MIDTERM EXAM 2PHYS 851 Quantum Mechanics I, Fall 20071. A system is described by the HamiltonianH=−planckover2pi122λ2(AA+A†A†−AA†−A†A), whereAandA†are the harmonic oscillator lowering and raising operators, which satisfy [A,A†] = 1.(a) [10 pts] Derive the Heisenberg equation of motion forAH.(b) [10 pts] What is the corresponding expression forddtA†H?(c) [10 pts] From these two equations, what isddtPH, wherePH=−iplanckover2pi1√2λparenleftBigAH−A†HparenrightBig? Write thesolution to this equation in terms ofPS, as well astand whatever constants are required .(d) [10 pts] Using your answers to parts (a), (b) and (c) what isddtXH, whereXH=λ√2parenleftBigAH+A†HparenrightBig?What is the solution to this equation?(e) [10 pts] Re-express the Hamiltonian in terms ofXandP.Do your previous answers makesense?2. We can define the momentum shift operatorTP(p0) viaTP(p0)|p)=|p+p0). The following questionsconcern this momentum shift operator:
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