pines-demon said:
The calculation is right, what step do you not understand?
Thanks, I have to explain the problem exactly. In a Hilbert space including two orbitals ##f## and ##c## we have the states as ##|n_f^{\uparrow} n_f^{\downarrow} n_c^{\uparrow} n_c^{\downarrow} \rangle ## where ##n## indicates the number of electron in each orbital. The total-spin operator for this system is defined as: $$\vec S =\frac {1} {2} \sum_{\mu \nu} (f_{\mu} ^{\dagger} \vec {\sigma}_{_{\mu \nu} } f_\nu+ c_{\mu} ^{\dagger} \vec {\sigma}_{_{\mu \nu} } c_\nu),$$ where ##\vec {\sigma}## is the vector of Pauli matrices and ##\mu, \nu## the spin indices.
With expanding this formula, we have:
\begin{align}
s_x=\frac{1}{2} ({f^{\dagger}_{\uparrow} f_{\downarrow} + c^{\dagger}_{\uparrow} c_{\downarrow} + f^{\dagger}_{\downarrow} f_{\uparrow} + c^{\dagger}_{\downarrow} c_{\uparrow} }) \\
s_y=\frac{1}{2} ({(-i)(f^{\dagger}_{\uparrow} f_{\downarrow} + c^{\dagger}_{\uparrow} c_{\downarrow}) +i(f^{\dagger}_{\downarrow} f_{\uparrow} + c^{\dagger}_{\downarrow} c_{\uparrow}) }) \\
s_z=\frac{1}{2} ({f^{\dagger}_{\uparrow} f_{\uparrow} + c^{\dagger}_{\uparrow} c_{\uparrow} - f^{\dagger}_{\downarrow} f_{\downarrow} - c^{\dagger}_{\downarrow} c_{\downarrow} }).
\end{align}
Now, I want to calculate ##S^2 |0 1 1 0\rangle = (S_x^2+ S_y^2 + S_z^2 )|0 1 1 0\rangle ## which is expected to be zero because we have two opposite spins, one in ##f## and the other in ##c## orbitals. However, my direct calculation doesn’t result in this expectation. My calculations are briefly as: $$s_z |0110\rangle =0$$ and
\begin{align}
s_x |0110\rangle=\frac{1}{2} (|1010\rangle +|0101\rangle) \\
s_y |0110\rangle=\frac{1}{2} (-i|1010\rangle +i|0101\rangle) \\
s_x |1010\rangle=\frac{1}{2} (|0110\rangle +|1001\rangle) \\
s_y |1010\rangle=\frac{1}{2} i (|0110\rangle +|1001\rangle)\\
s_x |0101\rangle=\frac{1}{2} (|1001\rangle +|0110\rangle) \\
s_y |0101\rangle=\frac{1}{2} (-i)(|1001\rangle +|0110\rangle)\\
\end{align}
then we have:
\begin{align}
s^2_x|0110\rangle=\frac{1}{2} s_x(|1010\rangle +|0101\rangle) =\frac{1}{2} (|1001\rangle +|0110\rangle) \\
\end{align}
and similarly:
\begin{align}
s^2_y|0110\rangle=\frac{1}{2} (|0110\rangle +|1001\rangle) \\
\end{align}
and
\begin{align}
s^2_z|0110\rangle=0
\end{align}
Finally, we have:
$$s^2 |0110\rangle = (s^2_x+ s^2_y+ s^2_z)|0110\rangle =|0110\rangle +|0110\rangle$$ which obviously is not zero.
I would be grateful if you could please provide any help on this.