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In class I've recently been taught the schrodinger equation and about spin(couple weeks apart). My questions are-

1. In laymans terms, what exactly does the S-eqn mean? What I have taken it to be so far, is an equation that determines the probability of the position of an e- in "orbit" around the nucleus

2. I get the full integer 1,2,3... spin, but what exactly does it physically mean to have 1/2 spin?

I searched a bit for clear answers but have come up with nil so far =)