Why can mesons have spin 2 if quarks are spin-1/2?

[latentcorpse]

hi,

my notes say that a meson is a quark anti quark pair and that all quarks are spin half fermions. fair enough. it then says that mesons are bosons with spin $0, 1 \hbar, 2 \hbar \dots$

this i don't get.

apparently, antimatter has the same mass, opposite electric charge and also the opposite of any additive quantum number of its corresponding matter particle.

spin is an additive quantum number so surely this means that all antiquarks have spin -1/2

therefore the total spin of the meson is either 1 or 0 (by the angular momentum addition theorem). how would it ever be possible to have a meson with spin 2 for example? or are my notes wrong here?

thanks.

 

[vela]

Spin is not an additive quantum number. Antiquarks, like quarks, have spin 1/2.

Remember that mesons consist of two particles bound together, so their orbital angular momentum contributes to the total angular momentum of the particle.

 

[latentcorpse]

Spin is not an additive quantum number. Antiquarks, like quarks, have spin 1/2.

Remember that mesons consist of two particles bound together, so their orbital angular momentum contributes to the total angular momentum of the particle.

okay. so just to clarify, what is an additive quantum number?

there are no restrictions on the l quantum number of a quark are there? besides that it must have an integer value according to the quantum state of that quark. for example a quark in the 1d state would have l=2 and if this were paired with an antiquark in a 1s (l=0) state then the total l would be 2 and the total s is going to be 1 or 0 depending on the alignment of the spin vectors of the component quarks within the meson. therefore the total angular momentum of the meson is going to be 3,2 or 1.
hopefully the above is correct but in my notes it was saying that the meson was a boson with spin 0,1,2 etc. When it says spin does it mean "total angular momentum, J=L+S" or does it mean the "total intrinsic spin, S"? It's pretty ambiguous.

 

[vela]

An additive quantum number is one like charge. To find the total charge of two particles, for instance, you just sum the individual charges. Spin doesn't combine like that. If you have two electrons, for example, each has s=1/2, but they can form a singlet state with s=0 or a triplet state with s=1.

It's the pair of quarks that have an orbital angular momentum, not each individual quark, but other than that, what you said is right. I think "spin" refers to the total angular momentum. Looking from the outside, you see a single particle with "intrinsic" angular momentum J. It's only when you look closer that you see the meson is really a collection of particles.

 

[latentcorpse]

thanks. but why do the quarks not have individual orbital angular momenta?

 

[vela]

Orbital angular momentum is a property of their bound state, not the individual quarks.

 

[latentcorpse]

i don't really understand why this is the case though. should i just accept it as a fact?

 

[vela]

It's probably easiest to see this by considering the case of two non-relativistic, interacting particles. Their Hamiltonian is given by

$$H=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+V(|r_1-r_2|)$$

You should recall from classical mechanics that you can rewrite it in terms of the motion of the center of mass of the system and motion relative to the center of mass:

$$H=\frac{p_{cm}}{2(m_1+m_2)}+\left(\frac{p^2}{2\mu}+V(|r|)\right)$$

where μ is the reduced mass, p_cm=p₁+p₂, p=(m₂p₁-m₁p₂)/(m₁+m₂), and r=r₂-r₁. The first term in the Hamiltonian describes how the whole system moves. The orbital angular momentum comes from the second part of the Hamiltonian, the part that has to do with how the particles are bound together.