In this case the energy for \( E_1 \) is different, we obtain namely $$ \begin{equation*} E_1=E_{\uparrow\uparrow}=-2J. \end{equation*} $$ The other cases do also differ and we have $$ \begin{equation*} E_2=E_{\uparrow\downarrow}=+2J, \end{equation*} $$ $$ \begin{equation*} E_3=E_{\downarrow\uparrow}=+2J, \end{equation*} $$ and $$ \begin{equation*} E_4=E_{\downarrow\downarrow}=-2J. \end{equation*} $$