200 The Poetry of Physics and The Physics of Poetry
is basically a wave equation. The diffraction pattern observed when a
beam of electrons passes through two slits is due to the interference of
the wave function passing through the two slits with itself. It is not due
to the interference of a beam of electrons passing through one slit with
the beam of electrons passing through the other slit. This has been
experimentally demonstrated by reducing the flux of electrons so that
only one electron at a time passed through the two-slit system. The
position on the screen where the electron landed after passing through
the slits is recorded and tabulated. After a sufficient time passes the
pattern that emerged was the same characteristic diffraction pattern one
obtained with a high flux beam of electrons. There were positions on the
screen where electrons would go only if both slits were open and would
not go it slit 1 was open and slit 2 was closed or if slit 1 was closed and
slit 2 was open. There were also positions where the electrons would not
go if both slits were open but would go if one or the other slit was open.
This is extremely mysterious. In the latter case I open slit 1 and close
slit 2 and observe electrons at position Y on the screen. If I now open
both slit 1 and slit 2, electrons no longer go to position Y. It is impossible
to understand in terms of a particle how the opening of slit 2 suddenly
prevents electrons from going to position Y via slit 1.
In view of the fact the results hold even with one electron at a time
the only way of interpreting this is to assume that the electron passes
through both slits and interferes with itself. But how is this possible if the
dimensions of the electron are smaller than the distance between the two
slits? The way this phenomenon is understood in terms of our quantum
mechanical description is to recognize that the probability amplitude, ψ,
describing the electron is interfering with itself.
Let us call the wave function at slits 1 and 2, ψ 1 and ψ 2 , respectively.
If only slit 1 is open then one obtains the pattern given by |ψ 1 |^2 and
if slit 2 is only open one obtains the distribution given by |ψ 2 |^2 as is
shown in Fig. 20.2(a). If both slits are open then the probability of
finding the electron is given by |(ψ 1 + ψ 2 )|^2 = |ψ 1 |^2 + ψ 1 ψ 2 + ψ 1 ψ 2 + |ψ 2 |^2.
This distribution is shown in Fig. 20.2(b), and is not simply the
sum of patterns due to slit 1 and slit 2 being open separately. The reason
it is not simply the sum is because of the interference of the wave
functions at slit 1 and 2, namely, ψ 1 and ψ 2. The square of the sum ψ 1 + ψ 2
contains more than just the terms |ψ 1 |^2 and |ψ 2 |^2 but also the interference
terms ψ 1 ψ 2 + ψ 1 ψ 2 , which are responsible for the diffraction pattern.
It is the presence of these terms that explain how it is possible that
certain positions on the screen are struck by electrons with only one