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Properties of Hermitian operators in complex vector spaces

  1. Sep 15, 2011 #1
    1. The problem statement, all variables and given/known data

    Given a Hermitian operator A = [itex]\sum \left|a\right\rangle a \left\langle a\right|[/itex] and B any operator (in general, not Hermitian) such that [itex]\left[A,B\right][/itex] = [itex]\lambda[/itex]B show that B[itex]\left|a\right\rangle[/itex] = const. [itex]\left|a\right\rangle[/itex]

    2. Relevant equations

    Basically those listed above plus possibly the Hermitian condition and eigenvalue definition which I will not list since they are well known.

    3. The attempt at a solution

    I have tried expanding it out in terms of the commutator, but this seems like the wrong approach. I am not sure that there is a way to calculate it directly. I do not think the proof is very involved but I am approaching it the wrong way and cant seem to get anywhere. This is listed as a fundamental property of a Hermitian operator in a complex vector space in my text, though it does not prove it and other references seem unconcerned. I am also not very comfortable using this kind of spectral decomposition, which is a bit of a problem.
     
  2. jcsd
  3. Sep 15, 2011 #2
    I should point out that the end of the above problem is to show that B[itex]\left|a\right\rangle[/itex] = const. [itex]\left|a +\lambda\right\rangle[/itex]
     
  4. Sep 16, 2011 #3

    vela

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    These two statements are inconsistent. In both cases, you're applying B to |a>, but you're getting different answers.

    You can prove the second statement using the commutation relation.
     
  5. Sep 16, 2011 #4
    They are inconsistent because the statement in the first post is incorrect. The correct relation to prove is the second.
     
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