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Completeness relations

  1. Aug 12, 2007 #1
    I am confused about completeness relations. I thought a completeness relation was something like:

    [tex] I = \sum_{i = 1}^n |i><i| = \sum_{i=1}^n P_i [[/tex]

    where P_i is the projection operator onto i. Now I saw this called a completeness relation as well:

    [tex]\delta(x - x') = \sum_{n=0}^\infty \Psi_n(x) \Psi_n(x')[/tex]

    How is that the same as my first equation? What is the difference between x and x'? The second equation can be found at http://en.wikipedia.org/wiki/Green's_function
     
  2. jcsd
  3. Aug 12, 2007 #2
    As set of functions [itex]\psi_n[/itex] being complete means that you can write down arbitrary function (of some kind, so not really arbitrary) as

    [tex]
    f(x)=\sum_{n=0}^{\infty} c_n \psi_n(x)
    [/tex]

    where the coefficients are given by an inner product

    [tex]
    c_n = \int_{-\infty}^{\infty} dx'\; \psi^*_n(x')f(x').
    [/tex]

    But you can rewrite this as

    [tex]
    f(x) = \sum_{n=0}^{\infty} \Big(\int_{-\infty}^{\infty} dx'\; \psi^*_n(x')f(x')\Big) \psi_n(x) = \int_{-\infty}^{\infty} dx'\; \Big(\sum_{n=0}^{\infty} \psi^*_n(x')\psi_n(x)\Big) f(x')
    [/tex]

    so there you see that it is pretty much the same as the sum being a delta function.
     
  4. Aug 12, 2007 #3
    I see how it works mathematically thanks. Still a little confused about "how to think about" x and x'...
     
  5. Aug 12, 2007 #4
    In the completeness relation there is really nothing to think about, as far as I'm aware... For the Green's function there is an interpretation which is something like "the wavefunction at [itex]x[/itex] resulting from a unit excitation applied at [itex]x^{\prime}[/itex]."
     
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