Undergrad Hamiltonian matrix - Eigenvectors

[Konte]

Hello everybody,

From a complete set of orthogonal basis vector $|i\rangle$ $\in$ Hilbert space ($i$ = $1$ to $n$), I construct and obtain a nondiagonal Hamiltonian matrix
$$
\left( \begin{array}{cccccc}
\langle1|H|1\rangle & \langle1|H|2\rangle & . &. &.& \langle1|H|n\rangle \\
\langle2|H|1\rangle & . & . &. &.&. \\
. & . & . &. &.&. \\
. & . & . &. &.&. \\
. & . & . &. &.&. \\
. & . &. &. &.& \langle n|H|n\rangle \end{array} \right)
$$

After diagonalization, I obtain diagonal $n\times n$ matrix that represent the eigenvalues of the Hamiltonian, and another $n\times n$ matrix $V$ composed of scalars that represent the eigenvectors of the same Hamiltonian.

My question is, what is the link between the scalar matrix $V$ and the complete set of orthogonal basis vector $|i\rangle$ that I choose in the beginning ?

Thank you very much everybody.

Konte

 

[BvU]

As far as I know the $\left | {i} \right \rangle$ are the base vectors for $V$.

so $V_{ni}$ are the coefficients $\left\langle e_n \middle | i \right \rangle$ ; in other words:

Column n of $V$ are the coordinates of eigenvector $e_n$ on the basis $\left | {i} \right \rangle$

 

[Konte]

Column n of $V$ are the coordinates of eigenvector $e_n$ on the basis $\left | {i} \right \rangle$

Thanks.

Konte