In conventional coordinate systems (anything you solve a simple Newtonian mechanics problem with), up and down are + and - z. A vector pointing up and a vector pointing down are anti-parallel. But in qm, we have up and down spinors making and orthonormal basis. These basis vectors are also called...

Consider first, the simple case of a disc rolling horizontally on the x-axis, having angular velocity $\mathbf{\omega}$ Let p be any point on the edge of the disc. Let $\mathbf{r_{c}}$ be the position of the center, and $\mathbf{X}$ be the vector joining the centre to point p. Then, if the posit...