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Showing that general operator equations are hermitian

  1. Oct 16, 2012 #1
    1. The problem statement, all variables and given/known data

    a)For a general operator A, show that and i(A-A+) are hermitian?

    b) If operators A and B are hermitian, show that the operator (A+B)^n is Hermitian.

    2. Relevant equations


    3. The attempt at a solution

    The first part I did,

    (A+A+)+=(A++A)=(A+A+)

    i(A-A+)=[i(A-A+)]+=(iA)+-(iA+)+)=i(A-A+)

    The second part I used binomial expansion (induction) but I was told they do not commute so this cannot be done?

    Is my answer to the first part correct and what route should I take for the second? Thanks
     
  2. jcsd
  3. Oct 16, 2012 #2
    First part looks about right.

    For the second part, do you really need to multiply things out? After all, if A and B are Hermitian then A+B is also Hermitian...
     
  4. Oct 16, 2012 #3

    dextercioby

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    The problems seems to have no connection with <Advanced Physics> and neither with mathematics, since it's not mathematically (i.e. in agreement with mathematics) formulated and not even semantically.
     
  5. Oct 17, 2012 #4

    dextercioby

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    For example, the mathematical problem for point a) should have been

    "Let A be a densly defined linear operator on a complex separable Hilbert space [itex] \mathcal{H}[/itex].

    Prove that if [itex] D(A)\cap D\left(A^{\dagger}\right) [/itex] is dense everywhere in [itex] \mathcal{H} [/itex],

    then the operator [itex] i\left(A-A^{\dagger}\right) [/itex] is symmetric.
     
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