The three-slit experiment
Jonah Thomas
Law_. He described the two-slit experiment simply.
It left me with a question that will take me a little time to describe.
Feynman described the situation roughly as follows (leaving out
everything I don't immediately need): You send electrons toward a
tungsten plate that has two holes in it. The probability that an
electron will pass through hole #1 is p1^2. The probability that an
electron will pass through hole #2 is p2^2. The probability that an
electron will get through the tungsten plate is (p1+p2)^2. But if you do
something to find out which hole the electron goes through, then the
probability instead drops to p1^2 + p2^2.
I know some about probability theory. If you start with a clear idea how
something is organised but you don't know some details, you can build a
probability model that fits your idea. But if you find out that the
reality fits your probability model that does not mean that the reality
is organised the way your idea was. It could be different and still fit
the model.
Still, I want to go backward that way and see how it comes out. To me,
the most obvious way to interpret the probability model that Feynman
describes is like this:
1. For an electron to get through the tungsten plate requires two
independent events, each of them with probability (p1+p2).
2. Each of the two events can "succeed" (where "succeed" means it gives
the result that allows the electron to pass through the plate) by one of
two mutually exclusive ways. One of those ways has probability p1 and
the other has probability p2. They can't both happen.
It's just as if the electron has two parts, and each part can go through
either hole independently but neither part can go through both holes. If
you do anything to "find out which hole the electron goes through" then
you eliminate the cases where the two parts go through different holes,
and leave only the cases where both parts go through the same hole.
This is a simple and obvious way to interpret the probabilities, but it
does not have to be right. I want to keep stressing that. But when I
assume it is right, then I find myself asking what happens if there are
three holes? Say there are three holes arranged in an equilateral
triangle.
o
o o
What happens then? One possible way it could happen is like this.
(p1 + p2 + p3)^2
Nine terms, and six of them are interference terms. The same 2:1 ratio
as the two-hole case.
Another way it could happen is like this.
k(p1 + p2 + p3)^3
where k is a fudge factor. If p1 and p2 and p3 are all around .001, then
why should the intensity drop by a thousand because you have an extra
hole? If k is somewhere around 1/(p1+p2+p3) then the result is more
reasonable.
Now there are 27 terms and 24 of them are interference terms. An 8:1
ratio.
The first way, the electron has two parts that each independently pass
through one of the three holes.
The second way, the electron could have three parts that each
independently pass through one of the three holes.
If it comes out the first way, it means that electrons intrinsicly have
two parts and only two parts that travel independently. But the second
way, the electrons could have as many parts as they like depending on
circumstance and we don't know as much about them as we do the first
way.
It might be possible to tell the difference between these two cases. Of
course, in the second approach if you get no recognizable result when
two parts of the electron go through one hole and the third part goes
through a different hole, then it goes back to a 2:1 ratio which is
harder to distinguish.
Does anybody know what happens in the three-hole experiment? Can we rule
out one of these probability distributions? Can we rule out both of
them?
Androcles
"Jonah Thomas" <jeth...@gmail.com> wrote in message
news:20091123024151.5...@gmail.com...
Yes. There are no three-hole experiments, and nobody ever diffracted
electrons at slits or holes. The nearest you'll get to that is the TV tube's
shadowmask. Electron diffraction is conducted using crystalline arrays
of molecules or atoms, so all your hard thinking about probabilities is
quite useless.
> Can we rule
> out one of these probability distributions? Yes.
> Can we rule out both of them? Yes.
Jonah Thomas
> "Jonah Thomas" <jeth...@gmail.com> wrote
You're calling Feynman a liar. He said you could do a two-hole
experiment with electrons and a tungsten plate with two holes in it.
"The source is a filament, the barriers tungsten plates, these are holes
in the tungsten plate, and for a detector we have any electrical system
which is sufficiently sensitive to pick up the charge of an electron
arriving with whatever energy the source has. If you prefer it, we
could use photons with black paper instead of the tungsten plate -- in
fact black paper is not very good because the fibres do not make sharp
holes, so we would have to have something better -- and for a detector a
photo-multiplier capable of detecting the individual photons arriving.
What happens with either case? I will discuss only the electron case,
since the case with photons is exactly the same." About halfway into
lecture 6.
Well OK, Feynman might very well be a liar. Are you certain he is?
dlzc
On Nov 23, 12:41 am, Jonah Thomas <jethom...@gmail.com> wrote:
...
> 1. For an electron to get through the tungsten
> plate requires two independent events, each of
> them with probability (p1+p2).
> 2. Each of the two events can "succeed" (where
> "succeed" means it gives the result that allows
> the electron to pass through the plate) by one
> of two mutually exclusive ways. One of those
> ways has probability p1 and the other has
> probability p2. They can't both happen.
Why not?
> It's just as if the electron has two parts,
> and each part can go through either hole
> independently but neither part can go through
> both holes.
Why not?
> If you do anything to "find out which hole the
> electron goes through" then you eliminate the
> cases where the two parts go through different
> holes, and leave only the cases where both
> parts go through the same hole.
Yes.
> This is a simple and obvious way to interpret
> the probabilities, but it does not have to be
> right. I want to keep stressing that.
Because people want to believe that electrons are billiard balls, and
slits / holes are well defined? Or that "models" are "reality"?
> But when I assume it is right, then I find
> myself asking what happens if there are
> three holes?
Or an infinite number of slits. Like a diffraction grating.
> Does anybody know what happens in the
> three-hole experiment? Can we rule out one
> of these probability distributions? Can we
> rule out both of them?
... first couple of hits with Google.
http://en.wikipedia.org/wiki/Low-energy_electron_diffraction
http://www.physics.pdx.edu/~pmoeck/pdf/ElectronJonsson.PDF
David A. Smith
Jonah Thomas
> Jonah Thomas <jethom...@gmail.com> wrote:
> ...
> > 1. For an electron to get through the tungsten
> > plate requires two independent events, each of
> > them with probability (p1+p2).
>
> > 2. Each of the two events can "succeed" (where
> > "succeed" means it gives the result that allows
> > the electron to pass through the plate) by one
> > of two mutually exclusive ways. One of those
> > ways has probability p1 and the other has
> > probability p2. They can't both happen.
>
> Why not?
Because that's how Feynman wrote the formula. (p1+p2) means mutually
exclusive.
If they can both happen at once then the probability of success is
(p1+p2- p2&p1)
That is, you get a success if p1 happens or if p2 happens, but now we've
double-counted the possibility that they both happen together. So we
subtract out one copy of the probability they both happen. If they're
completely independent then it turns easy, the chance they both happen
is p1*p2 and the formula is (p1+p2 - p1*p2). This is not the formula
Feynman used, he used (p1+p2).
So the formula is compatible with an event that can go one of two ways.
But of course, just because that's the natural interpretation of a
probability doesn't mean it has to be that way. It's just the only
obvious way to interpret it.
> > It's just as if the electron has two parts,
> > and each part can go through either hole
> > independently but neither part can go through
> > both holes.
>
> Why not?
(p1+p2) is an event that can go one way or the other.
(p1+p2)*(p1+p2) is two independent events that can each go one way or
the other.
Again, it's the only obvious interpretation, but it doesn't have to be
right.
> > If you do anything to "find out which hole the
> > electron goes through" then you eliminate the
> > cases where the two parts go through different
> > holes, and leave only the cases where both
> > parts go through the same hole.
>
> Yes.
>
> > This is a simple and obvious way to interpret
> > the probabilities, but it does not have to be
> > right. I want to keep stressing that.
>
> Because people want to believe that electrons are billiard balls, and
> slits / holes are well defined? Or that "models" are "reality"?
Because I don't want to claim my model is reality.
> > But when I assume it is right, then I find
> > myself asking what happens if there are
> > three holes?
>
> Or an infinite number of slits. Like a diffraction grating.
>
> > Does anybody know what happens in the
> > three-hole experiment? Can we rule out one
> > of these probability distributions? Can we
> > rule out both of them?
>
> ... first couple of hits with Google.
> http://en.wikipedia.org/wiki/Low-energy_electron_diffraction
This one is about reflecting low-energy electrons off a metal surface.
It is not about the electrons that go through the holes, though it might
say something interesting about the electrons that don't go through the
holes.
> http://www.physics.pdx.edu/~pmoeck/pdf/ElectronJonsson.PDF
This one gives a formula that looks like it's compatible with their
data. The only thing that changes with the number of slits N is
"sin^2[(N*pi*d/lambda)*sin(phi)]
That would tend to imply that each electron comes in two parts each of
which goes through one slit completely independent of which slit the
other one goes through. If there are five slits then one electron does
not go through three or four or five of them but only one or two of
them. It would tend to imply that this is a quality of the electron, and
not just a function of the number of slits.
But I can't be sure how much the formula was created on a priori grounds
and then the photographs were determined to fit the formula because they
didn't violate it so badly that the differences couldn't be explained
away. I might get a better sense of that with more study, or maybe not.
Thank you for the link!
Androcles
was describing how an experiment might be done. He even says
"if you prefer it" giving you a choice of electrons with a tungsten
plate or photons with black paper, and then whinges about the
sharpness of the fibrous edges of the paper. Since it is already
known that electron microscopes have a better resolution than
optical microscopes I fail to understand why he doesn't suggest
photons and razor blades -- even tungsten razor blades if the
material is that important. However, a suggestion -- even a poor
one -- is not a lie. Therefore I resent your ugly accusation.
YOU are calling Feynman a liar, not me. Are you certain he is?
dlzc
On Nov 23, 10:18 am, Jonah Thomas <jethom...@gmail.com> wrote:
...
> >http://www.physics.pdx.edu/~pmoeck/pdf/ElectronJonsson.PDF
>
> This one gives a formula that looks like it's
> compatible with their data. The only thing
> that changes with the number of slits N is
> "sin^2[(N*pi*d/lambda)*sin(phi)]
>
> That would tend to imply that each electron
> comes in two parts each of which goes through
> one slit completely independent of which
> slit the other one goes through. If there
> are five slits then one electron does not go
> through three or four or five of them but
> only one or two of them.
How would the formulation be different if it went through all n slots,
in some proportion?
> It would tend to imply that this is a quality
> of the electron, and not just a function of
> the number of slits.
How about a quality of the slits?
> But I can't be sure how much the formula
> was created on a priori grounds and then
> the photographs were determined to fit
> the formula because they didn't violate
> it so badly that the differences couldn't
> be explained away. I might get a better
> sense of that with more study, or maybe not.
>
> Thank you for the link!
You are welcome. Chances are, you can contact the authors directly,
and get answers to your questions...
David A. Smith
Jonah Thomas
> Jonah Thomas <jethom...@gmail.com> wrote:
> ...
> > >http://www.physics.pdx.edu/~pmoeck/pdf/ElectronJonsson.PDF
> >
> > This one gives a formula that looks like it's
> > compatible with their data. The only thing
> > that changes with the number of slits N is
> > "sin^2[(N*pi*d/lambda)*sin(phi)]
> >
> > That would tend to imply that each electron
> > comes in two parts each of which goes through
> > one slit completely independent of which
> > slit the other one goes through. If there
> > are five slits then one electron does not go
> > through three or four or five of them but
> > only one or two of them.
>
> How would the formulation be different if it went through all n slots,
> in some proportion?
If the electron split into N parts which each "chose" a slot
independently of the others, then it would have the form
"sin^N[ ....]
where the inside of the brackets would depend on things I don't have
thought out. When it's sin^2[ ....] then the natural interpretation is
that it's two parts, and not N parts.
> > It would tend to imply that this is a quality
> > of the electron, and not just a function of
> > the number of slits.
>
> How about a quality of the slits?
Maybe, in some non-obvious way. When it's one electron and two slits
then I naturally think the electron might be sometimes going through
both slits because there are two of them. When instead there are three
slits and the electron still goes through two of them and never three, I
naturally think it's the electron and not the slits. But that's just the
obvious approach, and I don't have a proof it's the only way that works.
Jonah Thomas
> Jonah Thomas <jethom...@gmail.com> wrote:
> ...
> > >http://www.physics.pdx.edu/~pmoeck/pdf/ElectronJonsson.PDF
> >
> > This one gives a formula that looks like it's
> > compatible with their data. The only thing
> > that changes with the number of slits N is
> > "sin^2[(N*pi*d/lambda)*sin(phi)]
> >
> > That would tend to imply that each electron
> > comes in two parts each of which goes through
> > one slit completely independent of which
> > slit the other one goes through. If there
> > are five slits then one electron does not go
> > through three or four or five of them but
> > only one or two of them.
http://en.wikipedia.org/wiki/Probability_amplitude
[sigh]
It turns out that what they are using is not a probability at all, but a
"probability amplitude". it's a complex number, and when you take the
norm of the square then you get the probability. So it didn't mean what
I thought in the first place. It isn't at all clear what it means.