Spin exchange operator for s=1/2

[Hakkinen]

Homework Statement

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
$\vec{\hat{\sigma _{1}}} , \vec{\hat{\sigma _{2}}}$

a)Prove that the operator $\hat{A]}\equiv \vec{\hat{\sigma _{1}}}\cdot \vec{\hat{\sigma _{2}}}$ 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
$\hat{D}\equiv \frac{1}{2}(1+\vec{\hat{\sigma _{1}}}\cdot \vec{\hat{\sigma _{2}}})$
is the spin-exchange operator for two spins – that is, it exchanges the spins of the two
particles.

Homework Equations

The Attempt at a Solution

I know that an operator is Hermitian if
$<f|Ag> = <Af|g >$ 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 what's 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 $\hat{S^2}$ ) the operator like this
$\vec{\hat{\sigma _{1}}}\cdot \vec{\hat{\sigma _{2}}}=\hbar^{-1}(\hat{S^2}-\hat{\vec{S^2_{1}}}-\hat{\vec{S^2_{2}}})$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.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[Nate810]

a) The operator \hat{A]}\equiv \vec{\hat{\sigma _{1}}}\cdot \vec{\hat{\sigma _{2}}} is Hermitian because it is the dot product of two Pauli spin matrices, which means that it can be written as \hat{A]}\equiv \vec{\hat{\sigma _{1}}}\cdot \vec{\hat{\sigma _{2}}} = \hbar^{-1}(\hat{S^2}-\hat{\vec{S^2_{1}}}-\hat{\vec{S^2_{2}}})This is a Hermitian operator since it is the difference of the squares of two Hermitian operators.The eigenvalues of this operator can be found by expanding the coupled state in the uncoupled basis and operating on it with the operator. This will give four eigenvalues, -1, 0, 1, and 2. b) The operator \hat{D}\equiv \frac{1}{2}(1+\vec{\hat{\sigma _{1}}}\cdot \vec{\hat{\sigma _{2}}}) is the spin-exchange operator for two spins. This can be seen by looking at the operation it performs on the coupled state. It exchanges the spins of the two particles, as the coefficients in front of the coupled states are swapped. This can be demonstrated by looking at its matrix representation.