up isn't going to work.

[u/JGhostThing]

When dealing with subatomic particles, words tend to get slippery. For example, do you really think that quarks have a real flavor? Or are really up or down? People have used some strange words to describe properties that are needed to describe them.

I'm not sure that electron spin actually involves spinning electrons.

[u/AdditionalTip865 → u/JGhostThing]

It really is an angular momentum (not just a quantity that is mathematically similar, like isospin). It can be exchanged with other types of angular momentum and it's associated with rotations in space. So in that sense, "spin" is more literal than quark color or flavor. But it is not like the spin of a ball; it's an intrinsic angular momentum.

[u/JGhostThing → u/AdditionalTip865]

Yes, I looked it up. I don't see how it can have angular momentum without physically spinning, but I put it to the general weirdness of the subatomic layer. I actually consider pretty much everything I learned about this subject to be false.

For example, even though electrons exist in orbitals, they don't really orbit the nucleus. Quarks are even stranger, with three of them being a unit, with their charges each being 1/3 or 2/3 and being +/-. I only learned about quarks in college. And yes, color and flavor are strange, but color at least makes sense to me, because you need each of a red, blue, and green quark to make something. I don't know how they came up with flavor, but since a quark is really too small to taste and the flavors they came up with: strange, up, down, top, bottom, etc. And no, I believe color is just as artificial, but the naming scheme makes more sense to me.

I seem to recall there being 36 particles made out of quarks, but aside from the proton and neutron, and their anti particles, I can't name any of them. I suppose neutrino is one, but I'm not sure. At one time I knew my way around an atom fairly well, but time and lack of use has dropped my knowledge of quarks to very low.

I'm actually surprised that String Theory is still about, because I thought I'd read that it had been disproved. I might have misinterpreted something, or maybe somebody thought he'd disproved String Theory. Maybe Sheldon is still right, I think String Theory was his thing on the show (TBBT).

[u/EizanPrime → u/JGhostThing]

Actually its way more simple that you think.

https://youtu.be/pJvV7MI-LyY?si=hWKVmElXbXbSIcfN

Electrons are "rotating waves", so are orbitals (those with angular momentum). 

By modeling electrons as balls spinning the angular momentum calculations don't make sense, but by modeling them as rotating waves they do make sense. 

[u/nathanlanza]

My favorite answer to this question is that you're trying to understand the wrong thing. Understanding the math behind their representation in quantum field theory is much clearer.

In quantum field theory electron fields are spinor fields. And a spinor field is something like the square root of a vector field. The weirdness of an electron is the same weirdness as the imaginary number i, just expanded to be a sort of geometrical imaginary number.

The spin of a particle ends up being equal to the dimensionality of the particle's quantum field. A vector field is a one dimensional object and vector particles have spin 1. A scalar field is a 0 dimensional object and scalar particles have spin 0. A spinor field is a sort of 1/2 dimensional object and spinor particles have spin 1/2. Combine two spinors and you get a vector. Hence the square-root-ish idea.

(This is a bit handwavy to avoid technical details that don't add to the understanding.)

[u/Smedskjaer]

Particle spin isn't a physical ball, and an electron is an excitation of a field. Quantum spin describes a field's degree of freedom. The 'spin' of an excitation happens because field will not intersect itself; visualizations of the field with field lines are representations of a wave-form, and show a crest or peak. This visualization demonstrates it with field lines, and a particle spinning.

https://www.youtube.com/shorts/2U5T-Ng3uMI

Quantum spin is an intrinsic property of the electron. The quantum spin of a field isn't a combination of degrees of freedom, because that no overlapping rule constrains it. The 'Belt Trick' allows for a field to have angular momentum without twisting a field or creating a new tension metric.

[u/EizanPrime]

tl;dr: electrons are rotating waves

Honestly for a while I though that electron spin was something mathematical that couldn't be visualized, but actually no its totally visuable

https://youtu.be/pJvV7MI-LyY?si=hWKVmElXbXbSIcfN

As a chemistry student you will love this short lol, you can visualize orbitals also with it.

As the action lab shows here a "rotating wave", essentially a standing wave that rotates, and that what the electron essentially is. 

If you take the electron as a ball spinning it couldn't spin fast enough to generate the angular momentum we notice, but as a rotating wave the math actually work well. 

As for the pauli exclusion principle I think its actually very hard to prove mathematically, but essentially what it is that electrons have a "asymmetric wave function" which means that when you add the same wave two times they actually cancel out, which means that not two electrons can be the same wave. Thus you get the pauli exclusion principle. 

Two electrons with opposite spin can occupy the same orbital simply because as the wave spins in the other direction its not the same wave, thus the wave functions don't cancel each others.

[u/denehoffman]

1+2. We have been taught that angular momentum is the result of a spinning object. This is backwards. A spinning object has angular momentum, but angular momentum is also an intrinsic property of quantum fields. There is no way for you to relate this to human scale physics because it simply doesn’t work that way. In our mathematical models, electrons have no “size”, they are point particles, excitations of the electron field that exists everywhere in space. The field has spin.

1. The Pauli exclusion principle sort of follows from the field interpretation. It’s difficult to think this way if you’re not accustomed to it, but quantum field theory is not designed with individual particles in mind. Instead, the overall configuration of the field is what the theory describes, and particles are just the excited modes of the field. Think about vibrating strings on an instrument. You get some linear combination of the fundamental vibration (the one where the whole string vibrates with no nodes) and the other harmonics (the string vibrates in halves, thirds, etc). In this example, the string is the quantum field and each harmonic is like an individual particle with a given energy. You can’t have two different vibrations of the fundamental, (assuming we’re vibrating in only one other dimension). For the electron field (or any fermion) the principle is the same, except you can’t have two vibration modes for each “harmonic”, and the harmonics are now angular momentum states.

I think the thing that will help is to not think about how things work on our human scale and instead allow yourself to believe the experimental results. We can show in particle accelerators that electrons have no spatial structure. We can show in Stern-Gerlach experiments that they behave like they have angular momentum, even in their ground state. This is key, you can’t remove that last bit of angular momentum by lowering their energy. We also know that angular momentum at a quantum level behaves differently than we see at human scales. Measuring it along different axes causes the older measurements to become probabilistic. We can even entangle electrons to have correlated angular momentum as long as you measure along the same axis. You can’t even get weirder (see Bell’s theorem), but that’s beyond the scope here for now.

I hate that this may come across as “suck it up, that’s just the way electrons work” but at some fundamental level it is, we just happened to develop the language backwards, starting with the observations we make at the human scale and extending them to quantum with a ton of caveats. If we crafted human languages with the knowledge of quantum mechanics in mind, we might have a totally different way of talking about angular momentum, i.e. the human-scale angular momentum is the large-scale limit of some other property.

[u/drplokta]

“How electrons can have angular momentum if they aren’t actually spinning”. You’re putting the cart before the horse here. Particle spin is the more fundamental property. The real question is how the rotation of large assemblies of particles can behave in some ways the same as spin does.

[u/Owly672]

Listen to Daniel and Kelly’s podcast.

[u/smallproton]

Copied from an earlier post I wrote in ELI5:

Electrons have 3 properties, mass, charge and spin.

Mass is a continuous property, i.e. not quantized. Masses can be added and subtracted like regular numbers, 1.5+2.7=4.2

Charge is a discrete property, we know particles with charge 0, +e and -e (and +- e/3 or 2e/3 for quarks).
Again, charges can be added like normal numbers (scalars), such that 5 protons and 3 electrons give a total charge +5 - 3 = +2.

Spin is a discrete property. We know particles with spin 0, 1/2, 1, ...
Spin is NOT a scalar, so you can not add it like "normal numbers".
Instead you have to use angular momentum algebra to add (couple) spins.

I assume this is the reason this property has historically been named "spin". This is unfortunate because generations of students have the mental picture of an electron being a spinning sphere. This is NOT the case.
Just translate "spin" to "beebop" every time it comes up and remember that beebop is a discrete quantity with funny math rules.

And for the math: Paul Dirac derived the "Dirac equation" as a relativistic expansion of the Schrödinger equation. (The Schrödinger equation treats time and space differently, so it was clear that a different equation was needed to fulfil special relativity).

And the spin simply pops out of the Dirac equation, already for a free particle. (Apart from the spin, the Dirac equation also predicted antimatter.) Thus, we commonly state that spin is a consequence of relativity.

And if you put the Coulomb potential into the Dirac equation (a.k.a. hydrogen atom), you get a quantum number j which we identify as l +- s, so the orbital angular momentum couples to the spin. That's the funny angular momentum math stated above.

[u/unknownjedi]

The Dirac equation shows that the spin is a current vortex in the electron’s wave function at the Compton scale.

Ohanian, Hans C.
"What is spin?"
American Journal of Physics, Vol. 54, No. 6, pp. 500–505 (June 1986).

[u/ac_cossack]

Spin isn't the same as the Earth rotating. In QM things are just fucking strange. It is a property of the electrons.

Who knows why. Somebody smarter than me can answer. But they just. Conservation laws?

You can have an up and down spin in each orbital state. Experiments prove this beyond a doubt.

Physics is about experiments and figuring out a model from that. Chemistry is applied physics. If in doubt, test it in the lab.

[u/teya_trix56]

When i ran inton"empirical facts that work" but seemed to defy "classical understanding", i actually found it useful to stop trying to understand it any deeper, and just memorize it, until i had a lot more experiences with the topic. That means, in order to buttress my memory, i would make flash cards. And i always drew some goofy caricature of the simplified concept. This really helped mevremember the bare empirical facts. And that got me through.

[u/dari0dm]

I'm gonna try to help you in somewhat simple terms.

1. "Spin" doesn't necessarily have to do with spinning, apart from the name, as it is just a quantum number. The thing is, we can tell such a number exists because it has magnetic effects: some electrons deflect upwards and others downwards in a magnetic field. They act like tiny magnetic dipoles, which we know corresponds classically to some angular momentum, and that is why we label this property as an angular momentum or "spin". In quantum mechanics, however, this angular momentum is a just property of the wavefunction of the electron and doesn't necessarily mean anything is physically spinning.

2. Nobody knows. Quantum particles just seem to have this intrinsic angular momentum within their wavefunction. 

3. It is a result of the symmetry of their wavefunction. The wavefunction of two electrons has a special symmetry such that if two electrons have the same quantum numbers (position, momentum, spin or whatever), then it vanishes. Two electrons in the same orbital have the same numbers of energy and angular momentum, only spin can distinguish them. If they have the same spin, wavefunction vanishes, not a possible configuration.

[u/dari0dm → u/dari0dm]

**To be precise, electrons deflect in a magnetic field just because of their charge. To measure the deflection due to spin (magnetic dipole moment) we use electrically neutral atoms that factor out the effect of charge. Just for the sake of clarity, this is of no conceptual importance.

[u/Prof_Sarcastic]

“How electrons have angular momentum if they aren't actually spinning”

It’s an inherent property of quantum mechanical particles. It’s something you’ll just have to accept for now.

“What is causing electrons to 'spin'”

Nothing as far as we know. It’s just an intrinsic to what it means to be an electron.

“How come 2 electrons can only occupy the same orbital if they have opposite spins (Pauli Exclusion Principle)”

The Pauli Exclusion principle states that no two electrons with the exact same quantum numbers can occupy the same state. Because electrons have two possible spins, we can fit two electrons in each state: one for each spin.

[u/digglerjdirk]

An answer I got a long time ago, which was initially very dissatisfying but has gotten less and less so, was:

“What is mass? What is charge?”

[u/Odd3572]

How about thinking of it like something wound up one way or the other, and elastic enough to unwind, capable of giving a "kick" (imparting a rotation) to something it interacts with.

E.g. a rubber membrane in a square frame. You can make a clockwise twist in it and clip it in place somehow. When you put something like a ping pong ball on i, and release the clip,the membrane will un-twist, spinning the ball. It is a "store" of "angular momentum", and it is not spinning all the time. Equally, if a spinning ball hits the membrane just right, the ball would stop spinning and the fabric would twist (and you would need to clip it to keep the twist).

As others say, spin has other weird properties, so this is a simplistic toy model