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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Rotation of operators

**Physics Forums | Science Articles, Homework Help, Discussion**