User:Tshibli/sandbox

Rotation of Operators
We define the Rotated operator A' by requiring that the expectation value of the original operator A with respect to the initial state be equal to the expectation value of the rotated operator with respect to the rotated state,


 * $$ \langle \psi' | A' | \psi' \rangle = \langle \psi | A | \psi \rangle $$

Now as,


 * $$ | \psi \rangle $$ → $$| \psi' \rangle = U(R) | \psi \rangle \,, \quad \langle \psi | $$ → $$ \langle \psi' | = \langle \psi | U^\dagger (R) $$

we have,


 * $$ \langle \psi | U^\dagger (R) A' U(R)| \psi \rangle = \langle \psi | A | \psi \rangle $$

since, $$ | \psi \rangle $$ is arbitrary,


 * $$ U^\dagger (R) A' U(R) = A $$

hence, on rotation, the operator A becomes,


 * $$ A $$ → U^\dagger (R) A' U(R)