Physics 751 Homework #2

Due Friday September 19, 11:00 am.

1. Defining the delta function as the limit of a narrow Gaussian wavepacket, prove it has the following properties:

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 p 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 explicit time-dependence, 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 |0>, and the first excited state by |1>, describe the behavior of the probability distribution s a function of time for the state |0> + |1> (appropriately normalized). Find the expectation value of the probability current at the midpoint of the well as a function of time. How would your analysis be different for the state |0> + i|1>?