## Complex Plane Waves (deBroglie Waves)

In Section 1a it was shown that a complex number with a
time-dependent phase could be represented by a rotating vector in the complex
plane. We will see in this section that a complex plane wave can be
represented by a phase that is time- and spatially-dependent.

The equation for a complex plane wave of constant amplitude and
traveling toward increasing x is of the form

Z(x,t) = |Z| exp( i (k x - w
t)

The applet below shows a complex plane wave moving in the
+x-direction. For a given x, the amplitude of the wave is constant in the
y-direction. The phase of the wave at any point along the x-axis is
represented by the color. For instance, all bands that are the same shade
of blue have the
same phase. The phase at the cursor can be more accurately read by
left-clicking in the applet.

Show Color Phase
Show Re Im

**Section 1b Exercises:**

- What is the wavelength of this wave?
- Does this wave have a single value for the wave number, k? If so,
what is it?
- Does this wave demonstrate localized behavior? In other words, would
a particle represented by this wave be more likely to be found in one place
over another?
- Follow a point of constant phase and determine the "phase"
velocity.

Click on the ** Show Re Im** link to view the real and imaginary parts of the
wave.

- What is the phase relation between the real and imaginary parts?

Perhaps the simplest solution to the Schroedinger equation is
the plane wave solution. These plane waves, known as "deBroglie
waves", however do not give a physical solution. They cannot be
normalized since they do not vanish at ±infinity
but continue to oscillate. Particle solutions, such as electrons and
protons, to the Schroedinger equation must be localized and able to be
normalized. The remainder of the sections help us examine the properties
of localized waves.

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