851HW6_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 6 1. [10 points] The quantum state of a free-particle of mass, M , at time t is a wave-packet of the form ψ ( x,t ) = 1 radicalbig Γ(5 / 4) σ e- ( x- x ) 4 2 σ 4 + ip x/ planckover2pi1 , We can safely predict that the width of the wave packet will grow in time. Clearly, the spreading velocity v s must be determined by the initial conditions x , p , and σ , as well as the mass, M , and Planck’s constant, planckover2pi1 , as there are no other parameters around to use. (a) [2 pts] If there are no external forces acting on the particle, which parameters can we rule out based on symmetry arguments ? (b) [2 pts] Of the remaining parameters, how many unique ways are there to combine them to make an object with units of a velocity? (c) [2 pts] Based on this result alone, give a units-based estimate for the velocity at which the wave-packet should spread....
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851HW6_09 - PHYS851 Quantum Mechanics I Fall 2009 HOMEWORK...

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