Re: Pedagogy of QM Double Slit Experiment



David Park wrote: 
Dear Experts,

I believe that the standard textbook presentations of the quantum mechanical
double slit experiment are often poor. They throw in complicating elements
that tend to confuse beginning students and obscure, at least in the
student's mind, the essential phenomenon. I would be interested in your
comments to make certain that I haven't missed the essential points myself.

1) Why 'double slit' instead of 'double pinhole'? 

No significant difference.

2) Doesn't a single pinhole, or single slit, experiment already show that
there is an essential quantum mechanical randomness that is introduced, and
that this randomness has a wavelike behavior?

Yes, you are right. A particle (electrons or photons) beam passing
through a single pinhole or a slit
forms a wide diffraction spot on the screen.
However, when the intensity of
the beam is very low, you can see that
the diffraction spot consists of a large number of small dots made by individual particles. This clearly demonstrates the main "paradox" of
quantum mechanics: one cannot predict where each individual particle
will hit the screen.

3) What exactly does a double slit experiment add to our knowledge that is
not already contained in the single slit experiment? (Is it just that it is
experimentally easier?)


Single-slit experiment illustrates the idea of diffraction (spreading of
the wave packet). Double-slit experiment illustrates another important
idea - the interference. Instead of summing up the probabilities of two
independent process we must sum their amplitudes. The diffraction spot
made by two pinholes is not just a superposition of two individual
diffraction spots.

You may find discussion of single-slit and double-slit experiments in
Chapter 3 of my book "Relativistic quantum dynamics"
http://arxiv.org/abs/physics/0504062



In the following, I will be thinking of a presentation such as Figure 21.4
(§21.4, page 504) in Penrose's 'Road to Reality' that shows, in 3D
perspective, an electron gun aimed at a screen with two vertical slits, and
a detection screen behind. I believe his is a pretty standard picture and
discussion. And in all of this we can assume the one particle at a time
case, since the experiment has actually been done.

Penrose first discusses the case when one slit is covered. We get a smeared
out distribution. Penrose writes: "No puzzle here." But I think that to the
beginning student there are a lot of puzzles! Classically, if the electron
gun was perfectly fixed and aimed at the center of the slit, then wouldn't
we get simply a single spot on the screen where all the electrons hit?
Instead there is a distribution of hits that is spread out both horizontally
AND vertically. Why does the distribution get spread out vertically? Perhaps
the gun is a long way away, the electron is approximated by a plane wave
function that would more than encompass the length of the slits, so it would
behave just like an optical slit case. But the student doesn't know anything
about that yet and this vertical spreading becomes a source of confusion.

So wouldn't it be better to begin with a pinhole, in which classically the
electrons were aimed right through the center of the pinhole? Classically,
they would all hit at one spot on the screen. But actually we obtain a
circular distribution of hits. Doesn't this already show that QM introduces
an essential randomness that has wavelike behavior?  We can see some of this
wavelike behavior by how the amount of spreading depends on the momentum of
the electrons.


Exactly.


(In the Feynman Lectures on Physics, Volume III, Feynman introduces the
analogy of a machine gun spraying bullets in all directions and supposedly
the single slit horizontal spreading is due to the random jerking around of
the gun, and maybe the ricocheting of bullets off the edge of the slit. So
we might also think of the electron gun having random classical motions that
spread the electrons around and shoots them in different directions. But
isn't this just introducing a confusing and complicated classical effect
that hides a true quantum mechanical effect?)


Exactly.


So I think it is misleading to suggest that there is nothing unusual about
the single slit, or single pinhole, experiment. I think it would be better
to give a careful discussion of the single pinhole case before going to the
double slit experiment. But the discussions should definitely be coupled
together.

If we do a double pinhole experiment, then we would obtain a complicated
2-dimensional interference pattern so perhaps going to the double slit
experiment where the phenomenon is essentially one dimensional is better.
But this switch should be justified to the student. (Or could we have a
vertical array of electron guns to match the vertical slits?)

What is it that the double slit case actually adds? Here I am not completely
certain and so am looking for clarification. With the single pinhole
experiment we might not be certain that the electrons do not have unique
trajectories. Couldn't all the electrons that hit a given spot have followed
a definite trajectory? So we might try to describe it by random but definite
trajectories.  (On the other hand, wouldn't the diffraction itself imply
that the electron had sampled all parts of the pinhole and therefore could
not have a definite trajectory?) The double slit case, with the clear
physical separation, and the clear change when the second slit is opened
shows that the electron must know about both slits and therefore can't be
given a definite trajectory. Also the interference pattern with its null
points is a more dramatic demonstration of wavelike character. But is this
just a clearer indication or is there something essential about the double
slit case that is not contained in the single pinhole case?

Great questions! In my view, the great enigma of QM is already demonstrated in the single pinhole experiment. This experiment alone
demands a radical change from classical determinism to quantum probabilities. The most consistent way (known to me) from classical
physics to QM is through
"quantum logic" (see Chapter 4 of my book). This way leads directly to
the Hilbert space description of quantum systems and to the "additivity
of amplitudes". This (rather technical) property of quantum theory
is demonstrated by the double-hole experiment.

Eugene.


.

Relevant Pages

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