We can in turn define the isospin Pauli matrices (in the same as we define the spin matrices) as $$ \hat{\tau}_x =\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), $$ $$ \hat{\tau}_y =\left(\begin{array}{cc} 0 & -\imath \\ \imath & 0 \end{array}\right), $$ and $$ \hat{\tau}_z =\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right), $$ and operating with \( \hat{\tau}_z \) on the proton state function we have $$ \hat{\tau}_z\psi^p(\mathbf{r})=-\frac{1}{2}\psi^p(\mathbf{r}), $$ and for neutrons we have $$ \hat{\tau}\psi^n(\mathbf{r})=\frac{1}{2}\psi^n(\mathbf{r}). $$