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Quantum Mechanics 2, Autumn 2011 CMI Problem set 3 Due by beginning of class on Monday September 5, 2011 Angular momentum and spin 1. Recall that the angular momentum raising operator is L+ = ~eiφ (∂θ + i cot θ ∂φ ). Use this to find L− . 2. Use the above formulae for L± to find the coordinate representation of the angular momentum basis states Y11 , Y10 and Y1,−1 up to normalization. 3. Write out the 9 equations summarized in the formula for products of Pauli matrices σi σ j = δi j + ii jk σk 4. Check that these formulae hold for the Pauli matrices ! ! 0 1 0 −i σx = , σy = , 1 0 i 0 (1) ! −1 0 σz = . 0 1 (2) 5. The hamiltonian for an electron in a vertical magnetic field Bẑ is ! g|e|~B 1 0 H=E , E= . 0 −1 4m Find the spin state ψ(t) if the initial spin wavefunction is ψ(t = 0) = 6. Compute hS z i, hS x i and hS y i at time t in the state ψ(t) 7. Physically interpret the obtained expectation values for S x , S y , S z . 1 (3) √1 2 (↑ + ↓).