We can create a wave function that represents an electron in a
small region of space. To do so we must add together many different
simple waves. Each simple wave has a different wavelength. Because
wavelength is related to momentum, each simple wave has a different
momentum.
Thus, we need many different momenta to create a wave function
for one electron. We interpret this result to mean that the electron
may have any one of the momenta. So, instead of having just one
momentum, these localized electrons have a probability of having
many different momenta. The probability of each momentum is related
to the amplitude of the wave function with that momentum (Figure
6).
Figure 6: The probability of each momentum depends on the amplitude on the amplitude-momentum graph.
We must make a similar statement about position. Even when we add lots of waves with lots of different momenta, we still get a wave function similar to Figure 10-7.
Figure 7: The relative probability of finding the electron at various locations for a localized wave function.
