# HW5 - the temperature in Kelvin for ±eld strengths of B =...

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PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 5: Two Level Systems: 1. [20 pts each] Cohen-Tannoudji pp 476-480, problems 4.1 and 4.5 2. [20 pts] First, write the density matrix of spin-1/2 particle whose state is a statistical mixture of equal parts spin-up and spin-down with respect to the z-axis. Then, ±nd the eigenstates of S x using the eigenstates of S z as a basis. Re-express this density operator using as a basis, the eigenstates of S x . Comment on your answer. As a function of magnetic ±eld strength B 0 , at what temperature does a spin-1/2 particle become ‘frozen’ in the spin-up state? Using the gyromagnetic ratio for an electron, estimate
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Unformatted text preview: the temperature in Kelvin for ±eld strengths of B = 10-3 Gauss, 1Gauss, and 10 3 Gauss. 3. [10 pts] Using the eigenvalues and eigenvectors from the “Review” page of Lecture 11, verify the “Far from Resonance” limiting expressions for Δ < 0 and Δ > 0. 4. [20 pts] As described in class, calculate the quantum-resonance response curve de±ned as the amplitude of the Rabi-Oscillations, as a general function of the detuning and Rabi frequency. Verify that the curve is a Lorentzian distribution, and compute the width of the resonance (some algebra will be required, but expect algebraic miracles to occur)....
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