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Hermitian Operator

  1. Feb 28, 2013 #1

    My task is to prove that the angular momentum operator is hermitian. I started out as follows:


    Where the above quantities are vector operators. Taking the hermitian conjugate yields


    Here I have used double quotes to represent that the hermitian conjugate of the corresponding quantity.


    Here the fact that the momentum and position are hermitian operators were used. However

    There has to be a flaw somewhere but I was not able to catch it, though I was able to prove that the angular momentum operator is hermitian when inspected component by component. I am yet to understand the error in the above derivation. Any help is appreciated.

    Thanks in advance
  2. jcsd
  3. Feb 28, 2013 #2


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    L′′ = p′′ × r′′

    The Hermitian conjugate reverses the operator order, but you've also reversed the order of the arguments to the cross product, which should bring in a minus sign.
  4. Feb 28, 2013 #3
    Thanks for the reply. So as far as I am concerned you tell me that reversing the arguments of the cross product is superfluous. But how can I denote that I have reversed the orders of the operators without doing that? My point is to demonstrate that I have reversed the order of operators in vector notation.
  5. Feb 28, 2013 #4


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    In general, if [itex]\vec{A}[/itex] and [itex]\vec{B}[/itex] are operators, then [itex](\vec{A} \times \vec{B})^\dagger = - \vec{B}^\dagger \times \vec{A}^\dagger[/itex]
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