But what corresponds to the
amplitude
or
displacement
of a matter wave? The ampli-
tude of an electromagnetic wave is represented by the electric and magnetic fields,
E
and
B
. In quantum mechanics, this role is played by the
wave function
, which
is given the symbol
(the Greek capital letter psi, pronounced “sigh”). Thus
represents the wave displacement, as a function of time and position, of a new
kind of field which we might call a “matter” field or a matter wave.
To understand how to interpret the wave function
we make an analogy
with light using the wave
–
particle duality.
We saw in Chapter 11 that the intensity
I
of any wave is proportional to the
square of the amplitude. This holds true for light waves as well, as we saw in
Chapter 22. That is,
where
E
is the electric field strength. From the
particle
point of view, the intensity
of a light beam (of given frequency) is proportional to the number of photons,
N
,
that pass through a given area per unit time. The more photons there are, the
greater the intensity. Thus
This proportion can be turned around so that we have
That is, the number of photons (striking a page of this book, say) is proportional
to the square of the electric field strength.
If the light beam is very weak, only a few photons will be involved. Indeed, it
is possible to “build up” a photograph in a camera using very weak light so the
effect of photons arriving can be seen. If we are dealing with only one photon,
the relationship above
can be interpreted in a slightly different way. At
any point, the square of the electric field strength
is a measure of the
probability
that a photon will be at that location. At points where
is large, there is a high
probability the photon will be there; where
is small, the probability is low.
We can interpret matter waves in the same way, as was first suggested by
Max Born (1882
–
1970) in 1927. The wave function
may vary in magnitude
from point to point in space and time. If
describes a collection of many electrons,
then
at any point will be proportional to the number of electrons expected to
be found at that point. When dealing with small numbers of electrons we can’t
make very exact predictions, so
takes on the character of a probability. If
which depends on time and position, represents a single electron (say, in an atom),
then
is interpreted like this:
at a certain point in space and time represents
the probability of finding the electron at the gi
v
en position and time.
Thus
is
often referred to as the
probability density
or
probability distribution
.
Double-Slit Interference Experiment for Electrons
To understand this better, we take as a thought experiment the familiar double-slit
experiment, and consider it both for light and for electrons.
Consider two slits whose size and separation are on the order of the wave-
length of whatever we direct at them, either light or electrons, Fig. 28
–
3. We
know very well what would happen in this case for light, since this is just Young’s
double-slit experiment (Section 24
–

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