Transforming Observables, Missunderstanding <PERSON>, Intro. to QM, or a Possible Error
In <PERSON>' Intro. to QM 3rd, Sec. 6.2, transforming an observable $Q$ by the translation operator $T$ is found to be $$ Q' = T^\dagger Q\ T $$ the same for the parity operator $\Pi$ instead of $T$ we have $Q' = \Pi^\dagger Q\ \Pi$.
But in other texts, e.g., <PERSON>, QM, 2nd ed, Vol. I, Complements of chapter VI, Complement B$_{VI}$, 5. Rotation of observables, and also in other questions here and here the transformation on the observable $A$ by a unitary transformation $U$ should be $$ A' = UA\ U^\dagger $$ where $U$, as I understand, should be an active transformation, as $T$ above and I expected that the two equations should be the same. 346
But it seems that the two definitions are not equivalent, or is there any mistake?
ADDED
<PERSON> Definition:
The transformed operator $\hat Q'$ is deﬁned to be the operator that gives the same expectation value in the untranslated state $\psi$ as does the operator $\hat Q$ in the translated state $\psi'$ $$ \langle\psi'|\hat Q|\psi'\rangle = \langle \psi | \hat Q' |\psi \rangle $$ There are two ways to calculate the effect of a translation on an expectation value. One could actually shift the wave function over some distance (this is called an active transformation) or one could leave the wave function where it was and shift the origin of our coordinate system by the same amount in the opposite direction (a passive transformation). The operator $\hat Q'$ is the operator in this shifted coordinate system.
Using Eq. 6.1, $$ \langle\psi|T^\dagger\hat Q\ \hat T|\psi\rangle = \langle \psi | \hat Q' |\psi \rangle $$
<PERSON> Definition:
Let us assume the system to be in the eigenstate $|u_n\rangle$ of $A$: the device for measuring $A$ in this system will give the result $a_n$ without fail. But just before performing the measurement, we apply a rotation $\scr R$ to the physical system and, simultaneously, to the measurement device; their relative positions are unchanged. Consequently, if the observable $A$ which we are considering describes a physical quantity attached only to the system which we have rotated (that is, independent of other systems or devices which we have not rotated), then, in its new position, the measurement device will still give the same result $a_n$ without fail. Now, after rotation, the device, by deﬁnition, measures $A'$, and the system is in the state: $$ |u_n'\rangle = R|u_n\rangle $$ We must therefore have: $$ A|u_n\rangle = a_n|u_n\rangle \implies A'|u_n'\rangle = a_n|u_n'\rangle $$ that is: $$ R^\dagger A' R |u_n\rangle = a_n|u_n\rangle $$
Note that $\scr R$ is the rotation the physical 3-dimensional space and $R$ is the its representative operator in the Hilbert space. 669