# Difference between Delta particles and protons and neutrons

## Main Question or Discussion Point

I am trying to figure out the difference between Δ0 and Δ- and the proton and neutron since both appear to have the same combinations of up/down quarks.

Deltas have isospin 3/2 and spin angular momentum 3/2 whereas protons and neutrons have isospin 1/2 and spin angular momentum 1/2. I'm thinking that the difference between the two is created via different linear combinations of up and down quarks that meet the isospin and spin angular requirements. For example, Δ0 has the form (neglecting coefficients) of

uud + udu + duu

and Δ- has the form

udd + dud + ddu

and both yield isospin = spin = 3/2

Is my thinking correct? I am having a hard time finding a simple description of the quark combinations for the proton and neutronso if anyone has a pointer that would be appreciated. I have been looking at the Wikipedia article on isospin http://en.wikipedia.org/wiki/Isospinon [Broken] but am not sure how to interpret the matrices.

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## Answers and Replies

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ChrisVer
Gold Member
Is my thinking correct? I am having a hard time finding a simple description of the quark combinations for the proton and neutronso if anyone has a pointer that would be appreciated. I have been looking at the Wikipedia article on isospin http://en.wikipedia.org/wiki/Isospinon [Broken] but am not sure how to interpret the matrices.
What is so confusing in the matrices? They give you the proton function as a state of three quarks and their spins... It's just in a more compact form, because otherwise the proton and neutron wavefunctions would look terrible... In particular if you have (after doing the multiplications of matrices) something like:
$|u u d> | \uparrow \uparrow \downarrow> = | u \uparrow, u \uparrow , d \downarrow >$ and so on...

As for the Deltas, you already pointed out their difference having different spins... They can be seen as excited states of proton/neutron.

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Thanks. Now it makes sense. This is my first venture into the Standard Model. My background is in QM so I understand the notations but the concept of isospin is new to me.

ChrisVer
Gold Member
The isospin for the two quarks, is in the notation of $u,d$ not to the arrows which denote the spin... It's better then to avoid writing things like isospin=spin=...
Spin and Isospin are different spaces.

Staff Emeritus
2019 Award
both appear to have the same combinations of up/down quarks.
Why should that mean they are the same particles? Look at chemistry: C3H8O could be propanol, isopropanol, or methyl ethyl ether.

I am trying to figure out the difference between Δ0 and Δ- and the proton and neutron since both appear to have the same combinations of up/down quarks.

Deltas have isospin 3/2 and spin angular momentum 3/2 whereas protons and neutrons have isospin 1/2 and spin angular momentum 1/2.

Not so: http://en.wikipedia.org/wiki/Delta_baryon. Δ0 and Δ- have isospin +/- 1/2, the same as the nucleons. As the others say, the isospin just comes from the quark content, so states with the same quark content must also have the same isospin. Only the spin is different; the Deltas have all their spins pointing in the same direction, while nucleons have one spin pointing antiparallel. Turns out that flipping that one spin around takes quite a lot of energy though, so there is a significant mass difference between the two states.

Staff Emeritus
2019 Award
Δ0 and Δ- have isospin +/- 1/2, the same as the nucleons.
No, they have the 3rd component of isospin, I3 of +1/2 and -1/2, the same as the nucleus. But they have isospin 3/2, not isospin 1/2. I am assuming you mean Δ+ and not Δ-. Δ- has I3 = -3/2.

No, they have the 3rd component of isospin, I3 of +1/2 and -1/2, the same as the nucleus. But they have isospin 3/2, not isospin 1/2. I am assuming you mean Δ+ and not Δ-. Δ- has I3 = -3/2.
Sorry yes I meant Δ+; I assume OP meant this also. And yes I mean 3rd component of isospin. I do not know what you are talking about when you say "But they have isospin 3/2"; isospin isn't a scalar quantity, that's why we usually talk about I3. I guess you mean the magnitude? Ok sure it is 3/2 for the Delta baryons. But actually I am not sure of the physical significance of this. I3 tells you what u and d quark content the state has, but what does it mean for a particle to have I=3/2, other than to say that it belongs to a particular multiplet? There must be a more fundamental QCD explanation.

Staff Emeritus
2019 Award
Isospin differences is what separates the deltas from the N's and the sigmas from the lambdas. Yes they "only' belong to a particular multiplet, but which multiplet they belong to is important.

Yes but *why* is important and what does it mean? In the case of fundamental particles I get it, it is clear to me the physical significance of the Higgs SU(2) doublets or quark SU(3) triplets, but what does it mean in these approximate strong sector symmetries? I assume there must be some description in terms of fundamental particles.

E.g. for the actual spins it is obvious enough why they are grouped a certain way, and if we have three parallel spins then the total J is 3/2, and there are possible z projections -3/2,-1/2,1/2,3/2, and somehow the isospins are analogous to this in that we have 3 "u/d" quarks, but if their isospins were "parallel" I would expect that to mean the quark content must be uuu or ddd or some superposition of these, not things like uud. Yet we do have uud in the Delta baryons. So I apparently don't understand the analogy correctly.

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Staff Emeritus
2019 Award
Isospin is a fundamental property of a hadron. For example, it determines their decays: a ρ0 and an ω0 have the same charge, mass (to a good approximation) and quark content. But one decays to two pions and the other decays to three pions.

ChrisVer
Gold Member
E.g. for the actual spins it is obvious enough why they are grouped a certain way, and if we have three parallel spins then the total J is 3/2, and there are possible z projections -3/2,-1/2,1/2,3/2, and somehow the isospins are analogous to this in that we have 3 "u/d" quarks, but if their isospins were "parallel" I would expect that to mean the quark content must be uuu or ddd or some superposition of these, not things like uud. Yet we do have uud in the Delta baryons. So I apparently don't understand the analogy correctly.
Take for example the spin J=3/2...
You can have the $\uparrow \uparrow \uparrow$ state in it, but you can also have:
$\uparrow \uparrow \downarrow +\uparrow \downarrow \uparrow + \downarrow \uparrow \uparrow$ has spin J=3/2 with projection +1/2.

So I suppose you can understand that the uuu can belong to I=3/2 as well as uud.

Take for example the spin J=3/2...
You can have the $\uparrow \uparrow \uparrow$ state in it, but you can also have:
$\uparrow \uparrow \downarrow +\uparrow \downarrow \uparrow + \downarrow \uparrow \uparrow$ has spin J=3/2 with projection +1/2.

So I suppose you can understand that the uuu can belong to I=3/2 as well as uud.
Oh, yes of course, I forgot that this happens. Intuitively I still don't understand such states, but ok the analogy works just fine :).

ChrisVer
Gold Member
Oh, yes of course, I forgot that this happens. Intuitively I still don't understand such states, but ok the analogy works just fine :).
It's intuitively-fine, if you understand that the projection +1/2 and -1/2 for the 3rd component of spin, exist in the representation of J=3/2 .. So you can see that the arrows up+up+down= up.
They are just linearly independent from those in repr J=1/2 ...

It's intuitively-fine, if you understand that the projection +1/2 and -1/2 for the 3rd component of spin, exist in the representation of J=3/2 .. So you can see that the arrows up+up+down= up.
They are just linearly independent from those in repr J=1/2 ...
The +1/2 -1/2 z-projections make sense, sure, it is more the fact that a weird combination of states with two ups and a down can nevertheless have a total spin of 3/2 that I have no good mental model for. I suppose it should be thought of differently; i.e. those "ups" and "downs" in the component states are really z-projections themselves. So the full spin-vector with length 3/2 "points" in a strange direction away from the z axis, such that if you measure the z-projections of each individual spin you will get two ups and a down (i.e. the full state vector points more in the up-z direction than the down-z direction). I.e. we might say the three spins "are" fully aligned in the J=3/2 case, (so that a 3/2=1/2+1/2+1/2 understanding of the total spin makes sense), but if they are not aligned with the z-axis then the z-projected spins are not all aligned. So the z-projected spin is then less than the total spin. Or something like that. It is even more abstract in the u/d case...

ChrisVer
Gold Member
Well that's write, the arrows only show the orientation of spin and not the representation.
The irreducible reps are obtained by taking the tensor products of the spins you combine (i.e. $2 \otimes 2 \otimes 2= 4 \oplus 2$ ). The only good matches are the very largest/lowest states, for which you have $J_z = \pm J$ and that's (I guess) why you can easily figure them out.
Then start acting with lowering/raising operators and you build the space.