Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 33D / i / Solution

In the spin-$z$ orthonormal basis, $|\uparrow ⟩=(1,0{)}^T$ and $|\downarrow ⟩=(0,1{)}^T$. The spin angular momentum operators are $s_i=(\hslash /2){\sigma }_i$. The Pauli matrices obey $[{\sigma }_i,{\sigma }_j]=2i{ϵ}_{ijk}{\sigma }_k$, soThe second identity is the correct $s(s+1){\hslash }^2$ eigenvalue for $s=1/2$. Matrix multiplication givesThus $s_x$ and $s_y$ flip the spin-$z$ basis states, with different complex phases.