# Matrix representation composite system

I am considering an assignment involving the Hubbard model. A state is given by $$|\Phi\rangle=-|2\uparrow1\downarrow\rangle-|1\uparrow2\downarrow\rangle$$ where particle 1 and 2 are electrons. The state is then written in the matrix representation $$|\Phi\rangle=\begin{bmatrix}0\\0\\-1\\1\end{bmatrix}$$. I do not really know how you get to that matrix representation ? I think my problem is that I do not generally know how to go from the ket to the matrix. I know that for a spin up electron we have the matrix (1,0) and (0,1) for spin down. But for this composite systems I am not sure. \

The basis are given by the 4 states: $$|1\uparrow1\downarrow\rangle$$,$$|2\uparrow2\downarrow\rangle$$, $$|1\uparrow2\downarrow\rangle$$,$$|1\downarrow2\uparrow\rangle$$.

• Can you provide more background on the problem? Do you know anything about the basis of your column vector? – Hanting Zhang Dec 9 '18 at 20:18
• I edited my question by adding the basis in the bottom. – Elias S. Dec 9 '18 at 20:39
• Is your 4th basis vector correct? Seems like it should have arrows up and then down like the others. – doublefelix Dec 9 '18 at 20:44
• It is written like that in the notes – Elias S. Dec 9 '18 at 20:49
• I can only see this making sense if $\left|1\uparrow2\downarrow\right>=-\left|1\downarrow2\uparrow\right>$. – anonymous Dec 9 '18 at 21:03

3. If there are $$N$$ basis kets you need $$N$$ vectors, each with $$N$$ components. You simply write the $$N$$ vectors which have just a single element 1 and the rest zero (i.e. columns of the $$N \times N$$ identity matrix), and assign them one by one to represent your basis kets. One can add further mathematical argument to show why this is what you have to do, but I am skipping that.
I can't understand the list of basis kets you gave, so I will have to illustrate the method by showing what happens for the standard four states for a pair of spin half particles. I have ordered them so as to try to match the vector you quoted. Thus I get $$|\uparrow\uparrow\rangle \leftrightarrow \left[ \begin{array}{c} 1\\0\\0\\0 \end{array} \right] ,\;\; |\downarrow\downarrow\rangle \leftrightarrow \left[ \begin{array}{c} 0\\1\\0\\0 \end{array} \right]$$ $$|\downarrow\uparrow\rangle \leftrightarrow \left[ \begin{array}{c} 0\\0\\1\\0 \end{array} \right] ,\;\; |\uparrow\downarrow\rangle \leftrightarrow \left[ \begin{array}{c} 0\\0\\0\\1 \end{array} \right]$$