PHY471 Fall 2000                                                 Quantum Physics I

Homework set 2                                 Due date: Monday, 9/13/2000





1.      [2pt] Series expansion

Ohanian writes on page 28 that the “factor  equals zero if n ≠ m, and it equals 1 if n=m.  State the assumptions under which this is true and then prove Ohanian’s statement.

2.      [1pt+2pt+2pt+3pt]      Dirac Delta “function”

Prove the following identities:

(i)                  d(-x) = d(x)

(ii)                x d(x-a) = a d(x-a)

(iii)               d(cx) = d(x)/|c| for c ≠ 0


3.      [5pt] Fourier series

Consider the function
In the interval -p/(2b) < x < p/(2b)) represent the function f as a fourier series and evaluate the coefficients.  Plot the fourier series of order n=1, n=2, n=3 (i.e. truncate the summation after the first, second and third terms, respectively).

4.      [5pt] Fourier transform

Find the fourier transform  of the same function f(x) considered in problem 3 and plot the fourier transform.  Show explicitly that