Calculate p.sigma

Pauli Matrices:

TableForm[sigmax = {{0, 1}, {1, 0}}]TableForm[sigmay = {{0, -I}, {I, 0}}]TableForm[sigmaz = {{1, 0}, {0, -1}}]

Calculate dot product:

TableForm[psigma = px sigmax + py sigmay + pz sigmaz]

Square it:

TableForm[Expand[psigma . psigma]]

Normal form of spin raising operator:

TableForm[(1/2) (sigmax + I sigmay)]