Note that both of the aboverotation matricesreduce to the identity matrix for rotations of 2π
radians. For a rotation ofπradians,Ryinterchanges the plus and minus components (and changes
the sign of the zero component), which is consistent with what we expect. Note also that the above
rotation matrices are quite different than the ones used to transform vectors and tensors in normal
Euclidean space. Hence, the states here are of a new type and arereferred to asspinors.
- See Example 18.10.5:A 90 degree rotation about the z axis.*
18.7 A Rotated Stern-Gerlach Apparatus*
Imagine a Stern-Gerlach apparatus that first separates anℓ= 1 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 ofLzwith eigenvalue + ̄h. We
follow this with an apparatus which separates in the u-direction, which is at an angleθfrom 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 followingdiagram.
Oven→
+
0 |
−|
z→
+D+
0 D 0
−D−
uWe put a detector in each of the beams split inuto determine the intensity.
To solve this with the rotation matrices, we first determine thestate after the first apparatus.
It is justψ(+z)=
1
0
0
with the usual basis. Now werotate to a new (primed) set of basisstateswith thez′along theudirection. This means a rotation through an angleθabout the y
direction. The problem didn’t clearly define whether it is +θor−θ, but, if we only need to know
the intensities, it doesn’t matter. So thestate coming out of the second apparatusis
Ry(θ)ψ(+z) =
1
2 (1 + cos(θy))
√^1
2 sin(θy)1
2 (1−cos(θy))
−√^12 sin(θy) cos(θy) √^12 sin(θy)
1
2 (1−cos(θy)) −
√^1
2 sin(θy)1
2 (1 + cos(θy))
1
0
0
=
1
2 (1 + cos(θy))
−√^12 sin(θy)
1
2 (1−cos(θy))
The 3 amplitudes in this vector just need to be (absolute)squared to get the 3 intensities.
I+=
1
4
(1 + cos(θy))^2 I 0 =1
2
sin^2 (θy) I−=1
4
(1−cos(θy))^2These add up to 1.
An alternate solution would be to use theLu= ˆu·L~ = cosθLz+ sinθLxoperator. Find the
eigenvectors of this operator, likeψ+(u). The intensity in the + beam is thenI+=|〈ψ+(u)|ψ+(z)〉|^2.