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 For θ=π/2: Rotation by angle θ about z axis: For small angles: For arbitrary axis: J

i

are “generators of infinitesimal rotations” 3

Commutation relations

From picture: For any two axes: Using Rotate green around x, blue around y 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

A

is a 2x2 matrix 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

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: 6

Quantum rotations

 Determining U (R) Start by demanding that relations as R

U

(R) satisfies same commutation The commutation relations specify

J

, and thus

U

(R) E.g., for spin ½: That's it!

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 δ 2 term Same R before z matrix as 8

D

-functions

Matrix elements of the rotation operator Rotations do not change

j .

z-rotations are simple: so we use Euler angles (

z y-z

):

D

-function 9