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