Q: What is “spin” in particle physics? Why is it different from just ordinary rotation?

Physicist: “Spin” or sometimes “nuclear spin” or “intrinsic spin” is the quantum version of angular momentum.  Unlike regular angular momentum, spin has nothing to do with actual spinning.

Normally angular momentum takes the form of an object’s tendency to continue rotating at a particular rate.  Conservation of regular, in-a-straight-line momentum is often described as “an object in motion stays in motion, and an object at rest stays at rest”, conservation of angular momentum is often described as “an object that’s rotating stays rotating, and an object that’s not rotating keeps not rotating”.

Any sane person thinking about angular momentum is thinking about rotation.  However, at the atomic scale you start to find some strange, contradictory results, and intuition becomes about as useful as a pogo stick in a chess game.  Here’s the idea behind one of the impossibilities:

Anytime you take a current and run it in a loop or, equivalently, take an electrically charged object and spin it, you get a magnetic field.  This magnetic field takes the usual, bar-magnet-looking form, with a north pole and a south pole.  There’s a glut of detail on that over here.

A spinning charged object carries charge in circles, which is just another way of describing a current loop. Current loops create “dipole” magnetic fields.

If you know how the charge is distributed in an object, and you know fast that object is spinning, you can figure out how strong the magnetic field is.  But in general, more charge and more speed means more magnetism.  Happily, you can also back-solve: for a given size, magnetic field, and electric charge, you can figure out the minimum speed that something must be spinning.

It’s not too hard to find the magnetic field of electrons, as well as their size and electric charge. Btw, these experiments are among the prettiest experiments anywhere.  Suck on that biology!

Electrons do each have a magnetic field (called the “magnetic moment” for some damn-fool reason), as do protons and neutrons.  If enough of them “agree” and line up with each other you get a ferromagnetic material, or as most people call them: “regular magnets”.

Herein lies the problem.  For the charge and size of electrons in particular, their magnetic field is way too high.  They’d need to be spinning faster than the speed of light in order to produce the fields we see.  As fans of the physics are no doubt already aware: faster-than-light = no.  And yet, they definitely have the angular momentum necessary to create their fields.

It seems strange to abandon the idea of rotation when talking about angular momentum, but there it is.  Somehow particles have angular momentum, in almost every important sense, even acting like a gyroscope, but without doing all of the usual rotating.  Instead, a particle’s angular momentum is just another property that it has, like charge or mass.  Physicists use the word “spin” or “intrinsic spin” to distinguish the angular momentum that particles “just kinda have” from the regular angular momentum of physically rotating things.

Spin (for reasons that are horrible, but will be included anyway in the answer gravy below) can take on values like \cdots, , -\hbar, -\frac{1}{2}\hbar, 0, \frac{1}{2}\hbar, \hbar, \frac{3}{2}\hbar, \cdots where \hbar (“h bar“) is a physical constant.  This by the way, is a big part of where “quantum mechanics” gets its name.  A “quanta” is the smallest unit of something and, as it happens, there is a smallest unit of angular momentum (\frac{1}{2}\hbar)!

It may very well be that intrinsic spin is actually more fundamental than the form of rotation we’re used to.  The spin of a particle has a very real effect on what happens when it’s physically rotated around another, identical particle.  When you rotate two particles so that they change places you find that their quantum wave function is affected.  Without going into too much detail, for particles called fermions this leads to the “Pauli Exclusion principle” which is responsible for matter not being allowed to be in the same state (which includes place) at the same time.  For all other particles, which are known as “bosons”, it has no effect at all.


Answer gravy: Word of warning; this answer gravy is pretty thick.  A familiarity with vectors, and linear algebra would go a long way.

Not everything in the world commutes.  That is, AB≠BA.  In order to talk about how badly things don’t commute physicists (and other “scientists”) use commutators.  The commutator of A and B is written “[A,B] = AB-BA”.  When A and B don’t commute, then [A,B]≠0.

As it happens, the position measure in a particular direction, Rj, doesn’t commute with the momentum measure in the same direction, Pj (“j” can be the x, y, or z directions).  That is to say, it matters which you do first.  This is more popularly talked about in terms of the “uncertainty principle“.  On the other hand, momentum and position measurements in different directions commute no problem.

For example, [R_x,P_x]=i\hbar and [R_x,P_y]=0.  This is more succinctly written as [R_j,P_k]=i\hbar \delta_{jk}, where \delta_{jk}=\left\{\begin{array}{ll}1&when\,j=k\\0&when \,j\ne k\end{array}\right.  This is the “position/momentum canonical commutation relation“.

In both classical and quantum physics the angular momentum is given by \vec{R}\times\vec{P}.  This essentially describes angular momentum as the momentum of something (\vec{P}) at the end of a lever arm (\vec{R}).  Classically \vec{R} and \vec{P} are the position and momentum of a thing.  Quantum mechanically they’re measurements applied to the quantum state of a thing.

For “convenience”, define the “angular momentum operator”, \vec{L}, as \hbar\vec{L}=\vec{R}\times\vec{P} or equivalently \hbar\vec{L}_\ell=\sum_{jk}\epsilon_{jk\ell}\vec{R}_j\vec{P}_k, where \epsilon_{jk\ell} is the “alternating symbol“.  This is just a more brute force way of writing the cross product.

Now check this out!  (the following uses identities from here and here)

\begin{array}{ll}\hbar^2[L_j,L_k]\\=[\hbar L_j,\hbar L_k]\\=\left[\sum_{st}\epsilon_{stj}\vec{R}_s\vec{P}_t, \sum_{mn}\epsilon_{mnk}\vec{R}_m\vec{P}_n\right]\\=\sum_{stmn}\epsilon_{stj}\epsilon_{mnk}\left[\vec{R}_s\vec{P}_t,\vec{R}_m\vec{P}_n\right]\\=\sum_{stmn}\epsilon_{stj}\epsilon_{mnk}\left(\left[\vec{P}_t,\vec{R}_m\right]\vec{R}_s\vec{P}_n+\left[\vec{P}_t,\vec{P}_n\right]\vec{R}_s\vec{R}_m+\left[\vec{R}_s,\vec{R}_m\right]\vec{P}_t\vec{P}_n+\left[\vec{R}_s,\vec{P}_n\right]\vec{P}_t\vec{R}_m\right)\\=\sum_{stmn}\epsilon_{stj}\epsilon_{mnk}\left(-i\hbar\delta_{tm}\vec{R}_s\vec{P}_n+0+0+i\hbar\delta_{sn}\vec{P}_t\vec{R}_m\right)\\=i\hbar\sum_{stmn}\epsilon_{stj}\epsilon_{mnk}\left(\delta_{sn}\vec{P}_t\vec{R}_m-\delta_{tm}\vec{R}_s\vec{P}_n\right)\\=i\hbar\sum_{stmn}\epsilon_{stj}\epsilon_{mnk}\delta_{sn}\vec{R}_m\vec{P}_t-i\hbar\sum_{stmn}\epsilon_{stj}\epsilon_{mnk}\delta_{tm}\vec{R}_s\vec{P}_n\\=i\hbar\sum_{stm}\sum_n\epsilon_{stj}\epsilon_{mnk}\delta_{sn}\vec{R}_m\vec{P}_t-i\hbar\sum_{stn}\sum_m\epsilon_{stj}\epsilon_{mnk}\delta_{tm}\vec{R}_s\vec{P}_n\end{array} \begin{array}{ll}=i\hbar\sum_{stm}\epsilon_{stj}\epsilon_{msk}\vec{R}_m\vec{P}_t-i\hbar\sum_{stn}\epsilon_{stj}\epsilon_{tnk}\vec{R}_s\vec{P}_n\\=i\hbar\sum_{tm}\left(\sum_s\epsilon_{stj}\epsilon_{msk}\right)\vec{R}_m\vec{P}_t-i\hbar\sum_{sn}\left(\sum_t\epsilon_{stj}\epsilon_{tnk}\right)\vec{R}_s\vec{P}_n\\=i\hbar\sum_{tm}\left(\delta_{tk}\delta_{jm}-\delta_{tm}\delta_{jk}\right)\vec{R}_m\vec{P}_t-i\hbar\sum_{sn}\left(\delta_{sk}\delta_{jn}-\delta_{sn}\delta_{jk}\right)\vec{R}_s\vec{P}_n\\=i\hbar\sum_{tm}\delta_{tk}\delta_{jm}\vec{R}_m\vec{P}_t-i\hbar\sum_{tm}\delta_{tm}\delta_{jk}\vec{R}_m\vec{P}_t-i\hbar\sum_{sn}\delta_{sk}\delta_{jn}\vec{R}_s\vec{P}_n+i\hbar\sum_{sn}\delta_{sn}\delta_{jk}\vec{R}_s\vec{P}_n\\=i\hbar\vec{R}_j\vec{P}_k-i\hbar\sum_{t}\delta_{jk}\vec{R}_t\vec{P}_t-i\hbar\vec{R}_k\vec{P}_j+i\hbar\sum_{s}\delta_{jk}\vec{R}_s\vec{P}_s\\=i\hbar\vec{R}_j\vec{P}_k-i\hbar\vec{R}_k\vec{P}_j\\=i\hbar^2\epsilon_{jk\ell}L_{\ell}\end{array}

Therefore: [L_j,L_k]=i\epsilon_{jk\ell}L_{\ell}.

So what was the point of all that?  It creates a relationship between the angular momentum in any one direction, and the angular momenta in the other two.  Surprisingly, this allows you to create a “ladder operator” that steps the total angular momentum in a direction up or down, in quantized steps.  Here are the operators that raise and lower the angular momentum in the z direction:

\begin{array}{ll}L_+ = L_x+iL_y\\L_- = L_x-iL_y\\\end{array}

Notice that

\begin{array}{ll}[L_z,L_+]\\=[L_z,L_x\pm iL_y]\\=[L_z,L_x]\pm i[L_z,L_y]\\=iL_y\pm i(-iL_x)\\=iL_y\pm L_x\\=\pm(L_x \pm iL_y)\\=\pm L_\pm\end{array}

Here’s how we know they work.  Remember that Li is a measurement of the angular momentum in the “j” direction (any one of x, y, or z).  For the purpose of making the math slicker, the value of the angular momentum is the eigenvalue of the L operator.  If you’ve made it this far; this is where the linear algebra kicks in.

Define the “eigenstates” of Lz, |m\rangle, as those states such that L_z|m\rangle=m|m\rangle.  “m” is the amount of angular momentum (well… “m\hbar” is), and |m\rangle is defined as the state that has that amount of angular momentum.  Now take a look at what (for example) L_+ does to |m\rangle:

\begin{array}{ll}L_zL_+|m\rangle\\=\left(L_zL_+-L_+L_z+L_+L_z\right)|m\rangle\\=\left([L_z,L_+]+L_+L_z\right)|m\rangle\\=[L_z,L_+]|m\rangle+L_+L_z|m\rangle\\=L_+|m\rangle+L_+L_z|m\rangle\\=L_+|m\rangle+mL_+|m\rangle\\=(1+m)L_+|m\rangle\end{array}

Holy crap!  L_+|m\rangle is an eigenstate of L_z with eigenvalue 1+m.  This is because, in fact, L_+|m\rangle = |m+1\rangle!

Assuming that there’s a maximum angular momentum in any particular direction, say “j”, then the states range from |-j\rangle to |j\rangle in integer steps (using the raising and lowering operators).  That’s just because the universe doesn’t care about the difference between the z and the negative z directions.  So, the difference between j and negative j is some integer: j-(-j) = 2j = “some integer”.  For ease of math the \hbar were separated from the L’s in the definition.  The actual angular momentum is “j\hbar“.

By the way, notice that at no point has mass been mentioned!  This result applies to anything and everything.  Particles, groups of particles, your mom, whatevs!

So, the maximum or minimum angular momentum is always some multiple of half an integer.  When it’s an integer (0, 1, 2, …) you’ve got a boson, and when it’s not (1/2, 3/2, …) you’ve got a fermion.  Each of these types of particles have their own wildly different properties.  Most famously, fermions can’t be in the same state as other fermions (this leads to the “solidness” of ordinary matter), while bosons can (which is why light can pass through itself).

Notice that the entire ladder operator thing for any L_j is dependent on the L operators for the other two directions.  In three or more dimensions you have access to at least two other directions, so the argument holds and particles in 3 or more dimensions are always fermions or bosons.

In two dimensions there aren’t enough other directions to create the ladder operators (L_\pm).  It turns out that without that restriction particles in two dimensions can assume any spin value (not just integer and half-integer).  These particles are called “anyons”, as in “any spin”.  While actual 2-d particles can’t be created in our jerkwad 3-d space, we can create tiny electromagnetic vortices in highly constrained, flat sheets of plasma that have all of the weird spin properties of anyons.  As much as that sounds like sci-fi, s’not.

It’s one of the several proposed quantum computer architectures that’s been shown to work (small scale).

This entry was posted in -- By the Physicist, Equations, Math, Particle Physics, Physics, Quantum Theory. Bookmark the permalink.

26 Responses to Q: What is “spin” in particle physics? Why is it different from just ordinary rotation?

  1. Andrew says:

    I laughed out loud at “pogo stick in a chess game.”

  2. Sanjay says:

    Great response!

  3. Edd Writer says:

    Damn gravy, it’s too thick for me to swallow… :(

  4. Pingback: Higgs Boson Impostors? It’s Possible, Say Scientists | Geeklin

  5. Pingback: Higgs Boson Impostors? It’s Possible, Say Scientists | www.D-7000.co.uk

  6. Erfan says:

    hello i decided to create an Electricity storage Circular Lamp but in this place no body can Help me . i sent a message to mitsubishi but they don’t answer me yet.
    i know every section of my plan.i know How should it be Creates. i just needs comes To your Laboratory and Realize my idea & Learn every more about These projects
    can you Do somthing for me?

  7. Alyster says:

    Hi, can you please explain what spinning faster than the speed of light means? Forgive my naiveté, but I don’t get how spinning can be measured in m/s. Thanks.

  8. The Physicist The Physicist says:

    You’re completely right, rate of rotation is measured in “angle per time” instead of “distance per time”.
    What the article is referring to is “tangential velocity”. For example, while the Earth turns at a rate of 360° per day (give or take), on the equator the tangential velocity is about 1,000 mph.

  9. David says:

    Wait, so if the angular momentum of say, electrons can only be deduced through their magnetic dipoles, then what is the difference between “spin” and “magnetic moment”?

    Aren’t these each treated as separate, intrinsic qualities that that things have or am I (yet again) confused?

  10. The Physicist The Physicist says:

    Spin is important for a number of reasons, with the magnetic moment being only one of them. The post talks about it a lot because it’s a good example of why spin can’t be classical (must be quantum mechanical).

  11. Phyllis McLemore says:

    According to Barbara Brennan, former NASA physicist, Consciousness vibrates faster than the speed of light. Instruments don’t pick it up because instruments cannot be built to detect any vibration not within this particular reality. In fact, parallel realities vibrate at faster and slower speeds (frequencies and amps) than the reality we think we are in right now. That is how we are not aware of them, except when angels and ghosts are seen. They are seen because they slow down.
    Electrons are shown in pictures as making circles around the nucleus of an atom. If there is no spin of these electrons then why are the paths around the nucleus shown to be circular or oval? If there is no actual spin, then how is there energy? If spin has to be quantum mechanical that means it’s spin is so fast that it cannot be detected. It is just theory because the spin is faster than the speed of light.
    Since all atoms have spinning electrons that are quantum speed that means that everything everywhere is quantum, not just at the macro size. I don’t understand how quanta are supposed to act differently that at the macro level. Isn’t the macro level full of spinning atoms? Aren’t atoms spinning as everything? Absolutely everything everywhere has to have a spin, a vibration or it does not exist.
    Sorry I am not a physicist, but angels talked to me my whole childhood and since then. I have to figure out why they are not always seen. They must be vibrating faster than the speed of light or we would see them all the time. If spin was classical then that would mean we could see it, that it’s velocity is no faster than the speed of light?

  12. Mark McCulloch says:

    This is far and away the best explanation of quantum spin I have read. Well done in making a very un-intuitive subject accessible.

  13. Swapnil says:

    What does it mean by the spin of protons and neutrons? And why does it apparently cause magnetism to arise

  14. eyelive4everNOW says:

    If electrons do not spin, but rotate, then protons and neutrons must rotate.
    Centrifugal force is why there is magnetism and gravity. Isn’t magnetism another kind of gravity? Seems the faster something rotates, the stronger the pull, the further the expansion. The earth spins and rotates. Somebody wrote me yesterday and said that gravity is just a theory. Then I looked up gravity and read that the effects of gravity are indistinguishable from the affects of acceleration. When a magnet pulls or repels another magnet isn’t that an acceleration?
    Instruments have been built to measure gravity. A ship mounted gravimeter was developed by Lucien Lacoste and Arnold Romberg in 1936.
    Acceleration is a very telling word. So since magnetism pulls or it repels energy, it too could be said “the effects of magnetism are indistinguishable from the affects of acceleration.” The faster the rotation, the harder the pull, the faster the acceleration.
    Is pull the same as quantum entanglement? Is gravity and magnetism and quantum entanglement different words for the same force? or the same energy?
    I’ve read that consciousness is vibrating all rotations and spins. So I like to study why there is rotation and spin at all. There is energy behind all movement and energy has to be directed. Or it would be like that quantum soup where quanta and photons just seem to go about willy nilly with no direction, but still pulsating and eternal and magical and mind bending.
    But then, that willy nilly rotating suddenly turns, and then an idea, a creation is suddenly manufactured by these quanta/photons because they are in the building of the idea, they are there….and the idea turns them and then they picture what the idea is. It is holographic. The quanta/photons form a picture from the idea…..and then idea just seems to explode, to become a bigger picture.
    I have read the picture is actually a series of waves which are measured as frequencies. Since waves are the core of everything, then magnetism and gravity and rotation and spin are all tied up together, affecting each other, ONE with each other all the time; always knowing what the other is doing.
    No secrets there, just like between people. Thoughts are energy and they rotate and spin and accelerate, too. They flow through us and other people, changing the polarization of every quanta with magnetism, pull and acceleration.
    It is an electromagnetic ocean of quanta, etc that we breathe and vibrate in.

  15. As I understand string theory (and I do).

    Take a 100,000 mile long “string” and make it conductive by plugging the ends together (a standing wave)
    and you prove E=MC^2.

    When the current flows EMF causes the loop to twist back on itself and compact
    (a 5 ft. long loose coil of #32 copper wire and a 9v battery prove this).

    The energized “string” spins and shrinks and a sub atomic “particle” is born.
    (be careful it is spinning faster than the speed of light). Remember the string has a thickness of the plank length.

    The internal spin of the “particle” creates mass/gravity (like gyroscopic spin creates mass/gravity).
    The unit resonates and is stable vibrating at lower speeds which can be detected at about 10^-13m.

    If you want to make other “particles” use different length strings. If you choose the wrong length it will
    decay to a stable “particle” by shattering into radiation and/or by splitting into two or more stable “particles”.
    All mater and energy will be conserved in the process because the radiation becomes part of the
    UCBR (those tiny shreds from the past to short to make one of the original particles).
    Put a “particle” in a collider and you can shatter it into sub-particles and radiation all of which may live
    for a nano-second and poof they all disappear back into strings

    Is that all it takes to make matter? (If the answer is yes, you got it right!).

    The spin being discussed here is not spin but harmonic vibrations.
    Regards, IGBY International

  16. singing says:

    What causes the energy to vibrate and resonant so that a string exists to form a particle?
    Since we are a series of waves and strings why is the word “solid” used over and over just to confuse all of us? And it is a definite confusion. If we are solid, then we are left with no source, no purpose, no life after death. Then the world fights and murders and steals because the only purpose it sees is to gain more and more and more, whether it is oil, or land or love.
    But we are not solid and we do have a source at least as intelligent as our DNA. We do come from somewhere for a reason before entering these bodies and we do go somewhere for a reason after leaving these bodies. DNA does not vibrate and resonate in such complex patterns with no reason and without a source.
    Read the book “Hands of Light” written by the former NASA physicist Barbara Brennan. If I can, and I have, healed myself with vibrations instantaneously, then harmony was definitely present; the harmony of an intelligent source that vibrates my quarks and electrons and atoms, etc. Then, I am the source, perfectly oscillating with waves and strings and coils which are all in harmony.

  17. Robert McLaughlin says:

    Is it possible that the intrinsic ‘spin’ of elementary particles is a result of the vibrations of strings from string theory?

  18. Hypatia says:

    Is spin related to the difficulty of detecting a elementary particle? For example, the discovery of the Higg’s boson with a spin 0 has required an astounding amount of effort, and the graviton if it exists has a predicted spin 2.

  19. Mariano Quiroga says:

    Should we definitely rule out the possibility of superluminal rotation for particles? Nothing can outrun “c” traveling through space, but self rotating seems like a different situation. A seemingly impossible angular momentum might imply that, depending on the radius of the spinning object, its surface could be speeding at c, while the rest of the particle is not. Would this be relevant for particles that aren’t even considered 3 Dimensional? Could this make any sense among the weirdness of the quantum realm?

  20. Alistair says:

    I don’t have awareness of all current theory, or the necessary math to determine, whether it is possible for a universal bath of miniscule sub atomic particles to exist, that are omnipresent, and billionths or trillionths of times smaller than an atom. That they are like a gigantic McDonald’s ball bath, constantly spinning in unison, like rollers at the beer store. With different sized particles within it, reacting in different manners to the pressure, and spin, often by setting up patterns like clumping, orbiting, and such, because in these compositions, the ever present tiny balls, get themselves into an equilibrium again. Waves of “pressure” and “force” through these itty-bitty balls are the components of all “energy”. Everything in the universe becomes an explainable mechanical occurence at that point. no mystery forces. ???

  21. Xerenarcy says:

    @Mariano Quiroga
    i’m with you on this. the speed of light applies to the propagation speeds of electromagnetic waves, it is not an absolute limit per se just that so much of physics involves EM it’s hard to find a situation when you’re not bound by it. this looks like one of those situations in principle but i’m not entirely sure.

    furthermore, if spin is quantized then how does it change direction?

  22. yoron says:

    Would you agree to that spin isn’t anything similar to angular momentum? Or do you want it to be a equivalence? As for saying it is more important, is the idea that a macroscopic angular momentum could be a result from particles spin? As a ideal form of some type?

    It really hurts my head this one, trying to connect spin to angular momentum, but it is information, isn’t it? And in that motto meaningful to the eye, and brain? We can see polarizations, that’s what sunglasses is for. On the other hand, if I get it right that is a macroscopic result from how matter is arranged. And yes, as always your explanation tries to give the gist of it.

    but it still hurts my head.

  23. Cass says:

    It seems to me that when position and momentum are along the same axis, the result of RP-PR or PR-RP indicates movement through space in either the positive or negative direction along that axis, whereas when P,R commute, it indicates the particle is stationary in space.
    Is that correct?
    Thanks

  24. Pingback: The End of an Era: Varian and the Birth and Growth of NMR | The Unemployed Chemist

  25. M Abuzar bhatti says:

    i think spin in atomic level is d same as in angular momentun in macro level,but we cant see the spin because d atomic particles moves with more than the speed of light and hence treat as a wave…and when these waves combine to form a macro particle,they slowed down their speed…….m i right?

  26. Ziriax says:

    I recently learned about Grassmann algebra and geometric algebra.

    Using these algebra’s angular momentum is not defined using the cross product, but as a bivector, using the wedge product. The cross product is not invariant under reflection, while the wedge product is. Reflection plays an important part when it comes to rotations, since a rotation about an angle can be described as a double reflection. I wonder if the up/down electron spin is actually a side effect of using incorrect mathematical modelling, namely the usage of the cross product instead the wedge product?

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>