Past exam of the mathematical methods I course of the University of Oxford / 2017 / 5 / Solution

For a unit vector, the radial coordinate is $r=1$ and the spherical coordinates giveThe Pauli matrices therefore giveThe diagonal entries are real and the off-diagonal entries are complex conjugates, so $H^{\ast }=H$. Its trace is zero and its determinant is $-1$, giving eigenvalues $+1$ and $-1$. A convenient orthonormal eigenbasis isThe half-angle identities verify $H{v}_{\pm }=\pm v_{\pm }$, $v_{\pm }^{\ast }{v}_{\pm }=1$ and $v_{+}^{\ast }{v}_{-}=0$. Multiplication of either vector by any unit-modulus phase gives an equivalent normalized eigenvector. At a pole, the azimuth is undefined geometrically, but the remaining phase freedom makes the chosen expressions consistent with the same one-dimensional eigenspaces.