Question

what exactly 1/2 spin factor indicate in quantum numbers ?

Answer 1

The spin 1/2 factor in quantum numbers gives the quantised intrinsic angular momentum of a "spinning" particle, such that the angular momentum projected upon the z-axis (the z component of this angular momentum) is given by \((1/2)\hbar\). Unfortunately there is no classical analogy to this type of spin, which is often the case in quantum mechanics, meaning that it is a purely quantum mechanical concept.

To illustrate this, think of having a card with an arrow upon it pointing upwards. In classical terms, we would have to rotate the card around 360 (i.e. once) degrees before the arrow was back in the same position, and the card was identical to the way it looked before rotation. For spin 1/2 quantum particles, if we extend this analogy, we would have to rotate the card 720 degrees (i.e. two full turns) before it returns to its original configuration. That is a 360 degree turn results in the arrow pointing upside down.

When Paul Dirac tried to make a fully relativistic form of the Schrödinger equation, he found that to make things internally consistent that "spin" had to arise from the implementation of relativity. Thus it can be interpreted that spin is not a classical spin, but rather a consequence of special relativity in a quantum mechanical environment. Previously spin had been introduced to explain certain experimental observations (Stern-Gerlach results for example), but Dirac showed that spin seemed to come intrinsically from the marriage of quantum mechanics and special relativity.