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Hermitian Operators and Imaginary Numbers

  1. Sep 23, 2004 #1
    So I understand what a hermitian operator is and how if A and B are hermitian operators, then the product of AB is not necessarily Hermitian since


    *Note here + is dagger

    (AB)+=B+A+=BA

    I also recognize that (AB-BA) is not Hermitian since (AB-BA)+=B+A+-A+B+

    In addition, I know that any real number a is a Hermitian operator since <Psi l|a Psi n>=<aPsi l|Psi n>

    Now here comes my questions.
    Where A and B are both hermitian operators,
    1)how do we know if something like i(AB-BA) is a hermitian operator where i is an imaginary number? How do I show that this is not a hermitian operator because I am pretty sure it is not?

    2) and how would I show that (AB+BA/2) is Hermitian because I feel like it should be, but I dont know how to interchange the 2 with the A and B operators?

    And if operator A corresponds to observable A, and operator B corresponds to observable B, what is a "good" (i.e.Hermitian) operator that corresponds to the physically observable product AB?


    When I am dealing with two operators, I dont think I am confused on how to work with them, but when dealing with 3 I get a little iffy. Peace and love.
     
  2. jcsd
  3. Sep 23, 2004 #2

    vanesch

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    [ i(AB-BA)]+ = (-i)(B+ A+ - A+ B+) = (-i) (BA-AB) = i(AB-BA)


    (AB+BA/2)+ - (AB+BA/2) = BA + AB/2 - AB - BA/2 = BA/2 - AB/2 = [B,A] / 2

    So if A and B do not commute, the difference between your operator and its conjugate is not 0 (because equal to the commutator divided by 2).

    1/2 (AB + BA).

    (Wigner's prescription, I think it is called).

    It is not unique, of course ;
    you can have 1/2(AB + BA) + c i [A,B] with c an arbitrary real number for example.


    cheers,
    Patrick.
     
  4. Sep 23, 2004 #3
    :smile:
    Hi, here are the answers
    1) Just use the definition of "Hermitian" : A+ = A and B+ = B (they are hermitian)
    so you get :

    [ i(AB-BA)]+ = (-i)(B+ A+ - A+ B+) = (-i) (BA-AB) and this equals i(AB-BA)



    2) Same system, man, just use the definition of hermitian operators :

    (AB+BA/2)+ - (AB+BA/2) = (BA + AB/2) - AB - BA/2 = BA/2 - AB/2 = [B,A] / 2

    If A and B are not commuting operators then the difference between the given operator and its conjugate is not 0 . If A and B commute then the commutator will be 0 !!

    3) this is an easy one and the answer is

    1/2 (AB + BA)


    regards
    marlon
     
  5. Sep 23, 2004 #4
    Sorry, Patrick

    I did not see you already answered this question...

    I apologize for that...

    marlon
     
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