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Unformatted text preview: MIDTERM EXAM 2 PHYS 851 Quantum Mechanics I, Fall 2007 1. A system is described by the Hamiltonian H = planckover2pi1 2 2 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 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 = 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?...
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.
- Fall '08