What experiment demonstrates Heisenberg's uncertainty principle?

[mycotheology]

When I read about the uncertainty principle, I keep reading about these experiments where they fire electrons through a single or double slit and observe the diffraction but I can't these experiments relate to Heisenbergs uncertainty principle. So when they fire the electrons, they know their momenta, then when they reach the slit, they diffract and their momenta changes and eventually, when they hit the screens, they know their positions. I'm real confused, how do these experiments show that when you know the momentum of a particle, you can't know its position and vice versa?

 

[jk22]

I'm interested in that too. I imagined that if you make a laser ray passes through a slit it should modify the wave length distribution of the light ?

Else i found such an ecperiment paper but i haven't read it carefully yet : http://cds.cern.ch/record/499984/files/0105061.pdf

 

[bhobba]

When I read about the uncertainty principle, I keep reading about these experiments where they fire electrons through a single or double slit and observe the diffraction but I can't these experiments relate to Heisenbergs uncertainty principle.

Check out the following:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

The reason you get interference is the unknown momentum as it passes through the slits - see equation 3. The reason its unknown is the slit is a position measurement.

Thanks
Bill

 

[Wes Tausend]

mycotheology,

I think I can recommend observing a related experiment that I believe demonstrates the Uncertainy principle. It involves a 1/2 hour video shown for college students and the video is only available to network connections in the US and Canada, so I hope you reside in such an area.

The video is called Particles and Waves and delves into quantum physics just a bit. Near the end of the video, the instructor, Dr. David Goodstein, demonstrates how light behaves as both particles and waves using polarized lenses. When demo'ing the particle aspect, he clearly shows that light can pass through a third set of carefully aligned polarized lenses in a particle probability function only, a phenomenen that cannot be answered any other way but probability. The video is found at http://www.learner.org/resources/series42.html . You must allow temporary pop-ups, click on the VoD box to the right of the Particle and Waves lesson #50 near the bottom, and the resulting small video window pop-up can be "full-paged" for easy viewing like a youtube video.

As a review, light is normally transmitted as an envelope in all 3 dimensions, and a single polarized lens, with it's many tiny parallel scratches, allows only a corresponding orientation portion to pass, such as the vertical orientation. Continued passing through a second polarized lens may be accomplished only by orienting the second lens scratches also vertical. Thereafter turning the second lens 90° will then block all light with one lens blocking vertical and the other horizontal waves. The uncertainy trick (proof) is when Dr. Goodstein adds a third polarized lens (at about the 26 minute mark) that obviously passes light as a probability function only.

As a review, I like to think of a polarized lens like a picket fence. A jumprope held by two people on the ends can be shaken only up and down in a wave-like action between the vertical fence slats. Adding another picket fence behind the first allows this to continue, unless the second picket fence is turned 90° (horizontal). With one vertical and one horizontal, all rope waves are impeded. Since this is a simple wave experiment, there is no corresponding particle probability insight, but it does clearly explain the wave likeness regarding polarized lenses. Good luck and I hope you have access to a North American network.

Wes
...

 

[jk22]

Check out the following:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

The reason you get interference is the unknown momentum as it passes through the slits - see equation 3. The reason its unknown is the slit is a position measurement.

Thanks
Bill

Interferences comes from the wave aspect, but the spreading if you have only one slit could be explained classically imagining small balls bouncing against the wall in the slit. Of course this is possible only if the particle is extended in space, so I'm not sure if this spreading is a test of the hup.

 

[jk22]

Erm i meant if the slit border is for example modelized by a quarter of circle and the particle can be considered a point.

 

[bhobba]

Interferences comes from the wave aspect,

Why do you think it has a wave aspect? You do know that De-Broglies hypothesis was superseded at least by 1926 when Dirac came up with his transformation theory?

The paper I linked to explained it by the Heisenberg Uncertainty Principle and principle of superposition. Did you read it?

You will find many threads on this forum explaining the issues with the so called wave-particle duality. It's not something the quantum theory actually has (at least in the way beginner texts elucidate it) - for example my standard textbook Ballentine doesn't mention it. Its a hangover from the early days of QM and beginner texts.

Thanks
Bill

 

[ZapperZ]

When I read about the uncertainty principle, I keep reading about these experiments where they fire electrons through a single or double slit and observe the diffraction but I can't these experiments relate to Heisenbergs uncertainty principle. So when they fire the electrons, they know their momenta, then when they reach the slit, they diffract and their momenta changes and eventually, when they hit the screens, they know their positions. I'm real confused, how do these experiments show that when you know the momentum of a particle, you can't know its position and vice versa?

Most obvious experiment: the single slit:

https://www.physicsforums.com/threads/misconception-of-the-heisenberg-uncertainty-principle.765720/

Zz.

 

[bhobba]

https://www.physicsforums.com/threads/misconception-of-the-heisenberg-uncertainty-principle.765720/

The paper I linked examines the same thing but models an exact position measurement as a delta function.

Thanks
Bill

 

[ZapperZ]

The paper I linked examines the same thing but models an exact position measurement as a delta function.

Thanks
Bill

I'm familiar with the Marcella paper, having quoted it several times. However, the "HUP" effect here is buried within the QM derivation. His emphasis in that paper is more of deriving the diffraction/interference patterns without using the typical wave picture, but rather entirely using the quantum mechanical treatment, something that most books often overlook.

Zz.

 

[vanhees71]

Again I must stress that the Marcella paper is highly misleading, precisely because it doesn't include the HUP, because it doesn't work with proper states but plane waves! I guess, I've to write up something about this, using wave packets. It's really astonishing that nobody seems to have written up this paradigmatic example, used in almost all textbooks at the beginning in such a hand-waving way. It's usually treated in the sense of an heuristic argument a la Einstein-de Broglie, and this is didactically very dangerous and shouldn't be done in modern lectures/textbooks anymore.

Another in the textbook literature is the use of the photo-electric effect, where the wrong (!) claim is made that it proves the quantization of the electromagnetic field, although it's well known that it is completely understandable using the semiclassical approximation (classical radiation field interacting with quantized bound electrons) and 1st-order time-dependent perturbation theory.

The third nogo, and imho it's the worst of all, is the use of the Bohr-Sommerfeld model for the hydrogen atom. No comment is necessary about this one!

 

[ZapperZ]

Again I must stress that the Marcella paper is highly misleading, precisely because it doesn't include the HUP, because it doesn't work with proper states but plane waves! I guess, I've to write up something about this, using wave packets. It's really astonishing that nobody seems to have written up this paradigmatic example, used in almost all textbooks at the beginning in such a hand-waving way. It's usually treated in the sense of an heuristic argument a la Einstein-de Broglie, and this is didactically very dangerous and shouldn't be done in modern lectures/textbooks anymore.

I don't understand this criticism. It appears that you don't seem to get the CONTEXT of his paper, which is to clearly show that one can get such interference pattern without the use of the classical wave treatment, something that many books simply said can be done, but never showed. What is wrong with the use of plane wave states as a special case? There are many phenomena that make use of such states as the starting point.

Another in the textbook literature is the use of the photo-electric effect, where the wrong (!) claim is made that it proves the quantization of the electromagnetic field, although it's well known that it is completely understandable using the semiclassical approximation (classical radiation field interacting with quantized bound electrons) and 1st-order time-dependent perturbation theory.

We have gone over this numerous times (and really, it shouldn't be brought up here since it is highly off-topic). Many of the newer papers that addressed this will use the standard photoelectric effect as an evidence, but not the definitive smoking-gun, for photons. However, as has been repeated, no semi-classical model exists to explain multiphoton photoemission, resonant photoemission, etc. You're welcome to take a crack at it.

Zz.

 

[vanhees71]

I agree, the photon statement was off topic, but I stand to my criticism against the Marcella paper. It's ironic if you quote it in connection with the HUP! Not only uses it plane waves (which are not representing states, because they are not square integrable) but also a zero-width slit!

Of course, nowhere is a classical-wave (nor classical-particle) picture in the full QT treatment of the single-slit, double-slit or grating experiments with quanta. I never claimed something in this direction.

 

[ZapperZ]

I agree, the photon statement was off topic, but I stand to my criticism against the Marcella paper. It's ironic if you quote it in connection with the HUP! Not only uses it plane waves (which are not representing states, because they are not square integrable) but also a zero-width slit!

Of course, nowhere is a classical-wave (nor classical-particle) picture in the full QT treatment of the single-slit, double-slit or grating experiments with quanta. I never claimed something in this direction.

I did not quote the Marcella paper in this thread, nor have I used it in connection with the HUP. Check again!

I also do not see the issue with using delta functions for the slit width as a demonstration of one extreme limit of the situation. This was a pedagogical treatment to illustrate that it can be done without further added complications.

We're not going to resolve this here because this has become a matter of taste. When you publish your more realistic treatment of this very same topic (preferably in EJP where Marcella published his), we can discuss this further.

Zz.

 

[bhobba]

Again I must stress that the Marcella paper is highly misleading, precisely because it doesn't include the HUP,

Can you elaborate on that? From the paper:
'Because position and momentum are non-commuting observables, a particle passing through slits always has an uncertainty in its y-component of momentum. It can be scattered with anyone of the continuum of momentum eigenvalues py = psinθ , where −π /2 ≤ θ ≤ π /2. Measurement of a well-defined scattering angle θ constitutes a measurement of the observable p ˆ y and, therefore, the basis vectors in Hilbert space are the momentum eigenvectors py.'

I know that paper has issues - and they have been discussed in a few threads.

The trouble is the paper that examines those issues is not exactly what I would call elementary:
http://arxiv.org/pdf/1009.2408.pdf

Thanks
Bill

 

[vanhees71]

Yes, but don't listen to his words but look at his deeds. He's using plane wave, which represent a situation, where the momentum has no uncertainty. Of course, it's not a state. So it's no contradiction to what he correctly states in words, but then he should use wave packets. The same holds for idealizing the slit to a 0-width $\delta$ distribution, which implies 0 width for position at the slit.

Of course you can start with harmonic waves, using the Helmholtz equation to find the propagator (for the case of a finite-width slit of course), but then you have to fold it with an appropriate wave packet to get the story correct, and of course, the typical setup can be understood as a potential-scattering problem (asymptically free particles going in, asymptocially particles going out).

 

[ZapperZ]

Yes, but don't listen to his words but look at his deeds. He's using plane wave, which represent a situation, where the momentum has no uncertainty. Of course, it's not a state. So it's no contradiction to what he correctly states in words, but then he should use wave packets.

Again, I do not see this being a problem at all. The momentum in the direction of propagation may not have any uncertainty, but this momentum is actually quite irrelevant to this case. This is because the momentum that is involved in the HUP (and consequently, the formation of the interference pattern he's dealing with in his paper) is the transverse momentum, i.e. parallel to the plane of the slit!

I do not see any loss in the pedagogical illustration by using such plane waves.

Zz.

 

[vanhees71]

There's no spread in transverse momentum for a plane wave. It's a generalized (!) momentum (as a three-vector!) eigenstate of momentum!

 

[ZapperZ]

There's no spread in transverse momentum for a plane wave. It's a generalized (!) momentum (as a three-vector!) eigenstate of momentum!

There is AFTER the slit!

Zz.

 

[bhobba]

Yes, but don't listen to his words but look at his deeds. He's using plane wave, which represent a situation, where the momentum has no uncertainty.

But that sort of thing is used all over the place in QM eg Dirac's classic. It's not square integrable - and Von Neumann was correctly scathing of it. But we now have the Rigged Hilbert Space formalism - such states are viewed simply as an idealisation for mathematical convenience. The same with modelling a narrow slit by a Dirac Delta function.

Thanks
Bill

 

[vanhees71]

There is AFTER the slit!

Zz.

There's no "before and after" the slit for a plane wave either. It's spread over the entire space already!

 

[vanhees71]

But that sort of thing is used all over the place in QM eg Dirac's classic. It's not square integrable - and Von Neumann was correctly scathing of it. But we now have the Rigged Hilbert Space formalism - such states are viewed simply as an idealisation for mathematical convenience. The same with modelling a narrow slit by a Dirac Delta function.

Thanks
Bill

The plane waves should be used as what they are, namely "mode decompositions" of true states. The rigged-Hilbert space formalism formalizes nicely what Dirac and 99% of all pracitioners do using the physicists' hand-waving way of dealing with continuous spectra of self-adjoint operators.

 

[bhobba]

There's no "before and after" the slit for a plane wave either. It's spread over the entire space already!

Yes it is - but the state just before the screen is different to the state just after.

Thanks
Bill

 

[ZapperZ]

There's no "before and after" the slit for a plane wave either. It's spread over the entire space already!

If you are going to be picky about such boundary condition, then you have a bigger problem to deal with, because using your logic, the original BCS theory is wrong, the description of tunneling phenonenon is wrong, and a lot of the many-body wavefunction is wrong. Is this what you are claiming?

Zz.

 

[vanhees71]

I don't understand this statement. What do you mean with "there's a state just before the screen is different to the state just after"? There's just one wave function, representing the state, right?

 

[vanhees71]

If you are going to be picky about such boundary condition, then you have a bigger problem to deal with, because using your logic, the original BCS theory is wrong, the description of tunneling phenonenon is wrong, and a lot of the many-body wavefunction is wrong. Is this what you are claiming?

Zz.

No, why should I claim this?

For a correct use of wave packets in scattering theory, look, e.g., in the standard textbook by Peskin and Schroeder. There's nothing wrong with the use of plane-wave-mode decompositions (aka Fourier transformations). It's one of the most important calculational tools in physics!

 

[bhobba]

I don't understand this statement. What do you mean with "there's a state just before the screen is different to the state just after"? There's just one wave function, representing the state, right?

If you measure the momentum before the screen its the same ie its in an eigenstate of momentum - after its different because the slits are a position measurement - it has gone though a state preparation procedure.

Thanks
Bill

 

[ZapperZ]

No, why should I claim this?

For a correct use of wave packets in scattering theory, look, e.g., in the standard textbook by Peskin and Schroeder. There's nothing wrong with the use of plane-wave-mode decompositions (aka Fourier transformations). It's one of the most important calculational tools in physics!

Then why are you making an exception with the Marcella treatment? The same complain you had with that can be applied easily to the standard tunneling treatment.

Again, why don't you write a rebuttal to that paper so that we can all read a more detailed problem that you are having? This back-and-forth is not helping. It is making it worse.

Zz.

 

[Wes Tausend]

When I read about the uncertainty principle, I keep reading about these experiments where they fire electrons through a single or double slit and observe the diffraction but I can't these experiments relate to Heisenbergs uncertainty principle. So when they fire the electrons, they know their momenta, then when they reach the slit, they diffract and their momenta changes and eventually, when they hit the screens, they know their positions. I'm real confused, how do these experiments show that when you know the momentum of a particle, you can't know its position and vice versa?

In reviewing the other answers, I think the alternate experimental observation I recommended in post #4 better answers your question (if it can be viewed). I'm not sure if anyone else on this thread has viewed this, and if so, I fear a valuable opportunity is lost. A different perspective can be most helpful in many cases.

The video presented by ZapperZ in his informative PF post, https://www.physicsforums.com/threads/misconception-of-the-heisenberg-uncertainty-principle.765720/ , is very good and refers directly to the "slit" experiment you mention in your question... the one that did not clear your understanding of the observation. The youtube video also leaves some ambiguity at the end as to whether the blurred spread of light is due to the uncertain repeated position of particles... or due to wave diffraction, or at least this is admitted by the commentator/presenter at the end.

The polaroid lens experiment in my video starting at minute 20:55 (see post #4) is almost exactly a rendition of the slit experiment in that the parallel lines of a polaroid lens represent a multitude of slits side by side. This does not leave a clear single strike pattern on the screen in itself (part of the light is impeded which is part of the reason polaroid sunglass lens work so well). The "polaroid slit" easily demonstrates the wave likeness of the "wavical" nature of light.

The elegant beauty of Dr. Goodsteins demo is found towards the end. By having turned two polaroid lens perpendicular to one another, the lightwaves have been previously entirely blocked, which is to be expected of wave action. Moving on, some contrasting discussion of particle probability ensues concerning lens angles. But the final particle demo starting at about minute 26 is the most striking. SPOILER ALERT

The final particle demo starting at about minute 26 is done by simply inserting a third lens in between the other two which are at perpendicular angles to one another. Since the two are perpendicular, no light is passing. However by inserting the third lens at an oblique angle to the other two, suddenly light passes through all three lenses and mysteriously appears again on the screen.

There can be only one answer to this result of light reappearing and that is... light taken from an apparent stream of darkness... now specifically passes strictly because of probability and probability only. And that is what Heisenbergs uncertainty principle is all about as a tool, the wrench for assembling and disassembling atoms. We must use it, if our science and machines are to be able to predict and use, the smallest bolts of our universe.

Wes
...

 

[vanhees71]

The video is pretty good, but it's in fact not using single photons as claimed, it's using a laser beam, which quantum mechanically is a coherent state. It's well described as a classical electromagnetic wave for this purpose.

BTW: now, I try to get one example for a correct diffraction calculation with wave packets done. It's pretty complicated. I'll try to get diffraction on a circular hole now. I'm not yet sure, if I get an exact solution for the propagator yet. As simple as this looks, it's not!

 

[TrickyDicky]

Then why are you making an exception with the Marcella treatment? The same complain you had with that can be applied easily to the standard tunneling treatment.

It is not the same thing. It's one thing to use a highly idealized and unrealistic implementation of Schrodinger'equation, that happens to work in experimental applications that exploit the concept of tunneling, and another thing(the opposite) is modelling rigorously (or that is the intention) an experiment and also use a very unrealistic and idealized mathematical object(the plane wave) for what it tries to represent.

 

[ZapperZ]

It is not the same thing. It's one thing to use a highly idealized and unrealistic implementation of Schrodinger'equation, that happens to work in experimental applications that exploit the concept of tunneling, and another thing(the opposite) is modelling rigorously (or that is the intention) an experiment and also use a very unrealistic and idealized mathematical object(the plane wave) for what it tries to represent.

Sorry, but I don't see a difference between the two situations. If he had an issue with the presence of a slit that somehow negates the use of plane waves that should be 'everywhere', then the SAME issue arises with the presence of a potential barrier.

In fact, having had to deal with a lot of modeling of various experiments using tunneling phenomena (tunneling spectroscopy, field emission), I could easily take what you said towards the end and apply it to the mathematical treatment to describe those two. Don't believe me? Look at the formulation of the Fowler-Nordheim model.

Zz.

 

[TrickyDicky]

Sorry, but I don't see a difference between the two situations. If he had an issue with the presence of a slit that somehow negates the use of plane waves that should be 'everywhere', then the SAME issue arises with the presence of a potential barrier.

In fact, having had to deal with a lot of modeling of various experiments using tunneling phenomena (tunneling spectroscopy, field emission), I could easily take what you said towards the end and apply it to the mathematical treatment to describe those two. Don't believe me? Look at the formulation of the Fowler-Nordheim model.

Zz.

Yes, it's the same issue, the distinction I made was purely contextual. It's very easy, what works needs no more justification, but when the only purpose is to give a formal explanation of a experiment you need something more, otherwise one might as well let it be and stick to the usual textbook handwaving.

 

[bhobba]

Yes, it's the same issue, the distinction I made was purely contextual

I am scratching my head about the issue here. Its the typical approximation stuff that's done all the time in mathematical modelling. I taught myself physics, in particular quantum physics, but was formally trained in mathematical modelling. Before the screen the wave-function is approximated as being in an eigenstate of momentum and energy - it really can't be because its wave-function would extend through all space - but that's what you do in modelling. After the screen the thin slits are modeled by having the position as exact and using a Dirac delta function - of course it's not exactly like that - but modelling wise its what's done to get a mathematical grip on the situation.

Thanks
Bill

 

[atyy]

I don't understand this statement. What do you mean with "there's a state just before the screen is different to the state just after"? There's just one wave function, representing the state, right?

You are right, of course, but it seems standard to approximate the motion along the axis as "classical". For example, http://www.atomwave.org/rmparticle/ao%20refs/aifm%20pdfs%20by%20group%20leaders/pfau%20pdfs/PK97.pdf says "Because the longitudinal motion of the atoms at velocities v of several thousands of metres per second may be treated classically ..."

To do a full calculation without any classical thinking seems like a headache, first to choose to right observable, and especially the calculation of the asymptotic pattern where I don't see how one can avoid an ancilla (without the slick and correct but hard to justify absorbing boundary condition).

 

[vanhees71]

Great paper. I'll have a close look at it. I must say, I can understand now, why the diffraction is not treated in standard textbooks. I couldn't solve the appropriate Helmholtz equation with the exact boundary conditions exactly, even not for a circular apperture. The best, I can come up with is the standard Sommerfeld-corrected approximate Kirchhoff solution, which should be a good approximation in the limit, where the de Broglie wavelength is small compared to the dimensions of the apperture, using the Green's function for the infinite plane. At the moment I'm at the process to get some numerical calculations done :-).

Of course the free-particle wave packet can be described pretty well in a classical approximation. Since there's no force, the leading-order classical approximations give the classical uniform motion: Ehrenfest's theorem together with the fact that the equations of motion in the Heisenberg picture are linear in position and momentum operators and are of the classical form; taking the averages gives then the classical equation of motion also for the expectation values.

 

[atyy]

Here are some other related papers, but I think they all make use of the approximation of the longitudinal motion being classical, which is hard to justify completely from purely quantum postulates.

http://arxiv.org/abs/quant-ph/0407245
http://arxiv.org/abs/0804.3006
http://arxiv.org/abs/0902.0234

 

[TeethWhitener]

I'm having a hard time understanding why the use of a plane wave is a big deal. Since the Schrodinger equation is linear, any superposition of plane waves will be a solution. And you can make a Gaussian wavepacket from a superposition of plane waves (this is in fact the definition of a Fourier transform). So I'm not seeing how the time-dependent behavior of a wavepacket would be qualitatively different from that of a plane wave.

 

[TrickyDicky]

Before the screen the wave-function is approximated as being in an eigenstate of momentum and energy - it really can't be because its wave-function would extend through all space - but that's what you do in modelling. After the screen the thin slits are modeled by having the position as exact and using a Dirac delta function - of course it's not exactly like that - but modelling wise its what's done to get a mathematical grip on the situation.

I have no problem with approximate models, that's physics. But one must know when a certain approximation is licit and when it is not. Zapper mentioned the use of plane waves in quantum tunnelling, but in that case the time-independent SE allows one not to think about any before or after the barrier, that acts as a spatial boundary condition, one doesn't need to have a localized electron at any given time. The HUP allows you to have a state with acertain probability in a classically forbidden zone.
The Marcella paper is different, it is a time-dependent model of an electron with no uncertainty about momentum(plane wave) interacting with the slit at some point in time modeled as you say with exact position(Dirac delta). When your approximation demands to model a quantum phenomenon throwing away the HUP I believe something's not right.

I'm having a hard time understanding why the use of a plane wave is a big deal. Since the Schrodinger equation is linear, any superposition of plane waves will be a solution. And you can make a Gaussian wavepacket from a superposition of plane waves (this is in fact the definition of a Fourier transform). So I'm not seeing how the time-dependent behavior of a wavepacket would be qualitatively different from that of a plane wave.

I think it is not granted in this particular case, see above. But I suspect it can't be done with wave packets either, there must be a reason why nobody tries to do this rigorously after all. There are theorems(i.e.Revisiting the First Postulate of Quantum Mechanics:Invariance and Physical Reality by Ronde, Massri) that point to problems for the dynamics of QM for quantum systems with Hilbert spaces of more than 2 dimensions.
Maybe some QFT scattering approach can do it.

 

[bhobba]

The Marcella paper is different, it is a time-dependent model of an electron with no uncertainty about momentum(plane wave) interacting with the slit at some point in time modeled as you say with exact position(Dirac delta). When your approximation demands to model a quantum phenomenon throwing away the HUP I believe something's not right.

Throwing away the HUP? That a position measurement as done by a slit has an unknown momentum after is the HUP.

It simple. Let's consider electrons - photons are more problematical. It leaves the source with a definite momentum and energy (NOT exact of course since that would mean a wave-function over all space - but is obviously true to a very good approximation). It passes through the slit or gets absorbed. If it passes through the slit then its position is known, so from the HUP its momentum is unknown. But the energy is the same which means the magnitude of the momentum is unchanged hence its direction is unknown and its scattered.

When both slits are open the state is obviously a superposition of the state with a single slit open - that is the principle of superposition. The symmetry of the situation leads to equation 9 in that paper - and - wonder of wonders - you get an interference pattern.

There really isn't much to it. Of course approximations are made - but that's done all the time in modelling. I have zero problem with people pointing that out - it must really be a single wave function etc etc. But it doesn't invalidate the analysis any more than for example modelling a ball rolling down an incline by a point invalidates that.

Thanks
Bill

 

[bhobba]

I'm having a hard time understanding why the use of a plane wave is a big deal. Since the Schrodinger equation is linear, any superposition of plane waves will be a solution. And you can make a Gaussian wavepacket from a superposition of plane waves (this is in fact the definition of a Fourier transform). So I'm not seeing how the time-dependent behavior of a wavepacket would be qualitatively different from that of a plane wave.

The issue is that plane waves are not square integrable hence can't be a valid solution to the Schrödinger equation. We have to make a few reasonableness assumptions to mathematically get a handle on it. While true its done all the time, often without even saying that's what's being done eg:
http://www.physics.ox.ac.uk/Users/smithb/website/coursenotes/qi/QILectureNotes3.pdf

See the free particle solution. It's not square integrable so its not valid - but no-one really worries about it.

Thanks
Bill

 

[TrickyDicky]

Throwing away the HUP? That a position measurement as done by a slit has an unknown momentum after is the HUP.

It simple. Let's consider electrons - photons are more problematical. It leaves the source with a definite momentum and energy (NOT exact of course since that would mean a wave-function over all space - but is obviously true to a very good approximation). It passes through the slit or gets absorbed. If it passes through the slit then its position is known, so from the HUP its momentum is unknown. But the energy is the same which means the magnitude of the momentum is unchanged hence its direction is unknown and its scattered.

When both slits are open the state is obviously a superposition of the state with a single slit open - that is the principle of superposition. The symmetry of the situation leads to equation 9 in that paper - and - wonder of wonders - you get an interference pattern.

There really isn't much to it. Of course approximations are made - but that's done all the time in modelling. I have zero problem with people pointing that out - it must really be a single wave function etc etc. But it doesn't invalidate the analysis any more than for example modelling a ball rolling down an incline by a point invalidates that.

Thanks
Bill

If you read the paper you linked in post #15, which I read after posting above, you might understand what's wrong with your analysis: Marcella's paper makes a purely classical approximation to a quantum problem, the "after" is the classical optics treatment, so it is quite reasonable to say there is no HUP in his treatment, as vanhees told you: "look at what he does, not what he says", the HUP doesn't show up magically just by invoking it because this is QM, right?, you must justify it with the math and the physics in a specific problem, not just imply it by using Dirac formalism.
So again approximations are fine, when they don't directly affect the essence of what you want to model, in this case the quantum nature of the interference phenomenon. This is so basic that I cannot imagine how it can scape anyone.

 

[TrickyDicky]

The issue is that plane waves are not square integrable hence can't be a valid solution to the Schrödinger equation. We have to make a few reasonableness assumptions to mathematically get a handle on it. While true its done all the time, often without even saying that's what's being done eg:
http://www.physics.ox.ac.uk/Users/smithb/website/coursenotes/qi/QILectureNotes3.pdf

See the free particle solution. It's not square integrable so its not valid - but no-one really worries about it.

Thanks
Bill

It is enphasized in that section about free particles!, you either use the time-independent SE like it's done in tunnelling etc, or if you are going to deal with a time-dependent problem and use the traveling plane wave like we are here you can never localize the electron with a slit, only in the well known classical macroscopic optics case you can.

 

[bhobba]

it is quite reasonable to say there is no HUP in his treatment,

It obviously does - as I explained in my post. Its why the particle is scattered by the slit.

Thanks
Bill

 

[TrickyDicky]

It obviously does - as I explained in my post. Its why the particle is scattered by the slit.

Thanks
Bill

It is claimed in your post rather than explained, it is explained in the lecture notes you linked above: Plane waves are not proper solutions and are not valid approximations for position dynamical wave functions(you need wave packets there), they are perfectly valid approximations for energy solutions of the TISE, that's why no one worries about it in tunnelling kind of problems.

 

[bhobba]

It is claimed in your post rather than explained,

I carefully explained it - reread it if you don't get it. But to repeat it - particles that go through the slit have been subjected to a position measurement. By the HUP there momentum is now unknown. If you still don't get it - nothing much more I can do.

Thanks
Bill

 

[TrickyDicky]

particles that go through the slit have been subjected to a position measurement.

An exact position (Dirac delta point) measurement cannot be performed to a wave function.

By the HUP there momentum is now unknown.

By the HUP that you put by hand there is no state anymore either :), besides if momentum is unknown one cannot use the y component, momentum is either known or not, the rest is the classical optics treatment, which is what Marcella does disguised in Dirac formalism.

 

[vanhees71]

Yes, it's completely right what TrickyDicky wrote. Take a good standard textbook of quantum theory, where scattering is described correctly. Then you'll see that you need wave packets to define the cross section. In the generalized (!) momentum eigenstates, aka plane waves, you get
$$S_{fi}=\delta_{fi} - (2 \pi)^4 \mathrm{i} T_{fi} \delta^{(4)}(p_f-p_i).$$
The transition probability is the modulus squared of the amplitude, which naively is $S_{fi}$. Of course, this doesn't make sense, but you first have to fold it with the incoming wave packets take the square and then take the limit to vanishing momentum spread. The final result is that there's only one energy-momentum conserving $\delta^{(4)}$ distribution in the transition probability density, which can then be meaningfully integrated out.

Admittedly, here the whole issue is complicated also by the problem of how to define asymptotic free states, where you have to use an appropriate switching procedure. If I remember right, a good source for the discussion of this is Messiah's classical quantum mechanics text on this issue. For the relativistic case, it's nicely dealt with in Peskin&Schroeder.

Of course, there are shortcuts in the literature like in Landau/Lifshitz vol. 4, using a "box regularization", where the momenta become descrete, and there's no problem in squaring the S-matrix elements to get transition probabability rates. After deviding over the finite four-volume and then taking the limit of the four-volume to $\infty$ you get the same result as with the proper procedure with the wave packets. However, the latter is much more physical, although a bit more complicated mathematics wise. When discussing the physics, one should use the wave-packet approach.

 

[TeethWhitener]

The issue is that plane waves are not square integrable hence can't be a valid solution to the Schrödinger equation.

I think you meant to say that plane waves aren't a valid physical solution to the Schrodinger equation. But that's my point. You can sum them in a way that they are a valid physical solution, namely:\psi (x) = \int_{-\infty}^{\infty}f(k)e^{ikx}dk with a proper weighting function f(k) such that square integrability holds at all times for this solution. But since this solution is built up from plane waves, then solving the time-dependent behavior of a single plane wave going through a slit should give you an idea of how the time dependent behavior of a linear combination of plane waves will act.

 

[bhobba]

An exact position (Dirac delta point) measurement cannot be performed to a wave function.

That's the modelling bit. Its used all the time in applied math eg short electrical impulses are modeled by a Dirac delta function - that the slit is of very small width is modeled similarly.

Thanks
Bill