### Derive the Expression for Rotation Operator *

The laws of physics do not depend on what axes we choose for our coordinate system- There is rotational symmetry. If we make an infinitesimal rotation (through and angle ) about the z-axis, we get the transformed coordinates

We can Taylor expand any function ,

So the rotation operator for the function is

A finite rotation can be made by applying the operator for an infinitesimal rotation over and over. Let . Then

The last step, converting the limit to an exponential is a known identity. We can verify it by using the log of the quantity. First we expand about : .

So exponentiating, we get the identity.

Jim Branson 2013-04-22