Verifying Rotational Operator in Quantum Mechanics

[folgorant]

hi all,
I have a problem about rotation operator in QM.
I must verify that $$(\hat{U}(R)f)(\textbf{x})=f(R^{-1}\textbf{x})$$

with: $$\hat{U}(R) = exp({\frac{-i\varphi\textbf{nL}}{\hbar}})$$

R rotation on versor n of angle $$\varphi$$

I don't really know how to start, so please give me an advice!

 

[nrqed]

hi all,
I have a problem about rotation operator in QM.
I must verify that $$(\hat{U}(R)f)(\textbf{x})=f(R^{-1}\textbf{x})$$

with: $$\hat{U}(R) = exp({\frac{-i\varphi\textbf{nL}}{\hbar}})$$

R rotation on versor n of angle $$\varphi$$

I don't really know how to start, so please give me an advice!

In the exponential, write L as a differential operator. Then expand the exponential to first order.

On the rhs, write explicly the transformed coordinates $$R^{-1} x$$ to first order I am the rotation parameters. Next, Taylor expand $$f( R^{-1} x)$$ to first order in those parameters.

The two sides will be equal