Every quantum number has an integral value but only spin quantum number has a fractional value, why?

First of all, spin is a consequence of a combination of quantum physics and special relativity. The spin of some particles has an integral value while others have a spin equal to an odd natural number divided by two. It is a known fact that if the spin is fractional, then no more than two such identical particles can have the same set of values for all the other quantum numbers. In addition upon exchanging the coordinates of these two particles, the result is a multiplication of the wave function by minus one.

After solving the (special) relativistically correct wave equation for the electron, one obtains the wave function Y(r,q,f) = R(r)Y_l,m(q,f)e^isp. As you will be knowing,

s = +(1/2) and –(1/2) respectively for the two electrons. You can verify by inspection that the wave functions of the two electrons have the same magnitude, but opposite signs. Importantly, the magnitude of the spin angular momentum is equal to

(h/2p)(√s)√(s+1) . This spin angular momentum is a consequence of special relativity in exactly the same manner in which E = mc² arises.

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