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every time the system has a symmetry, there is a quantum number that labels which state of the symmetry that the system is in.
for example, there are many ways that rotations can act on a quantum system, and if the system is a rotationally symmetric one, then whenever the system starts in one of those states, it has to remain that state.
the spin quantum number is just a number to label which of those states the system is in.
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