Baryon - Properties - Spin, Orbital Angular Momentum, and Total Angular Momentum

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 anywhere.

Quarks are fermionic particles of spin 1⁄2 (S = 1⁄2). Because spin projections varies 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). If two quarks have unaligned spins, the spin vectors add up to make a vector of length S = 0 and has only one spin projection (Sz = 0), etc. Since baryons are made of three quarks, their spin vectors can add to make a vector of length S = 3⁄2, which has four spin projections (Sz = +3⁄2, Sz = +1⁄2, Sz = −1⁄2, and Sz = −3⁄2), or a vector of length S = 1⁄2 with two spin projections (Sz = +1⁄2, and Sz = −1⁄2).

There is another quantity of angular momentum, called the orbital angular momentum, (azimuthal 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 (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 = |LS| to J = |L + S|, in increments of 1.

Baryon angular momentum quantum numbers for L = 0, 1, 2, 3
Spin (S) Orbital angular momentum (L) Total angular momentum (J) Parity (P)
(See below)
Condensed notation (JP)
1⁄2 0 1⁄2 + 1⁄2+
1 3⁄2, 1⁄2 3⁄2−, 1⁄2
2 5⁄2, 3⁄2 + 5⁄2+, 3⁄2+
3 7⁄2, 5⁄2 7⁄2−, 5⁄2
3⁄2 0 3⁄2 + 3⁄2+
1 5⁄2, 3⁄2, 1⁄2 5⁄2−, 3⁄2−, 1⁄2
2 7⁄2, 5⁄2, 3⁄2, 1⁄2 + 7⁄2+, 5⁄2+, 3⁄2+, 1⁄2+
3 9⁄2, 7⁄2, 5⁄2, 3⁄2 9⁄2−, 7⁄2−, 5⁄2−, 3⁄2

Particle physicists are most interested in baryons with no orbital angular momentum (L = 0), as they correspond to ground states—states of minimal energy. Therefore the two groups of baryons most studied are the S = 1⁄2; L = 0 and S = 3⁄2; L = 0, which corresponds to J = 1⁄2+ and J = 3⁄2+, respectively, although they are not the only ones. It is also possible to obtain J = 3⁄2+ particles from S = 1⁄2 and L = 2, as well as S = 3⁄2 and L = 2. This phenomenon of having multiple particles in the same total angular momentum configuration is called degeneracy. How to distinguish between these degenerate baryons is an active area of research in baryon spectroscopy.