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Homework Help: Perturbation theory

  1. Feb 18, 2012 #1
    time-independent, non-degenerate. I am referring to the following text, which I am reading:
    On page 4, it represents the results of the 2nd order terms. In Eqs. (32), (33) and (34) I don't understand the second equality, i.e. basing on which formula he has turned the potential terms into a sum.
    For example, in (32) how he got from [itex]\langle n^{(0)}|V|n^{(1)} \rangle [/itex] to [itex]-\sum_{m\neq 0}\frac{|V_{nm}|^2}{E_{mn}}[/itex]
  2. jcsd
  3. Feb 18, 2012 #2
    You substitute in the expression you found for the second order coefficients in the expansion.
    The sum is something along the lines of

    [itex]E_n^2 = \sum_{m \ne n} V_{n,m} c^1_m[/itex]

    and in the first approximation you find that [itex]c^1_m = \frac{V_n,m}{E_{m,n}}[/itex] and you realise that V is hermitian and you multiply them together and get what you have, the negative sign comes from interchanging the m and n in the [itex]E_{m,n} = -E_{n,m}[/itex] term after you hermitian conjugate [itex]c^1_m[/itex]
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