A 90 degree rotation about the z axis.

If we rotate our coordinate system by 90 degrees about the z axis,
the old x axis becomes the new -y axis.
So we would expect that the state with angular momentum
in the x direction,
,
will rotate into
within a phase factor.
Lets do the rotation.

Before rotation the state is

The rotated state is.

Now, what remains is to check whether this state is the one we expect.
What is
?
We find that state by solving the eigenvalue problem.

Setting
, we get the unnormalized result.

Normalizing, we get.

This is exactly the same as the rotated state. A 90 degree rotation about the z axis changes into .

Jim Branson 2013-04-22