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Problem with Eigenkets

  1. Oct 15, 2007 #1
    1. The problem statement, all variables and given/known data
    Show that if an operator A has an eigenket |a> to eigenvalue a then
    the adjoint operator A† has an eigenbra <a*| to eigenvalue a*. How
    is <a*| related to |a>?

    2. Relevant equations
    A|a> = |a>a
    | >† = < |

    3. The attempt at a solution
    I actually have no clue where to start this question. I am guessing it has something to do with A† would have an eigenket of |a>†. But I am unsure if this is correct at all.
    Would anyone be able to help me get started.
  2. jcsd
  3. Oct 15, 2007 #2


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    Take the dagger of (A |a>) This must be equal to the dagger of (a |a>).
  4. Oct 15, 2007 #3
    Ok, I do this. And I also took the dagge of both sides of the equation. So I got
    (A|a>)^(dagger) = (|a>a)^dagger which gets

    <a|A* = a*<a|.

    And using your statement form above I then do
    A*<a| = a|a> .
    How does this get me any closer to the answer?
  5. Oct 15, 2007 #4


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    Why would a*=a ??
  6. Oct 15, 2007 #5


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    So you are done! You have proved that <a| is an eigenbra of A* with eigenvalue a*!!
    ?????:confused: What statement from above? I did not say anything lik ethat!
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