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Property of the adjoint operator

  1. Jul 21, 2007 #1
    The adjoint of an operator A is defined as an operator A* s.t.

    [tex]<\phi|A\psi> = <A^{*}\phi|\psi>[/tex].

    How would you use the properties of inner products (skew-symmetry, positive semi-definiteness, and linearity in ket) to show that (cA)* = c*P*

    Note that I am using the conjugate and the adjoint symbol interchangeably. If anyone knows how to get a real adjoint symbol in LaTeX let me know.
  2. jcsd
  3. Jul 21, 2007 #2

    matt grime

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    what does \dag do? I assume that A and P are supposed to be the same letter... and then it is trivial. What have you attempted?
  4. Jul 21, 2007 #3
    Yes sorry. A and P are supposed to be the same letter.

    You could use the skew-symmetry property to show that:

    [tex]<\phi|cA\psi> = <cA\psi|\phi>^{*}[/tex]

    and that

    [tex]<cA^{\dag}\phi|\psi> = <\psi|cA^{\dag}\phi>^{*}[/tex]

    but I do not see how that helps.
  5. Jul 21, 2007 #4


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    ((cA)x | y) = ...

    Unless I'm mistaken, you have to use the definition of scalar multiplication with operators, and two peoperties of the inner product. In three (four) steps, you can show what (cA)* equals.
  6. Jul 21, 2007 #5
    I see. So, [tex] <(cA)x|y> = c*<Ax|y> = c*<x|A^{\dag}y> = < x|c*A^{\dag}y> [/tex].
  7. Jul 21, 2007 #6
    What about the property [tex](PQ)^{\dag} = Q^{\dag}P^{\dag}[/tex]? This one seems a bit more difficult.
  8. Jul 21, 2007 #7


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    I wouldn't really call it difficult. Again, ( (PQ)x | y ) = ...
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