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I Hamiltonian matrix - Eigenvectors

  1. Nov 9, 2016 #1
    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
     
  2. jcsd
  3. Nov 9, 2016 #2

    BvU

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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 ##
     
  4. Nov 9, 2016 #3
    Thanks.

    Konte
     
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