Past exam of the mathematical methods I course of the University of Oxford / 2015 / 12 / ii

For $v=(a,b{)}^T$, multiplying each Pauli matrix between $v^{†}$ and $v$ givesAll three components are real. Moreover $|x|=(|a{|}^2+|b{|}^2)/2=\Vert v{\Vert }^2/2$. Thus normalized two-component spinors map to the radius-$1/2$ version of the Bloch sphere; the usual unit-radius convention multiplies these coordinates by two.