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The spin quantum number is just used to differentiate between the two electrons in an orbital and it is needed to completely describe an electron state. One of the electrons in an orbital is facing up and the other is facing down. The spin quantum number just tells us which is which and makes it so that no electron in an atom has the same four quantum numbers as any other electron in the atom. The positive 1/2 is used for upward spin electrons and the downward facing spin is negative 1/2.
There are only two possibilities for the spin direction of an atom: upwards and downwards. Because of this, there are also only two possible spin magnetic quantum numbers. I believe the numbers +1/2 and -1/2 were chosen somewhat arbitrarily.
The spin number can only be +1/2 or -1/2. Lavelle didn't go into much detail into explaining the experiments and the way in which this was determined because it does not pertain to our class. All that we need to know is that +1/2 is spinning up while -1/2 is spinning down!
Yes, the concept of electron spin was discovered through the Stern Gerlach experiment. They fired silver atoms, which have 1 valence electron, through a pair of magnetized plates. The interaction between the local magnetic fields of the individual electrons and the plates' magnetic field caused the Ag atoms to land on the detector in two distinct regions. This lead to the conclusion that electrons have 2 spin states, +1/2 and -1/2. The choice of the 1/2 value could be arbitrary like l=0, 1, 2,3, etc. and ml=0, +/-1, +/-2, etc., but I'm not sure.
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