2008final - PHYS851 Quantum Mechanics I Fall 2008 FINAL...

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PHYS851 Quantum Mechanics I, Fall 2008 FINAL EXAM NAME: 1. Heisenberg Picture Consider a single-particle system described by the Hamiltonian H = i planckover2pi1 χ ( A A ) , where A = 1 2 parenleftbigg 1 λ X + i λ 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 ψ ( x, 0) = N e - (( x/σ ) sin( k 0 x )) 3 , where N is a normalization constant, and σ and k 0 are arbitrary constants. What is the wavefunction at any later time t ? Hint: recall that T ( d ) = e - i planckover2pi1 d P .
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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 = γ parenleftBig vector S 1 · vector B + vector S 2 · vector B parenrightBig , where vector
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  • Fall '09
  • M.Moore
  • mechanics, Fundamental physics concepts, Jx, uniform magnetic field

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