Problem in constructing Matrix representation in |↑↓> basis

[ck00]

If I want to derive the matrix representation for operator Q in the |S₁=1/2 ,m₁> |S₂=1/2 ,m₂ > basis, where |S_i,m_i> are common eigenstates of S² , S_i,z for the ith particle.

And I do it in this way:
<↑↑|Q|↑↑> <↑↑|Q|↑↓> <↑↓|Q|↓↑> <↑↑|Q|↓↓>
<↑↓|Q|↑↑> <↑↓|Q|↑↓> <↑↓|Q|↓↑> <↑↓|Q|↓↓>
... ... ... ...
... ... ... ...

|↑↑>=|S₁=1/2 ,m₁=+1/2> |S₂=1/2 ,m₂=+1/2 >
Is it correct?
THANKS

 

[ck00]

can anyone help?

 

[Fredrik]

Let's see...the ij component (row i, column j) in the basis \{e_i\} is Q_{ij}=(Qe_j)_i=\langle e_i,Qe_j\rangle. In bra-ket notation, Q_{ij}=\langle i|Q|j\rangle. So everything on your first row should have the same "bra", but one of them is different from the other three. I suspect it's just a typo, since the rest of it looks fine.

 

[ck00]

Let's see...the ij component (row i, column j) in the basis \{e_i\} is Q_{ij}=(Qe_j)_i=\langle e_i,Qe_j\rangle. In bra-ket notation, Q_{ij}=\langle i|Q|j\rangle. So everything on your first row should have the same "bra", but one of them is different from the other three. I suspect it's just a typo, since the rest of it looks fine.

oh,ya, you are right, it's just a typo. Thanks for teaching