# HW8 - PHYS851 Quantum Mechanics I Fall 2008 HOMEWORK...

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PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 8: 1. [20pts] Evolution of a gaussian wavepacket: Consider the time dependent Gaussian wavefunction ψ ( x,t ) = ( x | ψ ( t ) ) = bracketleftbigg π parenleftbigg σ + i planckover2pi1 t parenrightbiggbracketrightbigg - 1 / 2 exp bracketleftbigg ip 0 planckover2pi1 parenleftbigg x p 0 t 2 m parenrightbiggbracketrightbigg exp bracketleftbigg ( x p 0 t/m ) 2 2 σ 2 (1 + i planckover2pi1 t/ ( 2 )) bracketrightbigg . Show by direct differentiation that this satisfies Schrodinger’s equation for a free particle: i planckover2pi1 d dt ψ ( x,t ) = planckover2pi1 2 2 m d 2 dx 2 ψ ( x,t ) Then compute | ψ ( x,t ) | 2 and show that it is a normalized gaussian | ψ ( x,t ) | 2 = 1 πσ ( t ) e - ( x - x 0 ( t )) 2 σ 2 ( t ) What are the center, x 0 ( t ), and width, σ ( t ), as functions of time? 2. [10]Consider a baseball with a mass of .14 kg. Assume that it is localized to a size of 10 - 7 m. Assume that a pitcher throws a fast ball and it is clocked at 96 miles per hour. What does the Heisenberg uncertainty relation tell us that the fundamental quantum limit to the accuracy of this measurement
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