Formal demonstration that classical paths won't interfere

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polymer
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Formal demonstration that classical paths won't interfere

Quantum Mechanics is frequently motivated by observing that classical path's can't produce interference patterns. I realized that this statement is quite precise, and that, I've never seen a formal demonstration of this. Can somebody give, or link to, a proof?

I ask, because this particular question seems hard enough to require some mathematical muscle, yet at the same time requires incorporating physical laws( in contrast to kinematics, which is essentially just calculus ). I suspect this problem could act as a good vehicle for investigating the role of mathematics in physics.

EDIT:
Changed title from diffract to interfere
Last edited by polymer on Tue Apr 15, 2014 11:43 pm UTC, edited 1 time in total.

Schrollini
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Re: Formal demonstration that classical paths won't diffract

polymer wrote:Quantum Mechanics is frequently motivated by observing that classical path's can't produce diffraction patterns.

I'm not quite sure what you mean here. Classical waves certainly diffract, be they light, sound, water, etc. Is the question, why don't we describe everything in classical mechanics as waves? Well, that doesn't work too well. Is the question, why don't things other than the wave equation demonstrate diffraction? Well, I'm sure you could cook up a PDE that isn't a wave equation but still has solutions that look diffractionish. (Though I don't know what diffraction means if you don't have wave solutions....) Is the question, why don't such equations appear in classical mechanics? That's getting too existential for my blood. Or is this a question about path integrals?

Also, is this question about diffraction patterns or interference patterns? If I throw baseballs at a hole in the wall, they won't all land directly behind the hole. Instead, they will be somewhat more scattered, since some of them glanced off the edge of the hole. I would argue that this is, in a loose sense, a diffraction pattern. However, if I have two holes in the the wall, I'll just get two overlapping baseball diffraction patterns. I will not get an interference pattern, as I would with classical waves or quantum things.
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Re: Formal demonstration that classical paths won't diffract

I'd assume it was talking about classical particle paths.

In which case it's easy to prove. A classical particle only cares about what it touches (assuming it's neutral and there's no gravity) so, when it passes through a slit, it sees nothing and just keeps going. There're no forces acting so Newton's first law takes over.

If it hits the screen with the slit in then it's a simple (in)elastic collision which are pretty straightforward.

If you want your slit to have finite depth, you still don't get diffraction patterns, you'd just get the beam splitting into two parts, one which passes through and one which reflects off the internal wall of the slit (for large angles, you could eliminate the split again).

Whatever it is you're asking, I think the reason you don't see it proven is it's either false or trivial, neither of which are generally worth putting in textbooks.
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polymer
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Re: Formal demonstration that classical paths won't diffract

I didn't think I was that confusing. The solutions to Newton's equations for particles must be continuously differentiable functions from R to R3, where the domain corresponds to time and the range to position. An electron's position, pre 1900's, was presumed to satisfy Newton's equations. I call any, continuously differentiable, vector that is a function of a parameter a path( or perhaps a curve ).

My question amounts to asking for a demonstration, that no matter how complicated "nano forces" might be, you cannot create an interference pattern with classical paths. This observation implies the need for something like a probability state function instead.

This is part of the reason why I found the double slit experiment interesting, since it puts requirements on the kinds of functions(paths are impossible), rather then just the differential equation.

I know it's clear that a simple Newtonian model predicts two patterns on a back wall, but my question is more subtle then that. I'm asking for a demonstration that any differential equation of position must similarly fail. This discussion must include some physical law, since I can imagine drawing multiple paths that look like an interference pattern when a cross section of the paths is taken.

Goemon
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Re: Formal demonstration that classical paths won't interfer

Perhaps what you're looking for is the calculus of variations and geodesics? Those are the equations that prove that all particles (not acted on by any external force) travel in straight lines. Once that's been proven, all you have to do is draw a diagram depicting a point source, a slit, and the straight line paths of the particles travelling from the source through the slit to the detector to prove that the particles will be able to strike ALL points on the detector that fall between the "sides" of the slit, and NO points outside of that region. Which is to say, the region illuminated on the target will be a simple uniform rectangle; there can't be any sort of interference pattern.
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Schrollini
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Re: Formal demonstration that classical paths won't diffract

polymer wrote:I didn't think I was that confusing. The solutions to Newton's equations for particles must be continuously differentiable functions from R to R3, where the domain corresponds to time and the range to position. An electron's position, pre 1900's, was presumed to satisfy Newton's equations. I call any, continuously differentiable, vector that is a function of a parameter a path( or perhaps a curve ).

A word to the wise: If you're introducing your own terminology, you're being confusing. I believe what you're describing is a trajectory.

And in doing so, you've answered your question. Interference is a thing that fields do. It's the fact that the amplitude of the field in some places is significantly larger or smaller than you might expect from a naive estimation. Interference is not a thing at trajectories do. So classical trajectories don't -- can't -- interfere.

This is rather unsatisfying, I admit, so let's consider a few other questions:

Why is classical mechanics described by trajectories, while quantum mechanics is described by fields?
I dunno.

Can we construct a field out of those trajectories and look for interference in that?
Sure. Run many different trajectories (with randomly varying initial conditions, to make things interesting), and then define a field whose value is the density of trajectories at that point. But this field doesn't display signs of interference. This, I think, is the point you were driving at with your question about the double slit experiment.

Why not?
Because this field was just the simple sum of trajectories. If we cut out all the trajectories through one slit, for example, we simply lose their contribution to the total field in a straightforward, linear way. No interference.

And we circle back to the first question. How unsatisfying.

Perhaps considering the principle of least action will help. Classically, a particle that goes from A to B will do so along a trajectory that minimizes its action. If you change something that's not on the trajectory (closing the other slit, for example), it won't affect the trajectory. Speaking very loosely now, in quantum mechanics, the probability that the particles goes from A to B can be calculated by a sum over all possible trajectories with those end points. The trajectories are weighted in such a way that the trajectories near the classical one contribute heavily, while those far away contribute little. (Or slightly more correctly, their contributions interfere destructively.) Thus, even changes away from the classical trajectory (opening a slit, for example) can modify the probability field. Therefore, the probability field for two slits isn't just a linear superposition of two one-slit fields, and we may have interference.

A nice thing about this approach is that the contribution of the non-classical trajectories scales with ℏ. As you send ℏ → 0, all but the classical trajectory get canceled out by destructive interference.

This is the path integral formulation of quantum mechanics. My description was very handwavy and downright wrong in some places, I'm sure. But it can be made rigorous, and I suspect it may be the best answer to your question. Why don't we see interference in classical mechanics? We do -- all trajectories other than the one that minimizes the action destructively interfere!
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Re: Formal demonstration that classical paths won't diffract

polymer wrote: This discussion must include some physical law, since I can imagine drawing multiple paths that look like an interference pattern when a cross section of the paths is taken.

Yes, Newton's first law. An object no acted upon by an external force moves at a constant velocity. I.e. in a straight line.

It's a trivial geometry exercise then to show that straight-line paths do not produce diffraction-like effects.

I mean, you can do it with variational calculus, but I'm guessing you're more familiar with and convinced of the Newtonian formalism of classical mechanics than the Lagrangian or Hamiltonian formalisms.
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polymer
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Re: Formal demonstration that classical paths won't interfer

I must be far worse at explaining things then I think I am .

Of course straight line paths don't interfere, and it is fascinating that classical trajectories can be modeled by interfering paths. But that wasn't my question. My question amounts to asking for the argument you would give to this physicist Feynman is picking on

How can such an interference come about?... "Well, perhaps some of them go through silt 1, and then they go around through slit 2, and then around a few more times, or by some other complicated path... then by closing slit 2, we changed the chance that an electron that started out through hole 1 would finally get to the backstop..."
(Also for what it's worth, my usage of the word path is consistent with how Feynman used it here, although I think It bled into my language from the term "path of integration.")

Then Feynman observes that more particles appear on the interfered wall, and less particles appeared on the interfered wall, depending on where you looked, after blocking a slit. Which seems implausible if those high and low points were both the consequence of some preexisting paths. It's a classic case of making your old model too complicated to explain a strange result. My question amounts to demonstrating that it's not only complicated, but impossible.

So in a Newtonian context, you wouldn't be allowed to assume that the net force on the electrons was zero, it would have to be nonzero to give such complicated paths. But any proposed model, I suspect, would have some built in contradiction.

This also, technically, isn't a new question. Newton was aware that classical particles wouldn't interfere, and he was looking for that in his study of light.

Schrollini was closest to answering my question. If I were being difficult, I could argue that closing a slit would have a non-linear affect on the underlying forces forming the paths the electrons were making, explaining away the non-linear contribution. The thing is, if I actually tried to make that model formal enough to predict the future, I'm pretty sure I'd run in to trouble. An answer to my question amounts to explaining why there would be trouble.
Schrollini wrote:
Can we construct a field out of those trajectories and look for interference in that?
Sure. Run many different trajectories (with randomly varying initial conditions, to make things interesting), and then define a field whose value is the density of trajectories at that point. But this field doesn't display signs of interference. This, I think, is the point you were driving at with your question about the double slit experiment.

Why not?
Because this field was just the simple sum of trajectories. If we cut out all the trajectories through one slit, for example, we simply lose their contribution to the total field in a straightforward, linear way. No interference.

Twistar
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Re: Formal demonstration that classical paths won't interfer

Umm, I think I know what you're asking. You're asking why the diffraction pattern forces us to reject the notion that electrons can be described purely in a particle picture. So I mean, I guess technically it doesn't For example, I could construct classical forces that WOULD give me a diffraction pattern. For example, I can force a cathode ray tube tv to create a diffraction patter. This amounts to running currents which turn on magnets that deflect an electron beam and this is all governed by classical electromagnetism. So in other words, it is possible for classical forces to create a diffraction pattern. This is the analog to me carrying the baseballs through the slits one by one and making them hit the parts on the detector that I want them to to create the diffraction pattern and again it's clearly all classical forces.
But that's not what you were asking. I think it's clear that there's not magical forces guiding the electron after it passes through the slits. But here is the point I am trying to make. I think the answer to your question is this: these models make a few assumptions about how the slit interacts with the electrons. So here's the classical model broken down:

Say the electron source is at x=-1. the barrier with two slits is at x=0 and the screen is at x=1. the slits are located at (0,1) and (0,-1). and the electron gun is just spewing electrons at all angles. Also assume that the slits are small enough that only electrons shot at 45 degrees EXACTLY go through the slit (you could have a little spread if you wanted but it would make more math). Now, if the wall didn't give the electrons a kick we would just illuminate two points on the screen (1,2) and (1,-2). This is clear because of Newton's 1st law as has been pointed out. However, in this thought experiment people always say 'pretend the wall gives the electrons a "kick"'. What does this mean? This means you track the electrons path until the slit. Then when the electron gets to the slit you adjust it's momentum by a certain amount in the y direction say. The way this is usually done is you give it some random kick p_y which is normally distributed about p_y=0. If you do this you will see that you get a Gaussian looking distribution on the screen. Under these simple assumptions it is clear that you are going to get two Gaussian peaks and no interference pattern looking things.

Are you asking if we should introduce a more complicated model than the "kick" I have described? If so you would need to justify the more complicated model on experimental grounds.

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Re: Formal demonstration that classical paths won't interfer

polymer wrote:Quantum Mechanics is frequently motivated by observing that classical path's can't produce interference patterns. I realized that this statement is quite precise, and that, I've never seen a formal demonstration of this. Can somebody give, or link to, a proof?

I ask, because this particular question seems hard enough to require some mathematical muscle, yet at the same time requires incorporating physical laws( in contrast to kinematics, which is essentially just calculus ). I suspect this problem could act as a good vehicle for investigating the role of mathematics in physics.

EDIT:
Changed title from diffract to interfere

You can just *look* at it. If you try the double-slit experiment with something at classical scale, you'll get one of two results:

1. A diffraction pattern will appear, but when you put a detector in front of the slits, you detect it at both slits (because it's a wave), or:
2. No diffraction pattern will appear, and when you put a detector in front of the slits, you detect it at exactly one slit each time.

Only when something gets special enough to invoke quantum behavior do you end up with something that appears at only one of the slits at a time, but also causes interference patterns.
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Schrollini
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Re: Formal demonstration that classical paths won't interfer

Xanthir wrote:Only when something gets special enough to invoke quantum behavior do you end up with something that appears at only one of the slits at a time, but also causes interference patterns.

Careful -- if you measure carefully enough to determine which of the slits the electron goes through, you'll destroy the interference patterns.

The only way to get interference is if the electron wave function goes through both slits. In other words, the electron wave function must act like a field, having a value at each point in space. If you measure it carefully enough, you collapse the wave function to the point where it's much more like a trajectory - a single point in space - and you no longer have interference.
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Twistar
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Re: Formal demonstration that classical paths won't interfer

I think he means something special enough that it is able to at least demonstrate both of those properties during different experiments. A classical particle can never make an interference pattern and a classical wave can never be detected at only 1 slit.

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