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I don't quite understand how the following works, could anyone please explain?

Let $O_i,i=1,2,3$ be linear operators on a space of functions.

They satisfy the commutation relation $[O_a,O_b]=O_aO_b-O_bO_a=i\epsilon_{abc}O_c$

Why then is $$\exp(-i\phi O_2)O_1\exp(i\phi O_2)=O_1\cos\phi-O_3\sin \phi$$?

I don't understand how to deal with the exponentials. I thought that maybe it had to do with Euler's formula, but I don't quite know how that works with the operator.

Thanks in advance!


I believe it is related to angular momentum operators in quantum mechanics.

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    I think that what you need to use here first is that the exponential of a linear operator $A$ is given by $\exp(A)=1+A+A^2/2+\dots$ then play around with terms2012-11-19

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