Re: Why is spin 1/2 and -1/2

First of all, the "spin quantum number" needs to be multiplied by the fundamental unit of angular momentum, h_bar (planck's constant), to get the value for the "spin angular momentum". It turns out that the measured "spin angular momentum" of an electron is +/- h_bar/2, so we say the electron is a "spin 1/2 particle".

If you're wondering why we don't just make the fundamental unit of angular momentum "h_bar/2" and say that the electron has spin 1 (or -1)... This could be done but would also make for some ugly math. For example, the orbital angular momentum of an electron around a nucleus would no longer come in nice quantum increments (h_bar, 2h_bar, 3h_bar...), but would now come only in even numbers (2 h_bar/2, 4 h_bar/2, 6 h_bar/2...). It turns out that h_bar really is a fundamental counting unit.

Now, if you're asking the deeper question of WHY the electron has the h_bar/2 spin angular momentum in the first place... That's a tough one to answer without nasty math -- and may not even be completely understood. Here's my attempt anyway:

The beautiful Dirac equation must be solved by a 4x4 matrix which gives exactly four solutions for the electron. Two of them are electron solutions and two of them are positron solutions (positrons are the antimatter version of electrons.) The only difference between the two electron solutions is that they have to have different inherent angular momenta (spin)-- different by exactly the basic unit of h_bar. The only symmetric way to have them differ by h_bar is to give one solution +h_bar/2 spin and the other solution -h_bar/2 spin. The same goes for the positrons.

Hope that made some sense!