Angular-Momentum Theory
M. Auzinsh
D. Budker
S. Rochester
Optically polarized atoms: understanding
light-atom interactions
Ch. 3
Rotations

Classical rotations




Commutation relations
Quantum rotations

Finding U (R )

D – functions
Visualization
Irreducible tensors

Polarization moments
2
Classical rotations
Rotations use a 3x3 matrix R:
position or other vector
Rotation by
angle θ
about z axis:
For θ=π/2:
For small
angles:
For arbitrary
axis:
Ji are “generators of infinitesimal rotations”
3
Commutation
relations
Rotate green
around x, blue
around y
From picture:
For any two axes:
Using
Rotate blue
around x,
green around y
Difference is
a rotation
around z
4
Quantum rotations

Want to find U (R) that corresponds to R
U(R) should be unitary, and should rotate various objects as we expect
E.g., expectation
value of vector
operator:
Remember, for spin ½,
U is a 2x2 matrix
A is a 3-vector of 2x2 matrices
R is a 3x3 matrix
5
Quantum rotations

Infinitesimal rotations
Like classical
formula, except
i makes J Hermitian
For small θ:
minus sign is
conventional
 gives J units of
angular momentum
 The Ji are the generators of infinitesimal rotations
 They are the QM angular momentum operators.
 This is the most general definition for J
We can recover
arbitrary rotation:
6
Quantum rotations

Determining U (R)
Start by demanding that U(R)
satisfies same commutation
relations as R
The commutation relations
specify J, and thus U(R)
That's it!
E.g., for
spin ½:
7
Quantum rotations

Is it right?
We've specified U(R), but does it do what we want?
Want to check
J is an observable, so check
Do easy case: infinitesimal rotation around z
Neglect δ2
term
Same Rz
matrix as
before
8
D -functions
Matrix elements of the rotation operator
Rotations do
not change j .
D-function
z-rotations are
simple:
so we use
Euler angles (zy-z):
9