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Eigenkets of a function of a hermitian operator

  1. Sep 29, 2008 #1
    For a hermitian operator A, does the function f(A) have the same eigenkets as A?

    This has been bothering me as I try to solve Sakurai question (1.27, part a). Some of my class fellows decided that it was so and it greatly simplifies the equations and it helps in the next part too but I don't think so because I might add anything to A in order to make it non-hermitian.
  2. jcsd
  3. Sep 29, 2008 #2
    How is a function of an operator A defined? Have a look at page 51 of http://www.lsr.ph.ic.ac.uk/~plenio/lecture.pdf [Broken].
    Last edited by a moderator: May 3, 2017
  4. Sep 29, 2008 #3
    The operator is completely general but I think that definition 26 would apply. But I have a few question concerning Definition 27:

    1. What specifically are the Ai?
    2. Doesn't the power series expansion go under the continuous spectrum? I ask this because in the discrete spectrum at least, wouldn't the eigenkets of a hermitian operator be complete?

    Thanks for the link btw, I printed the first two chapters :)
    Last edited by a moderator: May 3, 2017
  5. Sep 29, 2008 #4


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    powers of A.

    E.g., 1, A, A^2, A^3, etc

    under? Don't know exactly what you mean by that...

    Often the symbol
    which "looks like" a discrete sum really means
    \sum_n+\int dn
    a sum over the discrete part of the spectrum and a integral over the continuous part.
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