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# oldexam2 - MIDTERM EXAM 2 PHYS 851 Quantum Mechanics I Fall...

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MIDTERM EXAM 2 PHYS 851 Quantum Mechanics I, Fall 2007 1. A system is described by the Hamiltonian H = planckover2pi1 2 2 λ 2 ( AA + A A AA A A ) , where A and A are the harmonic oscillator lowering and raising operators, which satisfy [ A,A ] = 1. (a) [10 pts] Derive the Heisenberg equation of motion for A H . (b) [10 pts] What is the corresponding expression for d dt A H ? (c) [10 pts] From these two equations, what is d dt P H , where P H = i planckover2pi1 2 λ parenleftBig A H A H parenrightBig ? Write the solution to this equation in terms of P S , as well as t and whatever constants are required . (d) [10 pts] Using your answers to parts (a), (b) and (c) what is d dt X H , where X H = λ 2 parenleftBig A H + A H parenrightBig ? What is the solution to this equation? (e) [10 pts] Re-express the Hamiltonian in terms of X and P . Do your previous answers make sense? 2. We can define the momentum shift operator T P ( p 0 ) via T P ( p 0 ) | p ) = | p + p 0 ) . The following questions concern this momentum shift operator:
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