Angular-Momentum Theory

1
Angular-Momentum Theory

• M. Auzinsh
• D. Budker
• S. Rochester
• Optically polarized atoms understanding
  light-atom interactions
• Ch. 3

2
Rotations

• Classical rotations
• Commutation relations
• Quantum rotations
• Finding U (R )?
• D functions?
• Visualization
• Irreducible tensors
• Polarization moments

3
Classical rotations
Rotations use a 3x3 matrix R
position or other vector
Rotation by angle ? about z axis
For ?p/2
For small angles
For arbitrary axis
Ji are generators of infinitesimal rotations
4
Commutation relations
Rotate green around x, blue around y
From picture
Rotate blue around x, green around y
For any two axes
Using
Difference is a rotation around z
5
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
6
Quantum rotations

• Infinitesimal rotations

Like classical formula, except
i makes J Hermitian
For small ?
? gives J units of angular momentum
minus sign is conventional

• 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
7
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
8
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 d2 term
Same Rz matrix as before
9
D -functions
Matrix elements of the rotation operator
Rotations do not change j .
D-function
z-rotations are simple
so we use Euler angles (z-y-z)