You probably noticed from your sketches that the locations of highest probability became small. The range of locations with medium and low probability became greater. (See Figure 8).
We can understand this result by returning to the momenta involved. We start with a wave function as in Figure 8(a). The electron can have many different momenta. (Re-member we needed all those momenta to construct its wave function.) Of course, it also has a range of possible locations. So, we don't quite know where it is, and we don't quite know how fast it is moving - an uncertainty in position; an uncertainty in momentum. (See Figure 9).
A short time later we look to see where the electron is. We expect to still have an uncertainty in position because there was such an uncertainty when we started. In addition, the electron had an uncertainty in its momentum at the start. This uncertainty means that we do not know exactly how fast the electron is moving. The location of an object at a later time depends on both its starting point and its speed. In the case of the electron both the start and the momentum have uncertainties. As time passes the initial uncertainty in position is compounded by the uncertainty in momentum. So, as an electron moves the uncertainty in position increases.
This change in uncertainty is represented in the program Quantum Motion by a change in the wave function. As time passes, the wave function spreads out. This spreading indicates a wider range of positions is probable as the electron moves.
The rate of the uncertainty in positions depends on the uncertainty at the start. Two possible starting wave functions are shown in Figure 10. Wave function (a) is restricted to a very small region of space. Wave function (b) is not as restricted.
(a) (b)
Figure 10: Wave functions that (a) represent an object restricted to a very small region of space and (b) not very restricted.
Which one needs a larger number of momenta to create it? Refer to your experiences with the program Wave Packet Explorer to explain why.
Use your answer above to determine which one will change the range of possible locations more quickly.
A very small uncertainty at the beginning results in a very rapid change in the wave function. A greater uncertainty at the start means the change in uncertainty is not so great. How well we know the position at one location determines how well we can know it later.