A linear combination of states that diagonalize the Hamiltonian

[kajmunso]

Homework Statement

Show that the linear combination of states that diagonalize the Hamilton of the hydrogen molecule are given by $$\mid \pm \rangle = \frac{1}{\sqrt{2 \pm 2 \langle 1 \mid 2 \rangle}}\left( \mid 1 \rangle \pm \mid 2 \rangle \right)$$
Verify these states are properly normalized and the corresponding energy expectation values are given by $$E_{\pm} = \frac{1}{1 \pm \langle 1 \mid 2 \rangle}\left( H_{11} \pm H_{12} \right)$$

Relevant Equations

We obviously have the states $$\mid \pm \rangle = \frac{1}{\sqrt{2 \pm 2 \langle 1 \mid 2 \rangle}}\left( \mid 1 \rangle \pm \mid 2 \rangle \right)$$
Now, we are given a couple of elements of a 2x2 Hamiltonian matrix, namely
$$ \begin{pmatrix}
H_{11} & H_{12} \\
H_{21} & H_{22}
\end{pmatrix}
$$
These are pretty long and messy- they involve an energy minus an integral. After talking with my professor, it sounds like I do not need to do these integrals out.
Also relevant is the equation to diagonalize a matrix $$ S^{-1} \hat H S$$

He told me I "need to show that the Hamiltonian matrix elements you get by using those states have nonzero elements only on the diagonal."
I understand what and how a diagonal matrix works, but what I don't understand is what those states are. Are they states I put in my "quantum mechanical sandwhich," that is like ⟨+∣H^∣+⟩ and so on, or are these somehow the states I multiply the 2x2 Hamiltonian with to get eigenvectors, which form the S matrix? Or am I thinking about this completely wrong? What does it mean for a state to diagonalize it?

 

[vanhees71]

Your matrix is representing the Hamiltonian in the basis $|1 \rangle$, $|2 \rangle$. Now diagonalize this matrix to get the eigenvectors, which are orthogonal and can be normalized, and the eigenvalues. You don't diagonalize a state but an operator, in this case the Hamiltonian (as acting in the 2D subspace under consideration).