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Angular-Momentum Theory M. Auzinsh D. Budker S. Rochester Optically polarized atoms: understanding light-atom interactions Ch. 3
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2 Classical rotations Commutation relations Quantum rotations Finding U (R ) D – functions Visualization Irreducible tensors Polarization moments Rotations
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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”
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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:
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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
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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:
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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:
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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
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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):
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