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A Imaginary position operator

  1. Dec 4, 2017 #1

    SeM

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    Hello, some operators seem to "add up" and give real eigenvalues only if they are applied on the imaginary position, ix, rather than the normal position operator, x, in the integral :

    \begin{equation}
    \langle Bx, x\rangle
    \end{equation}

    when replaced by:


    \begin{equation}
    \langle Bix, ix\rangle
    \end{equation}

    or


    \begin{equation}
    \langle Bix, x\rangle
    \end{equation}

    So using (2) or even (3) I get real (but negative) eigenvalues, instead of complex eigenvalues. I have not found any literature on such an absurd thing, as the imaginary position, however, being the only answer to these operators, I am wondering if anyone can point to some further literature on complex operators and complex eigenvalues in QM and whether (2) and (3) make any sense at all

    Thanks
     
  2. jcsd
  3. Dec 4, 2017 #2

    vanhees71

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    Science Advisor
    Gold Member
    2017 Award

    What's ##B##? Without clear definitions, I cannot make any sense of the expressions above. Where does this idea of complex/imaginary position eigenvalues come from?
     
  4. Dec 5, 2017 #3

    SeM

    User Avatar

    It's an idea that comes from a calculation that sums up only if I use an imaginary position operator. Apparently, it is so non-heard of that I will leave it as it is. It has to do with non-hermiticity, where B (not hermitian) only gives a real value on that integral if the position is imaginary.

    Thanks!
     
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