## Continuum Wave Packets

**Now we add a continuum of plane waves with different wave numbers to make a
localized free particle. **

Matter waves follow a quadratic dispersion:

**E = k**^{2}/2m = hw/2pi.

In the applet below, at t=0, all of the components are at zero phase and, thus, colored blue.
The area you see in the applet is all of space as far as the particle is
concerned. In other words, when the wave function spreads to the edges of
the applet, the wave function is reflected. The beautifully colored
patterns you will see as time progresses are due to self-interference.

Play the applet. Pause and reset the applet. Click one step forward and note the change in color of the wings of the
particle. Since the high momentum components travel faster than those with
low momentum, their relative phase changes more rapidly. These high
momentum components also have a relatively small amplitude.

**Student Exercises:**

- Pause and reset the applet. What is the Full Width at Half of the Maximum (FWHM)
at t = 0. Also measure it for t = 0.5, 1.0, 1.5, 2.0, 2.5 and
3.0. Plot these values. Is there a range for which the relation looks
linear?

Reset and Run the physlet while observing the rate of spread of the
particle.

- Click here to initialize Exercise 2. What
is the FWHM at t = 0? Run the physlet while observing the rate of
spread of the particle.
- Click here to initialize Exercise 3. What
is the FWHM at t = 0? Run the physlet while observing the rate of
spread of the particle.
- What generalization can you make about the rate of spread and the initial
FWHM? (You can view the original wave packet by clicking here.)

**© 2000 by Prentice-Hall, Inc. A Pearson Company**