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Wavefunction Collapse
Consider a spatially extended wavefunction,
. According to our
usual interpretation,
is proportional to the
probability of a measurement of the particle's position yielding
a value in the range
to
at time
. Thus, if the wavefunction is extended then there is a wide
range of likely values that such a measurement could give.
Suppose, however, that we make a measurement of the particle's position, and obtain the value
.
We now know that the particle is located at
.
If we make another measurement, immediately after the first one, then
what value would we expect to obtain? Common sense tells us that
we should obtain the same value,
, because the particle
cannot have shifted position appreciably in an infinitesimal time interval.
Thus, immediately after the first measurement, a measurement of
the particle's position is certain to give the value
, and has
no chance of giving any other value. This implies that the
wavefunction must have collapsed to some sort of ``spike'' function,
centered on
. This idea is illustrated in Figure 80.
As soon as the wavefunction collapses, it starts to
expand again, as described in the previous section. Thus, the second measurement
must be made reasonably quickly after the first one, otherwise the
same result will not necessarily be obtained.
Figure 80:
Collapse of the wavefunction upon measurement of
.

The preceding discussion illustrates an important point in wave
mechanics. That is, the wavefunction of a massive particle
changes discontinuously (in time) whenever a measurement of the particle's position is made. We conclude that there are two types of time
evolution of the wavefunction in wave mechanics. First, there is a smooth evolution that is governed
by Schrödinger's equation. This evolution takes place between measurements. Second, there is a discontinuous evolution that
takes place each time a measurement is made.
Next: Stationary States
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Previous: Heisenberg's Uncertainty Principle
Richard Fitzpatrick
20130408