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Hermitian Operators and the Commutator

  1. Oct 28, 2008 #1
    1. The problem statement, all variables and given/known data
    If A is a Hermitian operator, and [A,B]=0, must B necessarily be Hermitian as well?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 28, 2008 #2

    olgranpappy

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    Homework Helper

    attempt at solution?
     
  4. Oct 28, 2008 #3
    if Y is an eigenstate of both A and B with respective eigenvalues a and b and respective adjoints (A+) and (B+),

    <Y|AB|Y> = <Y|BA|Y>
    = <Y|Ab|Y> = <(B+)Y|A|Y>
    = b<Y|A|Y> = (b*)<Y|A|Y>

    Therefore, b=(b*), and so it follows that B=(B+), or B is Hermitian.
     
  5. Oct 28, 2008 #4

    olgranpappy

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    counter example:

    consider a hermitian operator H. H commutes with any function of H.

    For example, the function
    [tex]
    U=e^{-iHt}\;.
    [/tex]

    Does U commute with H?

    Is U hermitian?
     
  6. Oct 28, 2008 #5
    Much easier: how about B=iA?
     
  7. Oct 28, 2008 #6
    fair enough. thank you for answering my question, though that makes the problem a little more complicated... i hate it when that happens.
     
  8. Nov 1, 2008 #7
    Or the easiest of all: B=iI (with I the identity) :-)
     
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