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Derive the Expression for Rotation Operator
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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