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Spin exchange operator for s=1/2

  1. Mar 10, 2014 #1
    1. The problem statement, all variables and given/known data
    Consider a system of two spin 1/2 particles, labeled 1 and 2. The Pauli spin matrices
    associated with each particle may then be written as
    [itex] \vec{\hat{\sigma _{1}}} ,
    \vec{\hat{\sigma _{2}}}
    [/itex]

    a)Prove that the operator [itex]\hat{A]}\equiv \vec{\hat{\sigma _{1}}}\cdot \vec{\hat{\sigma _{2}}}[/itex] is Hermitian. Find its eigenvalues. (Hint : Consider its
    operation on spins in the coupled representation with well-defined total spin.)

    b)Show that the operator
    [itex]
    \hat{D}\equiv \frac{1}{2}(1+\vec{\hat{\sigma _{1}}}\cdot \vec{\hat{\sigma _{2}}})
    [/itex]
    is the spin-exchange operator for two spins – that is, it exchanges the spins of the two
    particles.


    2. Relevant equations



    3. The attempt at a solution

    I know that an operator is Hermitian if
    [itex]<f|Ag> = <Af|g > [/itex] and that its eigenvalues are real, the eigenvectors span the space and are orthogonal.

    I'm not sure how to use the first property to prove this operator is Hermitian, I've used it in the context of operators working on wavefunctions but not for an operator like this.

    I looked up a little bit about whats really going in the "dot" product of the two pauli vectors and it seems like there is some very deep stuff there with tensor products and whatnot, but I don't believe my Professor intended for us to solve it using that route.

    First I wrote (from the expression for [itex] \hat{S^2}[/itex] ) the operator like this
    [itex]\vec{\hat{\sigma _{1}}}\cdot \vec{\hat{\sigma _{2}}}=\hbar^{-1}(\hat{S^2}-\hat{\vec{S^2_{1}}}-\hat{\vec{S^2_{2}}})[/itex]


    Then I think I can use the operation of these terms on a spin state to find the eigenvalues of A. However I'm confused about how to write the spin state ket to be operated on. The hint my professor gives only confused me more. We looked at in class how to expand a coupled state in the uncoupled basis and he said to use this representation of the coupled state.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Mar 10, 2014
  2. jcsd
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