# Quantum Physics I PHY471, Fall
1999

# Homework set
3

# Due Monday,
9/20/1999

#

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

## 1.
[2+2 pt] Fourier
transform III

a)
Find the Fourier transform of the Dirac Delta “function”.

b)
Find the Fourier transform of the function *f(x) = C*, where *C* is a constant for all values of *x*.

## 2.
[1+1+1+1+3+3pt] Wave
packets

Consider a wave packet of the form

a) What
is the dispersion relation for this wave packet?

b)
What is the phase velocity of this wave packet?

c)
What is the group velocity for this wave packet?

d)
What is the classical velocity of the particle? Compare to the group and phase velocities.

e)
Now assume that the dispersion relation is a general function Expand this function
around *k*_{0} in a Taylor
series and ignore term of second order and higher. Show that the group velocity is , i.e. that the probability of the wave packet can be written
as .

f)
For a relativistic particle with , find the group velocity and compare with the particle
velocity.

## 3.
[4+2 pt] Gaussian
wavepacket

Show how equation (74) in Ohanian chapter 2 follows from equation
(73) in Ohanian chapter 2.

a) Use
an expression for from your integral table
and show that

b) Now
evaluate .