*Michael Fowler,
University of Virginia*

Here we review three kinds
of double slit experiments—*classical particles*, *classical waves*, and *quantum
things*.

The 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, ${S}_{1}$ and ${S}_{2},$ in a
sheet of armorplate, the slits may be covered one at a time with another piece
of armor. The bullets are “detected” by letting them plow 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 ${S}_{2},$ 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 ${P}_{1}\left(x\right)dx$ giving the probability that a bullet going through slit ${S}_{1}$ lands in the interval $x,x+dx.$ Similarly, covering slit ${S}_{1},$ we can find the probability distribution ${P}_{2}\left(x\right)$ for bullets going through slit ${S}_{2}.$ Now suppose we open both slits—this gives a probability distribution ${P}_{12}\left(x\right)$ and it is clear that this will be simply the sum:

${P}_{12}\left(x\right)={P}_{1}\left(x\right)+{P}_{2}\left(x\right).$ .

This of course follows
from the self-evident fact that any bullet getting to the screen must have gone
either through ${S}_{1}$ or ${S}_{2}.$ *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 ${S}_{1}$ and ${S}_{2},$ with ${S}_{2}$ 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. We could at
some instant in time measure the height distribution ${h}_{1}\left(x\right)$ of the corks from waves coming through ${S}_{1},$ ${S}_{2}$ 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 ${I}_{1}\left(x\right)$ for waves coming through ${S}_{1}$ has the
form

${I}_{1}\left(x\right)=A\overline{{h}_{1}^{2}\left(x\right)}.$ .

What happens when *both*
slits are open? Check out this animation!

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 ${h}_{12}\left(x\right),$ then

${h}_{12}\left(x\right)={h}_{1}\left(x\right)+{h}_{2}\left(x\right).$

In this harmonic motion, the corks bob downwards as much as upwards, so ${h}_{1}$ and ${h}_{2}$ are as likely to cancel as they are to reinforce each other.

The energy intensity when both slits are open,

${I}_{12}\left(x\right)=A\overline{{h}_{12}^{2}\left(x\right)}={I}_{1}\left(x\right)+{I}_{2}\left(x\right)+2A\overline{{h}_{1}\left(x\right){h}_{2}\left(x\right)}.$

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 lapping against 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 ${E}_{12}$ at any point on the screen, the electric field ${E}_{1}$ of the wave emanating from ${S}_{1}$ is added to ${E}_{2}$ from ${S}_{2}.$ The intensity of light ${I}_{12}$ reaching the screen is proportional to $\overline{{E}_{12}^{2}}.$ Just as for the water waves, ${E}_{1}$ and ${E}_{2}$ 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:
but when they arrive, they act like *particles*!