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 Sci Advisor P: 1,185 If we label the eigenstates of $S_z$ as $|{+}\rangle$ and $|{-}\rangle$, so that $S_z|{\pm}\rangle=\pm\frac12 |{\pm}\rangle$, then $$S_+|{-}\rangle=|{+}\rangle$$ $$S_-|{+}\rangle=|{-}\rangle$$ Also, $$S_+|{+}\rangle=0$$ $$S_-|{-}\rangle=0$$ That is, $S_+$ raises the value of $S_z$, and $S_-$ lowers it. That is how the raising and lowering operators are defined, and $S_x \pm i S_y$ is just what they work out to be.