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Homework Help: Strange product of integrals

  1. Jul 29, 2010 #1
    1. The problem statement, all variables and given/known data

    I'm trying to compute something of the form [tex]\langle \int_a^b{f(x) dx} \int_a^b{f(x)^{\dagger}dx} \rangle [/tex] where the dagger means complex conjugate and the brackets are ensemble average (f(x) is a statistical quantity). I'm supposed to use the relation that [tex] \langle f(x) f(x')^{\dagger} \rangle = c*\delta(f-f')[/tex] where c is some constant.


    2. Relevant equations

    [tex] \langle f(x) f(x')^{\dagger} \rangle = c*\delta(f-f')[/tex]

    3. The attempt at a solution

    I'm a bit perplexed. I have the function and its complex conjugate, but inside different integrals, which are being multiplied. And the ensemble average of a product isn't the same as the product of ensemble averages, either... is it? I'd be surprised.

    I thought maybe I could multiply the entire quantity by an extra f dagger, then somehow use the relation, but it didn't really get me anywhere.

    So I have no idea how to use the given relation. Can anyone help??
     
  2. jcsd
  3. Jul 30, 2010 #2

    nrqed

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    Are you sure that the x appearing in the second integral should not all have a prime on them?

    Are you sure that it is not [itex] \delta(x-x')[/itex] ??


    Check these two things and let us know. If I am correct about the two corrections, the problem becomes very easy.
     
  4. Jul 30, 2010 #3
    Oops, you're right. I've been working with functions of frequency and forgot. So yes, should be f(x) and f(x'), and the delta function should then be delta(x-x').
     
  5. Jul 30, 2010 #4

    nrqed

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    Great. Then are you all set? Replacing the product of the functions by a delta function makes the two integrations trivial.
     
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