Imagine a Stern-Gerlach apparatus that first separates an
atomic beam
with a strong B-field gradient in the z-direction.
Let's assume the beam has atoms moving in the y-direction.
The apparatus blocks two separated beams, leaving only the eigenstate of
with eigenvalue
.
We follow this with an apparatus which separates in the u-direction, which is at
an angle
from the z-direction, but still perpendicular to the direction of
travel of the beam, y.
What fraction of the (remaining) beam will go into each of the three beams which are
split in the u-direction?
We could represent this problem with the following diagram.
To solve this with the rotation matrices, we first determine the state after the first
apparatus. It is just
with the usual basis.
Now we rotate to a new (primed) set of basis states with the
along the
direction.
This means a rotation through an angle
about the y direction.
The problem didn't clearly define whether it is
or
, but, if we only
need to know the intensities, it doesn't matter.
So the state coming out of the second apparatus is
An alternate solution would be to use the
operator. Find the eigenvectors of this operator, like
. The intensity in the +
beam is then
.
Jim Branson 2013-04-22