Exercise 7.2

We calculate the matrix elements of \(\sigma_z \otimes \tau_x\) by building inner products:

\[ \begin{flalign*} \sigma_z \otimes \tau_x &= \begin{pmatrix} \braket{uu|\sigma_z \tau_x|uu} & \braket{uu|\sigma_z \tau_x|ud} & \braket{uu|\sigma_z \tau_x|du} & \braket{uu|\sigma_z \tau_x|dd} \\ \braket{ud|\sigma_z \tau_x|uu} & \braket{ud|\sigma_z \tau_x|ud} & \braket{ud|\sigma_z \tau_x|du} & \braket{ud|\sigma_z \tau_x|dd} \\ \braket{du|\sigma_z \tau_x|uu} & \braket{du|\sigma_z \tau_x|ud} & \braket{du|\sigma_z \tau_x|du} & \braket{du|\sigma_z \tau_x|dd} \\ \braket{dd|\sigma_z \tau_x|uu} & \braket{dd|\sigma_z \tau_x|ud} & \braket{dd|\sigma_z \tau_x|du} & \braket{dd|\sigma_z \tau_x|dd} \end{pmatrix} &\\ &= \begin{pmatrix} \braket{uu|ud} & \braket{uu|uu} & \braket{uu|-dd} & \braket{uu|-du} \\ \braket{ud|ud} & \braket{ud|uu} & \braket{ud|dd} & \braket{ud|du} \\ \braket{-du|ud} & \braket{-du|uu} & \braket{-du|dd} & \braket{-du|du} \\ \braket{-dd|ud} & \braket{-dd|uu} & \braket{-dd|dd} & \braket{-dd|du} \end{pmatrix} &\\ &= \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \end{pmatrix} \end{flalign*} \]

and get the same result as given in the textbook p. 190.