Understanding Spin 2 Particles: Examples and Intuition

[Wahid]

We've all skipped a few undergraduate QM lectures, right?

Right, and we all know how bad an idea that was, right?

Right, so when the time comes and you wished you hadn't skipped those lectures, you probably don't have enough time to swim around in your misery for long, because in 48 hours you've got an exam you can't afford to do badly in!

Such is the scene of my life right now, and I would really appreciate if you could help me with electron spin.

What is it? and where can I find the relevant conceptual and mathematical details on the internet?

Thank you in advance.

Wahid

 

[mormonator_rm]

The concept of basic electron spin is that electrons are spin-1/2 particles. That spin of 1/2 can be either positive of negative in magnitude, and the Pauli Exclusion Principle requires that no two electrons with the same spin magnitude can exist in one pair together.

Because this spin is not something that you can physically view an electron performing, you must visualize it abstractly. Imagine that you have a deck of cards, and that you take out one of the face cards. How far do you have to turn it before it looks the same again? 180 degrees, right? A face card is a spin-2 object, because it will appear the same twice in a 360 degree rotation. Imagine if you had to turn that face card 720 degrees to get it to look the same! It is not intuitive, nor does it seem natural! But that is what a spin-1/2 particle is like.

So, the spin number can essentially be described as a full rotation, 360 degrees, divided by an intrinsic symmetry angle. This is not a physical angle necessarily, but rather an abstract phase angle. But it is good to try to visualize it. We live in a vector-world, where everything intrinsically appears to have a spin-1 character, so spin-1/2 is especially hard to visualize.

You probably don't need to know that for the test, but it is just a way to understand what is going on. Beyond that, you probably need to understand spin coupling and degeneracy. You need to look in a textbook for those things; it would take a while to explain it in full here.

 

[lethe]

Originally posted by mormonator_rm
That spin of 1/2 can be either positive of negative in magnitude

more precisely, the result of the measurement of the component of spin along a specific axis can be positive or negative ($\pm\hbar/2$).

and the Pauli Exclusion Principle requires that no two electrons with the same spin magnitude can exist in one pair together.

more precisely, the Pauli exlusion principle states that no two fermions can be in the same quantum state. thus for two electrons in the same radial wavefunction of an atom, the spins of the two electrons must be different to satisfy the Pauli exclusion principle.

Because this spin is not something that you can physically view an electron performing, you must visualize it abstractly.

i think the name "intrinsic angular momentum" says it all. spin is that angular momentum that an object has when it is not moving.

Imagine that you have a deck of cards, and that you take out one of the face cards. How far do you have to turn it before it looks the same again? 180 degrees, right? A face card is a spin-2 object, because it will appear the same twice in a 360 degree rotation.

i m not sure what this example is supposed to illustrate, but i can assure you, you need to rotate a spin-2 object 360 degrees to get back where you started. not 180. any integer spin object.

Imagine if you had to turn that face card 720 degrees to get it to look the same! It is not intuitive, nor does it seem natural! But that is what a spin-1/2 particle is like.

it comes back to where it started after 360 degree rotation, up to a phase. insofar as any quantum state can seem intuitive (i.e. modding out by phase) then this agrees with intuition.

So, the spin number can essentially be described as a full rotation, 360 degrees, divided by an intrinsic symmetry angle. This is not a physical angle necessarily, but rather an abstract phase angle.

i don t exactly know what you re saying here, but it sounds fishy to me.

But it is good to try to visualize it. We live in a vector-world, where everything intrinsically appears to have a spin-1 character, so spin-1/2 is especially hard to visualize.

we do have some higher rank tensors in our macroscopic world, although i don t know how easy it is to "visualise" a stress tensor.

 

[lethe]

Originally posted by Wahid
We've all skipped a few undergraduate QM lectures, right?

you bet! and grad QM lectures as well! not to mention lots of other courses! i sympathize

Right, and we all know how bad an idea that was, right?

yeah, but it sure seemed like a good idea at the time.

Such is the scene of my life right now, and I would really appreciate if you could help me with electron spin.

alright, i ll say some stuff. i doubt that my rambling will help you more than sitting down with the textbook for 4 hours, but here it is, since you asked for it:



a spin j quantum state has 2j+1 dimensions, as a quantum mechanical Hilbert space. that is, it is spanned by a basis of 2j+1 vectors, one for each possible z-component of the spin (which i will denote with m).

$\sqrt{j(j+1)}$ is the value of the angular momentum of any such state.

the possible values of m range from -j to j (and you can easily see that there are 2j+1 possible values, agreeing with what i claimed above). so the maximum value you can measure for the

i can label any quantum state by these two numbers, then. they are so called quantum numbers. i can call the basis vectors in my Hilbert space |j,m>

so far, this is all very easy. where it gets hard is when you have to tensor product two angular momentum Hilbert spaces together. this happens, for example, when you put an electron (which has intrinsic angular momentum) in orbit around an atom (so it also has orbital angular momentum). another example is to just put two particles with spin together.

when you do this, let's say it is a particle with spin s and angular momentum l. then the total angular momentum operator is J=S+L. the z-component quantum numbers for the two separate hilbert spaces (intrinsic and orbital) are no longer good quantum numbers for this new Hilbert space. instead, i need to consider the z-component of the total angular momentum.

umm... i need to go to class. more later.

 

[mormonator_rm]

Originally posted by lethe
more precisely, the result of the measurement of the component of spin along a specific axis can be positive or negative ($\pm\hbar/2$).

I apologize for my generalization. In working with mesons I typically tend to think of all my spin momentum, angular momentum, and isospin quantities expressed in units of $\hbar/2$, or in other words just the quantum numbers. I am sorry I was awfully careless there.

Originally posted by lethe
more precisely, the Pauli exlusion principle states that no two fermions can be in the same quantum state. thus for two electrons in the same radial wavefunction of an atom, the spins of the two electrons must be different to satisfy the Pauli exclusion principle.

Correct. Thankyou for the clarification. I really fumble with my words alot, as you can all tell.

Originally posted by lethe
i think the name "intrinsic angular momentum" says it all. spin is that angular momentum that an object has when it is not moving.

I see that I over-clarified. Sorry again.

Originally posted by lethe
i m not sure what this example is supposed to illustrate, but i can assure you, you need to rotate a spin-2 object 360 degrees to get back where you started. not 180. any integer spin object.

To clarify, I used this example with reference to phase-space, not physical space. The rotation is not real, but is a phase in the wave-function. Often this effect can be seen in data such as Dalitz plots and histograms. I do acknowledge, though, that all integer-spin particles do have a node at a phase angle of 360 degrees.

Originally posted by lethe
it comes back to where it started after 360 degree rotation, up to a phase. insofar as any quantum state can seem intuitive (i.e. modding out by phase) then this agrees with intuition.

Yes, up a phase, but at a different point in its phase oscillation (if my wording makes any sense; I'm trying to figure out how to explain it with the right terminology but can't remember). It does agree with intuition if you think it through, it just might not appear intuitive at first.

Originally posted by lethe
i don t exactly know what you re saying here, but it sounds fishy to me.

It works when dealing with parity conservation and angular distribution data, and that's where I use it most.

Originally posted by lethe
we do have some higher rank tensors in our macroscopic world, although i don t know how easy it is to "visualise" a stress tensor.

Certainly, but I was just referring to the fact that we see in three dimensions with our eyes, and so everything we see appears in the form of vectors. I wasn't really thinking about tensors at the time I wrote this, but you are right; they are there, too.


As you can tell, my brain is just not really wired for thinking about fermions a whole lot, especially not in groups. By the way, your explanation of spin coupling is excellent and very clear. Way better than four hours with a textbook.

Wahid if you need anymore help, I think lethe is the guy to ask. After this raking, I'm definitely going to bow out now.

 

[lethe]

Originally posted by mormonator_rm

To clarify, I used this example with reference to phase-space, not physical space. The rotation is not real, but is a phase in the wave-function. Often this effect can be seen in data such as Dalitz plots and histograms. I do acknowledge, though, that all integer-spin particles do have a node at a phase angle of 360 degrees.

so are you saying that the phase of the wavefunction of a spin 2 particle returns to its starting point after a 180 degree rotation? hmm... let me think about that. if that is true, i didn t know that.

but my point here was, the angular momentum vector of a spin 2 particle will be pointing the opposite way after a 180 degree rotation, and requires a 360 rotation to come back to where it started.

the same will be true of any spin j particle for j an integer. (and 720 degrees for any half integer)

Yes, up a phase, but at a different point in its phase oscillation (if my wording makes any sense; I'm trying to figure out how to explain it with the right terminology but can't remember). It does agree with intuition if you think it through, it just might not appear intuitive at first.

well, i hear all the time that spin half particles just defy all intuition, and to that i make two objections:

1. there are classical configurations that you can make that can easily demonstrate that you really need 720 degrees, so this doesn t really hinge on half integral spin.

2. the real difference after a 360 degree rotation is only a difference in phase, and i claim that no one has any intuition about phase anyway. so insofar as anyone has any intuition about quantum mechanical objects (with arbitrary spin), that intuition is still valid for spin 1/2 objects as well.

but this is just my opinion against the party line of the textbooks, so i don t mean to be picking on you on this point. and feel free to disagree.

It works when dealing with parity conservation and angular distribution data, and that's where I use it most.

i ll take your word for it. i m just not sure what this "spin number" and "intrinsic symmetry angle" are. it sounded to me like you wanted to make some rule like this: a spin j particle has to be rotated 360/j degrees before it comes back to where it started, and to that, i object.

perhaps now you want to weaken your statement and say that when you rotated a spin j particle by 360/j degrees, the phase returns to where it started.

is this your claim? if so, i would be interested to hear more about it.

Certainly, but I was just referring to the fact that we see in three dimensions with our eyes, and so everything we see appears in the form of vectors. I wasn't really thinking about tensors at the time I wrote this, but you are right; they are there, too.

i guess i agree that all intuition that we have about rotation is about rotations of vectors. but there are plenty of everyday classical objects that one can find that demonstrate the failure of this intuition.

By the way, your explanation of spin coupling is excellent and very clear. Way better than four hours with a textbook.

thanks!

After this raking, I'm definitely going to bow out now.

hehehe... sorry about that...

 

[mormonator_rm]

Originally posted by lethe
so are you saying that the phase of the wavefunction of a spin 2 particle returns to its starting point after a 180 degree rotation? hmm... let me think about that. if that is true, i didn t know that.

but my point here was, the angular momentum vector of a spin 2 particle will be pointing the opposite way after a 180 degree rotation, and requires a 360 rotation to come back to where it started.

the same will be true of any spin j particle for j an integer. (and 720 degrees for any half integer)



well, i hear all the time that spin half particles just defy all intuition, and to that i make two objections:

1. there are classical configurations that you can make that can easily demonstrate that you really need 720 degrees, so this doesn t really hinge on half integral spin.

2. the real difference after a 360 degree rotation is only a difference in phase, and i claim that no one has any intuition about phase anyway. so insofar as anyone has any intuition about quantum mechanical objects (with arbitrary spin), that intuition is still valid for spin 1/2 objects as well.

but this is just my opinion against the party line of the textbooks, so i don t mean to be picking on you on this point. and feel free to disagree.

I'll do my best with what little time I have right now to give some basic examples. I think I'll demonstrate this using the first spin-triplet set of mesons, specifically the lighter isosinglets. They'll be a good example because they have the same angular momentum and spin momentum numbers, but couple to different values of J. So, we start with:

the scalar meson f₀(1370)
the pseudovector meson f₁(1285)
the tensor meson f₂(1270)

All three have L = 1, S = 1, and positive parity, but span the set of multiplets J^P = {0⁺,1⁺,2⁺}. First, f₀(1370) has spin and angular momentum coupling to J = 0. It can be viewed as having the angular momentum vector and spin momentum vector being equal and opposite at all times, so that the net result is zero no matter how you "turn" them in phase space. Therefore, no matter how small you make $d\theta$, the particle will always "look" the same (have the same nature in any phase). I should mention this, you'll like it; this is the case where the shortcut rule $\theta$_symmetry$= \frac {360}{J}$ breaks down (because it is undefined where it should be zero). The shortcut only works for J not equal to zero. So, for J = 0 the rule could be expressed as its inverse, $\theta$_symmetry$= \frac {J}{360}$, but what's the point of having a rule that works for only one situation. So I throw that one out at J = 0, and the usual shortcut works in every other case.

f₁(1285) could be viewed as having its angular momentum and spin momentum vectors coupled in the same direction and rotating together. The combined vector rotates about the axis only once every phase, hence J = 1. Simple enough to visualize.

f₂(1270) could be viewed as having angular momentum and spin momentum vectors that are rotating in opposite directions about the axis. Twice in every phase, the vectors will be opposite to each other, and twice in every phase they will coincide, hence J = 2. The particle will "look" the same twice in each phase, hence a symmetry angle of 180 degrees.

Now that I've nailed out those examples properly, I will try to think about the other things you have said and reply. I can see how you can make a spin-2 system classically out of two vector objects, like if you had a mechanical system that performed this, for example. I'll try edit this post again as soon as I can, and I hope I did the symbols properly; if not, then you will know how truly computer-illiterate I am...