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08_751_hw02 - Physics 751 Homework#2 Due Friday 11:00 am...

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Physics 751 Homework #2 Due Friday September 12, 2008 11:00 am. 1. (a) Defining the delta function as the limit of a narrow Gaussian wave packet (see the web notes on Fourier Series, etc.) prove it has the following properties: ( ) 1, ( ) 0 for 0. x dx x x 1 ( ) ( ), ( ) ( ), | | x x ax x a ( ) ( ) ( ). a x x b dx a b (b) Suppose you define the delta function by: 0 ( ) lim ( ), where ( ) 0 for | | / 2, ( ) 1/ for | | / 2. L L L L x x x x L x L x L Does this function have all the above properties? 2. Use Mathematica , Maple or Integral Tables to find the integral of (sin x )/ x from 0 to and from 0 to infinity. Use your result to estimate the overshoot that appears in a Fourier series representation of a step function (Gibbs’ phenomenon). 3. Suppose at t = 0, a free particle of mass m , in one dimension, has a Gaussian wavefunction 2 2 1/4 2 / 2 1 ( , 0) . x x t e
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