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Bra Ket question

  1. May 1, 2007 #1

    malawi_glenn

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    Is the complex conjugate to the outer product this? :

    ( |a> <b| ) * = ( |b> <a| )

    ?
     
  2. jcsd
  3. May 1, 2007 #2

    Dick

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    If you think of it as a matrix that's like saying A_ij*=Aji. Looks like an Hermitian conjugate to me.
     
    Last edited: May 1, 2007
  4. May 2, 2007 #3

    Meir Achuz

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    The cc does not interchange a and b.
     
  5. May 3, 2007 #4

    malawi_glenn

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    so

    (<a|b>) * = (<b|a>)

    but not

    ( |a> <b| ) * = ( |b> <a| )

    ?
     
  6. May 3, 2007 #5

    George Jones

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    First, note that <a|b> is a complex number, while |a><b| is a linear operator on a vector space.

    The answer to your question depends on what you mean by *. What do you mean by *?

    If you mean complex conjugation, then I don't know how to take (in a basis-independent way) the complex conjugate of a linear operator.

    If you mean adjoint, then (after changing * to [itex]{}^\dagger[/itex])

    [tex]\left< a | b \right>^\dagger = \left< b | a \right>[/tex]

    [tex]\left( \left| a \right> \left< b \right| \right)^\dagger = \left| b \right> \left< a \right|.[/tex]

    You should verify this from the definition of adjoint.
     
  7. May 4, 2007 #6

    malawi_glenn

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    hmm yeah i must have confused those two things.. thanx for all help =)
     
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