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Degenerate perturbation theory, vector derivative

  1. Feb 5, 2014 #1
    Stuff starts with this:

    [tex]
    \mathbb{R}\to\mathbb{C}^{N\times N},\quad t\mapsto A(t)
    [/tex]

    [tex]
    A(t)^{\dagger} = A(t)
    [/tex]

    [tex]
    A(t)v_n(t) = \lambda_n(t)v_n(t)
    [/tex]

    [tex]
    v_n(t)^{\dagger}v_{n'}(t) = \delta_{nn'}
    [/tex]

    [tex]
    \dot{A}(t)v_n(t) + A(t)\dot{v}_n(t) = \dot{\lambda}_n(t)v_n(t) + \lambda_n(t)\dot{v}_n(t)
    [/tex]

    Suppose that [itex]K>1[/itex] is the dimension of some eigenspace corresponding to some eigenvalue [itex]\lambda_n(0)[/itex]. We can assume the components arranged so that

    [tex]
    \lambda_n(0)=\lambda_{n+1}(0)=\cdots = \lambda_{n+K}(0)
    [/tex]

    I have not understood how to solve [itex]\dot{v}_n(0)[/itex].

    Isn't the main aim to solve the coefficients [itex]v_{n'}(0)^{\dagger}\dot{v}_n(0)[/itex], like in the non-degenerate case too? If these are known for all [itex]n'[/itex], then the whole vector can be written as

    [tex]
    \dot{v}_n(0) = \sum_{n'=1}^N \big(v_{n'}(0)^{\dagger}\dot{v}_n(0)\big)v_{n'}(0)
    [/tex]

    I understand that for those [itex]n'[/itex] such that [itex]\lambda_{n'}(0)\neq\lambda_n(0)[/itex] we have

    [tex]
    v_{n'}(0)^{\dagger}\dot{v}_n(0) = \frac{v_{n'}(0)^{\dagger}\dot{A}(0)v_n(0)}{\lambda_n(0) - \lambda_{n'}(0)}
    [/tex]

    I also understand that we can assume

    [tex]
    v_n(0)^{\dagger}\dot{v}_n(0) = 0
    [/tex]

    The real part will be zero due to the assumption [itex]\|v_n(t)\|=1[/itex], and the imaginary part can be chosen zero by choosing right complex phase.

    What I have not understood is how to obtain the coefficients for such [itex]n'[/itex] that [itex]n'\neq n[/itex] and [itex]\lambda_{n'}(0)=\lambda_n(0)[/itex].

    I have Quantum Mechanics Second Edition by Bransden and Joachain. After discussing how to obtain [itex]v_{n'}(0)[/itex] (by some rotation in the eigenspace) they state

    Well to me it seems that I cannot obtain all the coefficients in a way "similar" to the non-degenerate case.
     
  2. jcsd
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