Michael Fowler
University of Virginia
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Here we review three kinds of double slit experiments - classical particles, classical waves, and quantum things.
This lecture follows the very clear presentation in Feynman's Lectures on Physics.
First we consider a double slit experiment with bullets.
The source is a not very accurate machine gun, which sprays bullets
towards two parallel slits, 1 and 2, in a sheet of armorplate,
which may be covered one at a time with another piece of armor.
The bullets are "detected" by plowing into a soft (but
thick) wood screen some distance beyond the slits. We assume the
bullets don't break up, you always find whole bullets in the wood.
If we now cover slit 2, and fire a lot of bullets, by examining
the screen at the end we will see where the bullets went, and
be able to construct a probability distribution P1(x)dx
giving the probability that a bullet going through slit 1 lands
in the interval x, x + dx. Similarly, covering
slit 1, we can find the probability distribution P2(x)
for bullets going through slit 2. Now suppose we open both slits
(but don't fire so many bullets that they collide with each other
in the air after bouncing through the slits!). Then we will find
a probability distribution P12(x) which
is simply the sum:
This of course follows from the self-evident fact that any bullet
getting to the screen must have gone either through slit 1 or
slit 2. The probabilities add.
There is one other important point about bullets -- they are lumpy. You either find one or you don't. This is also true of photons and electrons.
Now think about a double slit experiment with waves, say, surface waves on water-the standard ripple tank experiment. Assume that plane waves of definite wavelength impinge from the left on a barrier having two slits S1 and S2, with S2 being initially closed. Then to the right of the barrier, waves emanate from the open slit with circular wavefronts. These waves are detected at a "screen" over to the right. The waves could be detected by having corks bobbing up and down in the water just in front of the screen, and perhaps we should have very small corks and an imaginary screen to get rid of waves being reflected backwards! We could at some instant in time measure the height distribution h1(x) of the corks from waves coming through slit 1 (slit 2 being closed). Of course, this height distribution is changing all the time, and is not really what interests us. What we really want to know is the intensity of the wave motion at each point on the "screen" (analogous to the brightness of light, for example). In any simple harmonic motion, the intensity is proportional to the square of the amplitude. For our surface waves on water, the amplitude is of course just the height of the wave (relative to still water), so the energy intensity I1(x) for waves coming through slit 1 has the form
What happens when both slits are open? It is well known that waves
superpose linearly -- that is, the heights add.
If at some instant of time with both slits open the height distribution
of the corks is given by h12(x), then
In this harmonic motion, the corks bob downwards as much as upwards,
so h1 and h2 are as likely
to cancel as they are to reinforce each other.
The energy intensity when both slits are open,
The crucial point here is that in contrast to the bullets, the
intensity when both slits are open is not the sum of the
separate intensities for the slits being open one at a time. This
is just the familiar wave interference effect.
Notice that in this classical wave picture, there is no lumpiness. The energy of the wave at the screen varies smoothly, and can have any value we choose.
Historically, an experiment equivalent to the double slit experiment for light was what convinced people it was indeed a wave. The light reaching a screen exhibited the identical intensity pattern observed for water waves. This was fully explained with the advent of Maxwell's equations: the light was simply an electromagnetic wave, the amplitude of the electric field being analogous to the height of the water wave in the discussion above. To find the total electric field E12 at any point on the screen, the electric field E1 of the wave emanating from S1 is added to E2 from S2. The intensity of light I12 reaching the screen is proportional to E122. Just as for the water waves, E1 and E2 oscillate about mean values of zero, so the analogy is complete.
The discovery that light is after all lumpy on a fine enough scale throws this picture into confusion. Experimentally, light of frequency f can only be detected in lumps of size hf.
To the classically trained mind, this naturally suggests that
the source of light is actually sending out these lumps, and each
lump presumably passes through one of the slits on its way to
the screen. However, this naïve picture leads to a contradiction
-- how could it generate the observed diffraction pattern? Experimentally,
the diffraction pattern builds up even if the light is so dim
that there is a time interval between each photon (lump) passing
through the apparatus. The picture of each photon going through
one of the slits cannot be correct, because the distance between
the slits governs the brightness pattern on the screen, and thus
the probability pattern of where the photons land.
The only known resolution to this conceptual problem is to give up trying to visualize the path of the photon, just solve Maxwell's equation for the wave propagation, then interpret the resulting wave intensity as giving the relative probability of detecting the photon at any point. This is how quantum mechanics is done. It does not correspond to a physical picture readily interpretable in terms of familiar concepts. However, it does accord well with what is observed in nature!
Electrons behave in the same way as photons -- there is little
to add to the description in the preceding section. The irony
is, of course, that before quantum mechanics electrons were seen
as bullets, light as waves, and it has turned out that they are
both basic quantum particles, with their propagation governed
by wave equations.