# Exponentials and linear operators

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.

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 terms –  tibL Nov 19 '12 at 12:46

The operators $O_1,O_2,O_3$ form a representation of the lie algebra $\mathfrak{so}(3)$. These are also the Pauli spin matrices.
The exponentials generate rotations in $\mathbb{R}^3$ so that $e^{i\phi O_2} \in SO(3)$. Any rotation in 3D is is specified by an axis and angle of rotation. Conjugation by another rotation, changes the axis by rotation, but doesn't change the angle.
This is similar to how two permutations conjugate one another $\sigma (1234)\sigma^{-1} = (\sigma(1)\sigma(2)\sigma(3)\sigma(4))$.