Spin, Orbital Angular Momentum, and Total Angular Momentum
Spin (quantum number S) is a vector quantity that represents the "intrinsic" angular momentum of a particle. It comes in increments of 1⁄2 ħ (pronounced "h-bar"). The ħ is often dropped because it is the "fundamental" unit of spin, and it is implied that "spin 1" means "spin 1 ħ". (In some systems of natural units, ħ is chosen to be 1, and therefore does not appear in equations).
Quarks are fermions—specifically in this case, particles having spin 1⁄2 (S = 1⁄2). Because spin projections vary in increments of 1 (that is 1 ħ), a single quark has a spin vector of length 1⁄2, and has two spin projections (Sz = +1⁄2 and Sz = −1⁄2). Two quarks can have their spins aligned, in which case the two spin vectors add to make a vector of length S = 1 and three spin projections (Sz = +1, Sz = 0, and Sz = −1), called the spin-1 triplet. If two quarks have unaligned spins, the spin vectors add up to make a vector of length S = 0 and only one spin projection (Sz = 0), called the spin-0 singlet. Since mesons are made of one quark and one antiquark, they can be found in triplet and singlet spin states.
There is another quantity of quantized angular momentum, called the orbital angular momentum (quantum number L), that comes in increments of 1 ħ, which represent the angular moment due to quarks orbiting around each other. The total angular momentum (quantum number J) of a particle is therefore the combination of intrinsic angular momentum (spin) and orbital angular momentum. It can take any value from J = |L − S| to J = |L + S|, in increments of 1.
| S | L | J | P (See below) |
JP |
|---|---|---|---|---|
| 0 | 0 | 0 | − | 0− |
| 1 | 1 | + | 1+ | |
| 2 | 2 | − | 2− | |
| 3 | 3 | + | 3+ | |
| 1 | 0 | 1 | − | 1− |
| 1 | 2, 1, 0 | + | 2+, 1+, 0+ | |
| 2 | 3, 2, 1 | − | 3−, 2−, 1− | |
| 3 | 4, 3, 2 | + | 4+, 3+, 2+ |
Particle physicists are most interested in mesons with no orbital angular momentum (L = 0), therefore the two groups of mesons most studied are the S = 1; L = 0 and S = 0; L = 0, which corresponds to J = 1 and J = 0, although they are not the only ones. It is also possible to obtain J = 1 particles from S = 0 and L = 1. How to distinguish between the S = 1, L = 0 and S = 0, L = 1 mesons is an active area of research in meson spectroscopy.
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