- 523

- 0

## Main Question or Discussion Point

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

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

Therefore

[itex]\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} [/itex].

Am I understanding this correctly?

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

Therefore

[itex]\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} [/itex].

Am I understanding this correctly?

Last edited: