1. Angular-Momentum Theory M. Auzinsh D. Budker S. Rochester Optically polarized atoms: understanding light-atom interactions Ch. 3

2. 2 Classical rotations Commutation relations Quantum rotations Finding U (R )‏ D – functions‏ Visualization Irreducible tensors Polarization moments Rotations

3. 3 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: J i are “generators of infinitesimal rotations”

4. 4 Commutation relations Rotate green around x, blue around y Rotate blue around x, green around y Difference is a rotation around z For any two axes: Using From picture:

5. 5 Quantum rotations Want to find U (R) that corresponds to R E.g., expectation value of vector operator: U(R) should be unitary, and should rotate various objects as we expect Remember, for spin ½, U is a 2x2 matrix A is a 3-vector of 2x2 matrices R is a 3x3 matrix

6. 6 Quantum rotations Infinitesimal rotations For small θ: i makes J Hermitian  gives J units of angular momentum minus sign is conventional Like classical formula, except The J i 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. 7 Quantum rotations Determining U (R)‏ Start by demanding that U(R)‏ satisfies same commutation relations as R That's it! The commutation relations specify J, and thus U(R)‏ E.g., for spin ½:

8. 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 δ 2 term Same R z matrix as before

9. 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):