# 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:

(b)  Suppose you define the delta function by:

Does this function have all the above properties?

2. Use Mathematica, Maple or Integral Tables to find the integral of (sinx)/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

By taking a Fourier transform and putting in the explicit time-dependence for each plane wave component, find the form of the wavefunction as a function of time, and provide a physical interpretation in terms of finding the particle somewhere.

4. Denoting the lowest energy eigenstate in an infinite square well by , and the first excited state by , describe the behavior of the probability distribution as a function of time for the state   (appropriately normalized).

Find the probability current at the midpoint of the well as a function of time.

How would your analysis be different for the state ?

5.  Prove Parseval’s Theorem:

6.  Prove the rule for the Fourier Transform of a convolution of two functions: