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I'm trying to understand a calculation my professor did in my quantum mechanics script. Here it is:

Each rotation $R \in O(3)$ induces a unitary transformation in $L^2(R^3)$, i.e. the space of square integrable real functions. $U( R ): \psi(\vec{x}) \rightarrow \psi(R^{-1} \vec{x}) $.

The professor goes on to show that $R \rightarrow U( R)$ is a unitary representation of O(3). He does that by showing $U( 1 ) = id$ and $U(R_2)U(R_1) = U(R_2 R_1)$. The first one is trivial while the second one seems to be wrong for the definition of $U(R )$ above.

$U(R_2)U(R_1) \psi(\vec{x}) = U(R_2)\psi(R_1^{-1} \vec{x}) = \psi(R_2^{-1} R_1^{-1} \vec{x}) = \psi((R_1 R_2)^{-1}\vec{x})$. Which isn't the same as $U(R_2 R_1)$.

Does anybody know where I went wrong?

Second question is:

It says in the script "$U(\lambda) = U(R(\lambda))$ produces a 1 parameter unitary group. Its generating function A is: $(A\psi)(\vec{x}) = i\hbar \frac{d}{d \lambda} \psi(R(-\lambda)\vec{x})$". Without really explaining where this comes from. What definition do I need to understand that equation?

Cheers and thanks in advance!

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