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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 FINAL EXAM NAME: 1. Heisenberg Picture Consider a single-particle system described by the Hamiltonian H = i planckover2pi1 ( A A ) , where A = 1 2 parenleftbigg 1 X + i planckover2pi1 P parenrightbigg so that [ A,A ] = 1. (a) Derive the Heisenberg equations of motion for A H ( t ) and A H ( t ). (b) Solve these equations and give the solutions in terms of A S and A S . (c) From these solutions, express X H ( t ) and P H ( t ) in terms of X S and P S . (d) At t = 0 the wavefunction of the particle is known to be ( x, 0) = N e- (( x/ ) sin( k x )) 3 , where N is a normalization constant, and and k are arbitrary constants. What is the wavefunction at any later time t ? Hint: recall that T ( d ) = e- i planckover2pi1 dP . 2. Consider a pair of identical spin-1/2 particles in a uniform magnetic field. Neglect the motion of the particles, and consider only their spin degrees of freedom. The Hamiltonian is then H =...
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This note was uploaded on 12/02/2010 for the course PHYSICS 851 taught by Professor M.moore during the Fall '09 term at Michigan State University.
- Fall '09