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Hermitian Conjugate of an Operator

  1. Sep 3, 2007 #1
    1. The problem statement, all variables and given/known data

    a = x + [tex]\frac{d}{dx}[/tex]

    Construct the Hermitian conjugate of a. Is a Hermitian?

    2. The attempt at a solution

    <[tex]\phi[/tex]|(x+[tex]\frac{d}{dx}[/tex])[tex]\Psi[/tex]>

    [tex]\int[/tex][tex]\phi[/tex][tex]^{*}[/tex](x[tex]\Psi[/tex])dx + <-[tex]\frac{d}{dx}[/tex][tex]\phi[/tex]|[tex]\Psi[/tex]>

    I figured out the second term already but need help with first term... am I on the right track?
     
  2. jcsd
  3. Sep 4, 2007 #2

    Dick

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    Well, x is real. It's the position operator. So x^*=x.
     
  4. Sep 4, 2007 #3

    dextercioby

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    HINTS:

    1.What's the domain of "a" as an operator in the [itex] L^{2}(\mathbb{R},dx) [/itex] ?
    2. Stick to that domain. Consider the matrix element of that operator among 2 vectors in that Hilbert space. What restrictions do you get when trying to find the adjoint ? Therefore ?
    3. Does the adjoint exist ?
    4. What's its domain ?
    5. Is the "a" operator hermitean/symmetric ?
     
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