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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