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Quantum Weirdness I:
The Double-Slit Experiment
This is the classic illustration of quantum wave-particle
duality, but it is also an good springboard for considering
the "collapse of the wave packet", nonlocality, and other
expressions of quantum wierdness.
Quantum systems often display both wave-like and
particle-like behaviors. To show exactly what this means,
let's consider an experiment performed with classical
particles (pellets), classical waves (water waves), and
quanta (electrons).
Experimental Set-up
The experiment consists of a source (a pellet launcher, a
paddle for producing water waves, or a hot filament for
producing electrons), a screen with two small slits in it,
and a detector at some distance from the source and screen.
The source is set up so that its intensity can be controlled
by some kind of "dimmer switch", making the rate of pellet
firing or the size of the water waves as small as desired.
The detector has construction appropriate for measuring the
energy conveyed to each point on the detector's surface. For
the pellets, you could imagine a flexible pad that is
permanently deformed when a pellet strikes it; for the water
waves, it might be a row of small floaters that can move up
and down with the water surface, and whose motion is
measured electronically. For the electrons, it might be a
photographic film that is exposed whenever it interacts with
an electron.
 What
Happens with Classical Particles
Most of the pellets launched from the source will be
stopped by the screen, but some will go through the slits
and eventually strike the detector. Most of these will
follow a more-or-less straight path through one of the
slits, and strike the detector at a spot opposite the slit
it went through. Some will be deflected slightly by grazing
collisions with the edge of the slit. Pellets going though
one slit will be completely unaffected by pellets going
through the other slit. So for any spot on the detector, the
number of pellets arriving is just the sum of those coming
by way of the first slit and those coming by way of the
second slit; and the energy deposited at each point on the
detector is proportional to the number of pellets arriving
there.
If we turn down the intensity of the source, the pellets
arrive at the detector very infrequently. If only one or two
arrive, the places where they hit the detector may seem
totally random. But if we wait long enough, the pattern
produced will show the predictable double peak at the spots
opposite the slits.
 What
Happens with Classical Waves
The water waves strike the screen, and are mostly
reflected back or scattered. The waves that go through the
slits, though, spread out in circular patterns centered on
the slits. The slits act like small "point sources" of
new water waves. As the waves from the two slits spread
outward, they overlap and interfere with other. In some
places, a crest from one wave meets a trough from the other,
and they cancel each other out, leaving the water
essentially still there. In other places, crest meets crest
and trough meets trough, so the motion of the water is
amplified. At the detector, the floaters that are located
where the two waves cancel each other will not move at all,
whereas those located where the waves reinforce each other
will show amplified motion. So a plot of the energy
transmitted to each spot on the detector shows an
interference pattern, an array of alternating highs
and lows.
This interference pattern is not the sum of the
pattern produced by each of the two slits separately. If one
slit is closed off, all the waves come from the other slit,
and they just spread out in a uniform circular pattern
without any interference. The detector pattern made by a
single slit is just a very broad peak opposite the open
slit, with none of the alternating highs and lows that are
seen when both slits are open.
If we turn down the intensity of the wave source, the
pattern seen on the detector does not change its shape, it
just gets weaker.
 What
Happens with Quanta
The electrons strike the detector and expose the
photographic film. The film will be strongly darkened where
it has received large amounts of energy from the electons,
and unexposed where it has received none. When this
experiment is done, the film records an interference
pattern, just like that seen with classical waves. And,
again as with classical waves, the interference pattern goes
away if only one slit is left open. If this were the end of
the story, we would conclude that electrons behave like
classical waves.
The surprise comes when the intensity of the source is
turned down. Now, instead of seeing an interference pattern,
we see just a few spots on the film, much like we would
expect from classical particles. Instead of the electron
energy being spread out over the entire film to make a faint
but complete interference pattern (which is what classical
waves would do), we see that the electrons interact with the
film at discrete sites. If we wait long enough, these
individual points will build up the interference pattern
seen when the intensity is high, just as the pellet
experiment builds up its own double-peak pattern over
time.
Some Lingo
Quantum mechanics has developed a generalized way of
talking about systems and their states, which will be
helpful later on as we extend the double-slit example to
other cases.
The classical particles in the first example are said to
be in a mixture state; this means that there are some
of them have gone through slit one and some have gone
through slit two. The particles arriving at the detector
include both. If you make measurements on a mixture, it is
the same as making measurements on both parts of the mixture
separately and then just adding the results together. The
parts of a mixture are independent; they don't interfere
with each other. [Note: Even a single particle can be in
a mixture state, meaning just that we don't happen to know
which of the slits it passed through.]
The classical waves in the second example are said to be
in a superposition state. This means that you must
combine the two waves coming through the two slits, taking
their interference into account. Then, after combining them,
you can use the total combined wave to predict things like
how the energy will be deposited to the detector. In a
superposition, the measurement results will generally
not be the same as if you take the measurement
results from the two parts and just add them together. You
have to take the interference into account.
A peculiar thing about quantum systems is that although
we often need to treat them as superpositions in order to
explain the experimental results, we do not actually
observe the superposition state. With classical
waves, we can see the total wave pattern before the waves
strike the detector. With the electrons, there is no
observation until they strike the detector, at which point
they appear to be localized particles, not an extended
superposition wave. If we try to observe the electrons
before they strike the detector (say, by placing small
detectors near each slit), we still do not observe the
superposition. We observe a mixture instead--some particles
observed going through slit one, others observed going
through slit two. After those observations are made, the
electrons continue to behave like a mixture, not like a
superposition, so the interference patten on the detector is
no longer seen; it is replaced by the double-peak pattern
consistent with the observation of individual particles,
some going through each slit.
The Interpretation Game
Does this observed wave-particle duality have a
straight-forward interpretation?
Perhaps electrons are really waves, with their substance
extended out through space, and some physical process cause
them to "scrunch up" when they are detected. It is this
image that is conjured up by the phrase collapse of the
wave packet to describe what happens when a quantum
system is subjected to a measurement operation. In this
interpretation, the superposition is a physically real
thing, just like a water wave. The problems with such an
interpretation are severe, though, and I don't know of any
physicists who seriously entertained it after alternatives
became avaible. One problem is that a superposition can be
enormous. We can detect individual photons from galaxies at
the other side of the universe. Until the detection happens,
those photons are part of an enormous spherical wave
billions of light years in diameter. It violates relativity
(and boggles the imagination), to think that
"pieces" of the photon separated by such enormous
distances could instantaneously be summoned to a single
point just because we happen to have a telescope sitting
there. And, of course, there is no conceivable mechanism for
peforming such an instantaneous condensation.
Perhaps electrons are really particles, and the wave
function is some other kind of phenomenon or field that
navigates the particle toward spots where the wave amplitude
is high. This interpretation was developed and publicized by
David Bohm, and has some faithful adherents. If it were
true, one might hope for some way to observe the wave
function and the particle independently, but this does not
appear to be possible. So this interpretation is open to
criticism by Occam's razor, since it postulates an
additional physical entity that is unobservable and has no
effect on what we do observe. (It was this sort of argument
that led to the rejection of idea of the lumiferous ether, a
postulated medium for the transmition of light.)
Furthermore, this interpretation must also violate
relativity theory, since it requires particles to move at
infinite speeds when crossing the "nulls" of the
interference pattern, where theory and observation say they
can never be detected.
The interpretation that most physicists have found most
satisfactory is the probability interpretation of the
wave function, proposed by Max Born and taken up by Bohr,
Heisenberg, and Schroedinger as a key component of the
Copenhagen Interpretation of quantum mechanics. In this
interpretation, the wave function is regarded as a
mathematical object that allows us to calculate the
probability that the quantum will interact at a
certain place or in a certain way. In this interpretation,
it is useless (or at least premature) to speculate on what
physical processes describe the motion or behavior of a
particle while it is in a superposition state, unless there
is some way to make direct observations on that state.
Instead, the interactions that constitute our measurement
observations, with their inherent randomness, are seen as
fundamental. The mathematical idea of a superposition allows
us to accurately calculate the probabilities of the various
possible measurement results; it is a human mental
construction, not a physical object.
The Copenhagen Interpretation thus refuses to answer the
question "what is the electron really doing before it
is detected?" In this view, the first point at which it
becomes possible to talk about the electron really
doing anything at all is when it is detected. It is
worth noting that there is no "quantum weirdness" in
the observations themselves (spots on a photographic film);
the weirdness comes when we try to tell a story of how the
spots got there in terms of things we are familiar with,
like pellets and water waves.
Go to Quantum Weirdness
II: Schroedinger's Cat
Return to Quantum Weirdness entry
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