# Hamiltonian matrix - Eigenvectors

• I

## Main Question or Discussion Point

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

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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 and extranjero
Column n of ##V## are the coordinates of eigenvector ##e_n## on the basis ##\left | {i} \right \rangle ##
Thanks.

Konte