Spin (physics)
Spin projection quantum number and multiplicity

In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the axis of rotation of the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component of angular momentum for a spin- s particle measured along any direction can only take on the values

where S is the spin component along the i -th axis (either x, y, or z), s is the spin projection quantum number along the i -th axis, and s is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the z axis:

where S is the spin component along the z axis, s is the spin projection quantum number along the z axis.

One can see that there are 2 s + 1 possible values of s. The number " 2 s + 1 " is the multiplicity of the spin system. For example, there are only two possible values for a spin- 1 / 2 particle: s = + 1 / 2 and s = − 1 / 2. These correspond to quantum states in which the spin component is pointing in the + z or − z directions respectively, and are often referred to as "spin up" and "spin down". For a spin- 3 / 2 particle, like a delta baryon, the possible values are + 3 / 2, + 1 / 2, − 1 / 2, − 3 / 2.
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