In the lecture, my professor wrote this on the board
$$
\begin{equation}
\begin{split}
(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2})|++\rangle &= |++\rangle \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\blacktriangledown)\\
(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2})(|+-\rangle+|-+\rangle) &= (|+-\rangle+|-+\rangle)\\
(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2})(|+-\rangle-|-+\rangle) &= -3(|+-\rangle+|-+\rangle)
\end{split}
\end{equation}
$$
but I don't get how these are correct. I know that
$$
\begin{equation}
\begin{split}
|1\;1\rangle &= |++\rangle \\
|1\;0\rangle &= \frac{1}{\sqrt{2}}(|+-\rangle+|-+\rangle) \\
|0\;0\rangle &= \frac{1}{\sqrt{2}}(|+-\rangle-|-+\rangle)
\end{split}
\end{equation}
$$
I will work out equation $(\blacktriangledown)$ in the usual matrix representation of the eigenstates of $S_z$ basis:
$$
|+\rangle=\begin{pmatrix}1\\
0
\end{pmatrix},\;\;\;\;\;\;\;\;\;\;\;\;\;\;|-\rangle=\begin{pmatrix}0\\
1
\end{pmatrix},
$$
So we have
$$
\begin{equation}
\begin{split}
(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2})|+\rangle_{1}\otimes|+\rangle_{2}&=&\vec{\sigma}_{1}|+\rangle_{1}\otimes\vec{\sigma}_{2}|+\rangle_{2}\\&=&\begin{pmatrix}1 & 1-i\\
1+i & -1
\end{pmatrix}_{1}\begin{pmatrix}1\\
0
\end{pmatrix}_{1}\otimes\begin{pmatrix}1 & 1-i\\
1+i & -1
\end{pmatrix}_{2}\begin{pmatrix}1\\
0
\end{pmatrix}_{2}\\&=&\begin{pmatrix}1\\
1+i
\end{pmatrix}_{1}\otimes\begin{pmatrix}1\\
1+i
\end{pmatrix}_{2}
\end{split}
\end{equation}
$$
but this is not $|++\rangle=|+\rangle\otimes|+\rangle$. What did I do wrong here? What have I misunderstood?
Best Answer
Your expression for: $$(\vec \sigma_1 \cdot \vec \sigma_2) |+\rangle_1 \otimes |+\rangle_2=\vec \sigma_1 |+\rangle\otimes \vec \sigma_2 |+\rangle_2$$ Is wrong. It sould read: $$(\vec \sigma_1 \cdot \vec \sigma_2) |+\rangle_1 \otimes |+\rangle_2=\sigma_{1x}|+\rangle_1\otimes \sigma_{2x}|+\rangle_2+\sigma_{1y}|+\rangle_1\otimes \sigma_{2y}|+\rangle_2+$$ $$\sigma_{1z}|+\rangle_1\otimes \sigma_{2z}|+\rangle_2$$ $$=\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}\otimes\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}$$ $$+\begin{pmatrix}0&-i\\i&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}\otimes\begin{pmatrix}0&-i\\i&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}$$ $$+\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}\otimes\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}$$ $$=\begin{pmatrix} 0 \\1 \end{pmatrix}\otimes\begin{pmatrix} 0 \\1 \end{pmatrix}$$ $$+\begin{pmatrix} 0 \\i \end{pmatrix}\otimes \begin{pmatrix} 0 \\i\end{pmatrix}$$ $$+\begin{pmatrix} 1 \\0 \end{pmatrix}\otimes \begin{pmatrix} 1\\0\end{pmatrix}$$ $$=\begin{pmatrix} 0 \\1 \end{pmatrix}\otimes\begin{pmatrix} 0 \\1 \end{pmatrix}$$ $$-\begin{pmatrix} 0 \\1 \end{pmatrix}\otimes \begin{pmatrix} 0 \\1\end{pmatrix}$$ $$+\begin{pmatrix} 1 \\0 \end{pmatrix}\otimes \begin{pmatrix} 1\\0\end{pmatrix}$$ $$=\begin{pmatrix} 1 \\0 \end{pmatrix}\otimes \begin{pmatrix} 1\\0\end{pmatrix}$$ i.e. I think you have to do the dot product between the Pauli matrices vectors first then put them through the tensor product.