A Rotated Stern-Gerlach Apparatus

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**.

We put a detector in each of the beams split in to determine the intensity.

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

The 3 amplitudes in this vector just need to be (absolute)

These add up to 1.

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