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HW14 - PHYS851 Quantum Mechanics I Fall 2008 HOMEWORK...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 14 1. [15 pts] Consider the two-state quantum system described by H = H + V , where H = planckover2pi1 δ 2 ( | 1 )( 1 | − | )( | ) , and V = planckover2pi1 Ω 2 ( | )( 1 | + | 1 )( | ) . In the Schr¨odinger picture we have | ψ S ( t ) ) = c ( t ) | ) + c 1 ( t ) | 1 ) . Solve Schr¨odinger’s equation for the coefficients c ( t ) and c 1 ( t ) with the initial conditions c (0) = 1 and c 1 (0) = 0. Let the operator W be defined in the Schr¨odinger picture as W = | 1 )( 1 | − | )( | . Compute ( W ) as a function of time. 2. [20 pts] For the same system as problem 1, compute ( W ) using the Heisenberg picture. Do this by first computing the commutator [ H,W ] and inserting the result into the Heisenberg equation of motion for W . Repeat this procedure to find the equations of motion for whatever operators appear on the RHS of the equation for dW/dt . Try to obtain a closed set of equations (there should be three in all). Solve the this set of equations for the appropriate initial conditions and compare yourthree in all)....
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