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Hermitian Operators

  1. Oct 25, 2007 #1
    I have some questions about the properties of a Hermitian Operators.
    1) Show that the expectaion value of a Hermitian Operator is real.
    2) Show that even though [tex]\hat{}Q[/tex] and [tex]\hat{}R[/tex] are Hermitian, [tex]\hat{}Q[/tex][tex]\hat{}R[/tex] is only hermitian if [[tex]\hat{}Q[/tex],[tex]\hat{}R[/tex]]=0

    2. Relevant equations

    3. The attempt at a solution

    1) Expectation Value <[tex]\hat{}Q[/tex]>= [tex]\int\Psi[/tex]*[tex]\hat{}Q[/tex][tex]\Psi[/tex] and for a Hermitian Operator [tex]\hat{}Q[/tex]*=[tex]\hat{}Q[/tex]
    Therefore does
    1) Expectation Value <[tex]\hat{}Q[/tex]>= [tex]\int\Psi[/tex]*[tex]\hat{}Q[/tex][tex]\Psi[/tex]=([tex]\int[/tex][tex]\Psi[/tex]*[tex]\hat{}Q*[/tex][tex]\Psi[/tex] )* prove that the expectaion value is real as the complex conjugate = the normal value?

    attempt at 2)
    now if A, B are hermitian this is only true if AB is also hermitian?
  2. jcsd
  3. Oct 25, 2007 #2


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    Homework Helper

    1. Use this TEX parse [tex] \hat{Q} [/tex].
    2. For a vector [itex] \psi [/tex], the expectation value of the linear operator A is [itex] \langle \psi, A\psi [/itex]. If A is hermitean, can you show that the exp. value is real ?

    The 3-rd point is a little bit involved.
  4. Nov 24, 2007 #3
    Another question about Hermitians..

    If A and B are Hermitian, then is AB also hermitian?

  5. Nov 24, 2007 #4
    think of what the hermitian conjugate is for AB...
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