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Subtracting constant energy

  1. Jan 10, 2006 #1
    correct me if im wrong..
    as i see it, i can subtract any constant from the diagonal of the hamiltonian without really changing the system it describes... am i right?

    if i got a two state system, the hamiltonian can look like this:

    [tex] \left(
    \begin{array}{cc}
    0 & P_{2->1} \\
    P_{1->2} & \Delta E \\
    \end{array}
    \right)[/tex]

    where [tex]\Delta E=H_{22}-H_{11}[/tex]

    if im right so far... then here's my problem - when you diagonize the matrix it doesnt have the same eigenvalues (i mean not even the same delta...) if you add or subtract from the diagonal... and that can't be right... so whats wrong in my view of things?
     
    Last edited: Jan 11, 2006
  2. jcsd
  3. Jan 11, 2006 #2

    vanesch

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    Staff Emeritus
    Science Advisor
    Gold Member


    You must be making a mistake.
    Imagine H0 the original hamiltonian (the one you quote, for instance), and |e1> an eigenvector.
    Then we have H0 |e1> = E1 |e1>

    Now consider that we add C 1 to H0 (unit matrix times constant C).

    Then of course C 1|e1> = C |e1>

    Define H1 = H0 + C 1

    H1 |e1> = H0 |e1> + C 1 |e1> = E1 |e1> + C |e1> = (E1 + C) |e1>

    So we see that |e1> is again an eigenvector of H1, but this time with eigenvalue E1 + C. All eigenvalues of H1 will be the eigenvalues of H0 plus C, and the eigenvectors will be the same. The differences between the eigenvalues will of course remain the same (C will be eliminated).

    cheers,
    Patrick.
     
  4. Jan 11, 2006 #3
    yeah, i found my mistake... the delta remains the same... :blushing:
     
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