*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
(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

_{}

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:

_{}