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852_2010hw13 - PHYS852 Quantum Mechanics II Spring 2010...

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PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 13 Topics covered: Hilbert-space Frame Transformations, Time-Dependent Perturbation Theory 1. The Hamiltonian for a driven two-level system is H = planckover2pi1 ω 0 | 2 )( 2 | + planckover2pi1 Ω cos( ωt ) ( | 1 )( 2 | + | 2 )( 1 | ) , (1) where ω 0 is the separation between the bare levels, and ω is the driving frequency. (a) Make a frame transformation generated by the operator G = planckover2pi1 ω | 2 )( 2 | , and determine the equation of motion for the state-vector in the new frame, defined by | ψ G ( t ) ) = U G ( t ) | ψ ( t ) ) . (b) Make the rotating wave approximation (RWA) by assuming that ω ω 0 , and dropping any terms that oscillate at or near 2 ω 0 . Write, in terms of the detuning Δ = ω 0 ω , the effective time-independent Hamiltonian, H G , that then governs the time evolution of | ψ G ( t ) ) . (c) Assume that the system begins at time t = 0 in the ground-state of H G , and calculate | ψ G ( t ) ) . Is this a stationary state in the rotating frame? Now use | ψ S ( t ) ) = U G ( t ) | ψ G ( t ) ) to see what this state looks like in the Scr¨odinger picture. Is it a stationary state in the Schr¨odinger picture?
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