Quantum Physics I PHY471, Fall 1999

Homework set 2

Due Monday, 9/13/1999

 

Please clearly state your assumptions, number the equations and indicate logical connections between different lines.

1.                  [2+2 pt] Schrödinger’s wave equation for the free particle

a)      What relationship must hold between w and k, for y(x,t)=Aei(kx-wt) to be a solution of Schrödinger’s wave equation for the free particle?

b)      Show that y(x,t)=Aei(kx-wt) is not square-integrable.

2.                   [2+2+1 pt] Fourier transform I

a)      Ohanian, #2.3, p. 54

b)      Plot the function y(x) and its Fourier transform over reasonable ranges.

c)      Show explicitly that Parseval’s theorem applies for these two functions.

3.                  [2+2 pt] Fourier transform II

a)      Ohanian, #2.4, p. 54

b)      Plot the function y(x) and its Fourier transform over reasonable ranges.

4.                  [2+1 pt] Ohanian, #2.5, p. 54

5.                  [1+1+1+1 pt] Dirac Delta “function”

Prove the following identities:

a)      d(-x) = d(x)

b)      x d(x-a) = a d(x-a)

c)      d(cx) = d(x)/|c| for c ≠ 0

d)      d(x-y) d(x-z) = d [d(y-z)]/dx