Rotation of operators

[jdstokes]

Suppose we know the matrix elements of an operator with respect a given cartesian reference frame L. If we know the sequence of rotations going from L to some other reference frame L', what is the expression for the operator in the new reference frame.

Let R be the required rotation and \mathcal{D}(R) the corresponding rotation operator. We know that the state of the systems changes under active rotation by multiplication | \psi \rangle \mapsto \mathcal{D}(R) |\psi\rangle. In our case we're rotating the environment so the basis states which make up the operator should transform according to |\phi_i \rangle \mapsto U|\phi_i\rangle.

Therefore

\hat{O} = \sum_{ij} o_{ij} | \phi_i \rangle\langle \phi_j | \mapsto \sum_{ij} o_{ij} U| \phi_i \rangle \langle \phi_j |U^{\dag} = U \hat{O} U^{\dag} .

Am I understanding this correctly?

[PRB147]

|\psi\rangle \longrightarrow D(R)|\psi\rangle
O|\psi\rangle\longrightarrow D(R)O|\psi\rangle=D(R)OD^{-1}(R)D(R)|\psi\rangle

[jdstokes]

So I gather what I said is correct, in other words I could obtain the spin-x operator from the spin-z operator in the following fashion:

\hat{S}_x = \mathcal{D}_y(\pi/2) \hat{S}_z \mathcal{D}_y(-\pi/2) e.g.??

[lbrits]

So I gather what I said is correct, in other words I could obtain the spin-x operator from the spin-z operator in the following fashion:

\hat{S}_x = \mathcal{D}_y(\pi/2) \hat{S}_z \mathcal{D}_y(-\pi/2) e.g.??

Yes, and this is also easily checked explicitly for spin \tfrac{1}{2} because of the Pauli matrix identity e^{i \theta \hat{n} \cdot \vec{\sigma} } = .... But a more group theoretical treatment is probably best.