If we block only them= 1 beam, the apparatus would be represented by
+|
0
−
z→=| 0 〉〈 0 | + |−〉〈−|.
- See Example 18.10.7:A series of Stern-Gerlachs.*
18.6 Rotation operators forℓ= 1*.
We have chosen thezaxis arbitrarily. We could choose any other direction to define our basis states.
We wish to know how totransform from one coordinate system to another.Experience has
shown that knowing how an object transforms under rotations is important in classifying the object:
scalars, vectors, tensors...
We canderive(see section 18.11.3)the operator for rotations about the z-axis. This operator
transforms an angular momentum state vector into an angular momentum state vector in the rotated
system.
Rz(θz) = eiθzLz/ ̄h
ψ′ = Rz(θz)ψSince there is nothing special about the z-axis, rotations about the other axes follow the same form.
Rx(θx) = eiθxLx/ ̄h
Ry(θy) = eiθyLy/ ̄hThe above formulas for therotation operatorsmust apply in both the matrix representation and
in the differential operator representation.
Redefining the coordinate axes cannot change any scalars, like dotproducts of state vectors. Oper-
ators which preserve dot products are calledunitary.We proved that operators of the above form,
(with hermitian matrices in the exponent) are unitary.
Acomputation(see section 18.11.4)of the operator for rotations about the z-axis gives
Rz(θz) =
eiθz 0 0
0 1 0
0 0 e−iθz
.
Acomputation(see section 18.11.5)of the operator for rotations about the y-axis yields
Ry(θy) =
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))
.
Try calculating the rotation operator for the x-axis yourself.