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I What does complex conjugate of a derivate mean?

  1. Nov 19, 2017 #1
    An exercise asks me to determine whether the following operator is Hermitian:
    [itex]
    {\left( {\frac{d}{{dx}}} \right)^ * }.
    [/itex]

    I dont even know what that expression means.
    a) Differentiate with respect to x, then take the complex conjugate of the result?
    b) Take the complex conjugate, then differentiate with respect to x?
    c) [itex]{\left( {\frac{d}{{dx}}} \right)^ * } = \frac{d}{{d{x^*}}} = \frac{d}{{dx}}[/itex]?

    Can someone clarify?
     
  2. jcsd
  3. Nov 19, 2017 #2

    andrewkirk

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    If ##x## is real then (a) and (b) are the same, so it probably means those.
     
  4. Nov 19, 2017 #3

    Mentz114

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    The derivative wrt x of ##i\,g\left( x\right) +f\left( x\right) ## is ##i\,\left( \frac{d}{d\,x}\,g\left( x\right) \right) +\frac{d}{d\,x}\,f\left( x\right) ##.
    Nuff said ?
     
  5. Nov 19, 2017 #4

    PeterDonis

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    In what textbook?
     
  6. Nov 20, 2017 #5

    strangerep

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    If the original question is the context of QM, then I'll bet it doesn't mean either of those. Rather, the exercise probably intends to determine whether ##d/dx## is self-adjoint on the space of square-integrable functions.

    In that case, @jstrunk: you should probably take a look at the Wikipedia page for "hermitian operators". :oldbiggrin:
     
  7. Nov 20, 2017 #6

    DrClaude

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  8. Nov 21, 2017 #7

    A. Neumaier

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    The star applied to operators generally means the adjoint operator. For differential operators you can find it using integration by parts.
     
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