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I'm stuck on the concept of the rotation operator in QM.

From what I understand, one constructs a representation of SO(3) on a Hilbert space by mapping a rotation matrix [itex]R\in SO(3)[/itex] specified by an angle [itex]\phi[/itex] and a unit vector [itex]\vec{n}[/itex] to

[itex]D(R) = \exp[-\frac{i \phi}{\hbar}\vec{J}\cdot \vec{n}] [/itex].

However, I know that

[itex] R = exp[-\frac{i \phi}{\hbar}\vec{J}\cdot \vec{n}] [/itex],

which is just the exponential map from [itex]\mathfrak{so}(3)[/itex] to [itex]SO(3)[/itex].

This would amount to saying

[itex]D(R)=R[/itex],

which confuses me. What is going on here? Are we viewing [itex]R[/itex] as an operator on the Hilbert space?

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# Rotation operator quantum mechanics

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