Some sins in physics didactics - comments

[bhobba]

The way he did it gives the right results. He used the relativistic mass not in F=ma, but in F=dp/dt.

I am not sure that resolves the issue - but I would need to check my copy of the lectures.

Thanks
Bill

 

[bhobba]

I'm not convinced Feynman's explanation was wrong. But yes, if it is wrong, we should not teach it. Of course there will be errors from time to time, but we should not teach things that are deliberately wrong. In this case, if Feynman is wrong, I'm pretty sure he made an unintended error.

Ok - at least you are consistent about it.

Thanks
Bill

 

[bhobba]

The Feynman-Wheeler absorber theory, to my knowledge, has never been put into a (semi-)consistent quantum theory, as was famously predicted by Pauli after listening to Feynman's talk at Princeton. It's a funny to read story in one of Feynman's autobiographical (story) books (I guess "Surely you are joking").

That's my understanding as well.

But some claim Paul Davies fixed that issue:
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2103380

Thanks
Bill

 

[atyy]

BTW, the reason I don't know whether Feynman's explanation is wrong is that I don't think it is the one ZapperZ argues against. ZapperZ argues against the slowing down being due to the delay of absorption and re-emission by atoms. If I remember correctly, Feynman's argument involved superposition and a change in phase. Heuristically, this seems to be correct, since it is more or less an attempt to apply QED to a material. It also seems similar to ZapperZ's phonon explanation, since a phonon is a superposition of localized atomic wave functions, so perhaps the explanations are "Fourier transform" pairs of each other. Of course it can't be so simple, but this is why I don't think Feynman's argument is obviously wrong.

Feynman did make mistakes in his lectures. A famous one is an error in the application of Gauss's law. http://www.feynmanlectures.info/flp_errata.html (See the story right at the bottom)

 

[bhobba]

BTW, the reason I don't know whether Feynman's explanation is wrong is that I don't think it is the one ZapperZ argues against.

I jusr checked it.

It's in chapter 3. He explains it due to the extra time its takes to traverse the medium from scattering by the electrons. He doesn't assume its absorbed and re-emitted - but scattered in an unknown direction.

I think its better than the usual explanation of absorption and remission - but its not entirely correct either.

Thanks
Bill

 

[stevendaryl]

I'm not convinced Feynman's explanation was wrong. But yes, if it is wrong, we should not teach it. Of course there will be errors from time to time, but we should not teach things that are deliberately wrong. In this case, if Feynman is wrong, I'm pretty sure he made an unintended error.

I think it depends on how you teach it. If you teach something as a model, rather than as the "truth", then there is nothing wrong (in my opinion) with using models that are known to have limited applicability.

 

[atyy]

I think it depends on how you teach it. If you teach something as a model, rather than as the "truth", then there is nothing wrong (in my opinion) with using models that are known to have limited applicability.

Almost everything is a model with limited applicability, so this is not any real criterion.

 

[stevendaryl]

Almost everything is a model with limited applicability, so this is not any real criterion.

I'm just saying that I disagree with your rule that you should never teach something that you know is false. That's true with everything.

As far as what models should be taught, I think that it's kind of subjective. Some models are definitely dead ends--nothing learned from them is of any use in more advanced treatments (the phlogiston model might be an example). Other models teach concepts that get refined by later models, and it's a matter of opinion whether knowing the model is a hindrance or help in understanding better models.

 

[atyy]

I'm just saying that I disagree with your rule that you should never teach something that you know is false. That's true with everything.

As far as what models should be taught, I think that it's kind of subjective. Some models are definitely dead ends--nothing learned from them is of any use in more advanced treatments (the phlogiston model might be an example). Other models teach concepts that get refined by later models, and it's a matter of opinion whether knowing the model is a hindrance or help in understanding better models.

But if you read my comment in context, that is not what I said at all. For example, I argued that you should not teach things that are wrong in the sense that they are misleading. But I immediately said that did not mean the old quantum theory photon explanation of the photoelectric should not be taught. In fact, I said exactly what you are saying as a away to advocate teaching the old quantum theory explanation of the photoelectric effect.

 

[stevendaryl]

But if you read my comment in context, that is not what I said at all. For example, I argued that you should not teach things that are wrong in the sense that they are misleading. But I immediately said that did not mean the old quantum theory photon explanation of the photoelectric should not be taught. In fact, I said exactly what you are saying.

Okay, I misunderstood. But I wouldn't use the word "wrong" here, because every model is wrong, in some sense. Misleading is more relevant, if we can objectively say what it means to be misleading. I guess I would say that an explanation, based on one model, is misleading if it is contradicted (as opposed to tweaked/refined?) by more accurate models?

 

[atyy]

Okay, I misunderstood. But I wouldn't use the word "wrong" here, because every model is wrong, in some sense. Misleading is more relevant, if we can objectively say what it means to be misleading. I guess I would say that an explanation, based on one model, is misleading if it is contradicted (as opposed to tweaked/refined?) by more accurate models?

Yes, which is why my comment really had to be read in context. There you can see I argued for teaching two wrong models - the photoelectric effect and possibly Feynman's explanation of the slow speed of light in a medium - because they capture ways of thinking that are powerful, even by the standards of our current best theories. I argued both that the wrong models should be taught, and that they should not be taught in a way that anything had to be unlearnt later.

Also, one doesn't have to use the idea of "not being contradicted" as the idea of not being misleading. We still teach Newtonian physics, yet it is contradicted and not just tweaked by general relativity and quantum mechanics. But teaching Newtonian mechanics is usually not considered misleading.

What is misleading is to teach the photoelectric effect as "proving" the necessity of photons. That was vanhees71's point. I agree with that. However, I don't agree that one should not to teach it as very powerful picture, aspects of which are formalized in quantum field theory, and that is still an efficient way of deriving Planck's blackbody formula, the Fowler-Dubridge theory still used in modern papers like the one pointed out by ZapperZ, and its use in modern devices for detecting single photons.

In the same way, I don't agree that "wave-particle duality" is a myth or misleading, since it is formalized into the particle nature of the quantum mechanical Hilbert space and the Fock space of non-rigourous quantum field theory and the wave nature of the equation of motion in the Schroedinger and Heisenberg pictures.

 

[martinbn]

@atyy: Sorry for the off topic questions, but what do you mean by the particle nature of the quantum Hilbert space and the wave nature of the equations of motion in the Heisenberg picture?

 

[atyy]

@atyy: Sorry for the off topic questions, but what do you mean by the particle nature of the quantum Hilbert space and the wave nature of the equations of motion in the Heisenberg picture?

Let's work in QM. There we have the Schroedinger equation which is a "wave" equation. For 1 particle, the Hilbert space basis is some set of wave functions. For two particles, the Hilbert space basis is made from the tensor products of the 1 particle basis functions. So particles define the Hilbert space. The only difference to a classical particle is that a quantum particle does not have simultaneous position and momentum at all times. However, in the classical limit, we do recover the classical equation of motion for classical particles, justifying the term "particle" for the quantum object.

Non-rigourous QFT is the same, except we use a second quantized language and work in Fock space, and the number of particles is not necessarily conserved in relativistic theory.

The other way that wave-particle duality is formlized in QM are the commutation relations. Position is particle and momentum is wave, and they do not commute.

So rather than saying wave-particle duality is a myth, I would rather say wave-particle duality is a vague notion that is formalized deep in QM in several ways.

It is like the equivalence principle. It started vaguely, with some idea that it is only "locally" true, but we don't have a definition of "local" before we have the mathematical theory. After we have the full theory, we find that the equivalence principle can be formalized, and local means "first order derivative".

 

[martinbn]

This is still not clear to me. You have to keep in mind that I am not a physicist and need things said explicitly. Perhaps this is too far from the topic to discuss it here.

 

[atyy]

This is still not clear to me. You have to keep in mind that I am not a physicist and need things said explicitly. Perhaps this is too far from the topic to discuss it here.

How do we know how to describe the Hilbert space?

1 particle basis functions: ψ_m(x)

2 particle basis functions: ψ_m(x₁)ψ_n(x₂)

So we define the Hilbert space by using particles.

 

[martinbn]

And what are these functions?

 

[atyy]

And what are these functions?

Let's take the particle in an infinite well. These are energy eigenfunctions of the Schroedinger equation.

 

[martinbn]

Ok, but you are already considering a space of functions (smooth, complex valued, solutions of the equation ect.). Then you build a Hilbert space out of them, which is just $L^2(\mathbb R^3)$. You can just start with it. What is its particle nature?

 

[atyy]

Ok, but you are already considering a space of functions (smooth, complex valued, solutions of the equation ect.). Then you build a Hilbert space out of them, which is just $L^2(\mathbb R^3)$. You can just start with it. What is its particle nature?

For one particle, the classical limit recovers the classical particle.

 

[martinbn]

What is a classical limit of a Hilbert space? And these Hilbert spaces, for one or two or many particles, are all isomorphic.

 

[atyy]

What is a classical limit of a Hilbert space? And these Hilbert spaces, for one or two or many particles, are all isomorphic.

Yes, of course, you can even have quantum gravity using a single particle. But there is a reason the we do call the Schroedinger equation for 1 particle by that name.

 

[martinbn]

Yes, of course, you can even have quantum gravity using a single particle. But there is a reason the we do call the Schroedinger equation for 1 particle by that name.

I don't doubt that there are reasons, but my confusion is not about the Schrodinger's equation but about the Hilbert space. I am just trying to understand your comment. I am still confused about the particle nature of Hilbert spaces, and the classical limit of a Hilbert space.

 

[atyy]

I don't doubt that there are reasons, but my confusion is not about the Schrodinger's equation but about the Hilbert space. I am just trying to understand your comment. I am still confused about the particle nature of Hilbert spaces, and the classical limit of a Hilbert space.

Of course there is no such thing. One takes the classical limit together with Schroedinger equation in the usual way.

 

[martinbn]

Ok, then, what is the particle nature of the Hilbert space then!?

 

[atyy]

Ok, then, what is the particle nature of the Hilbert space then!?

See post #85 :) That is how we write basis functions when we describe 2 particles.

 

[martinbn]

See post #85 :) That is how we write basis functions when we describe 2 particles.

I know that, but it does demystify for me the particle nature of the Hilbert space. (and it is just one way to write a basis) Anyway...

 

[atyy]

I know that, but it does demystify for me the particle nature of the Hilbert space. (and it is just one way to write a basis) Anyway...

Well, do you at least agree with terminology like the Schroedinger equation for 1 particle, or the Schroedinger equation for two particles?

 

[vanhees71]

I'm sorry that I can't follow the very interesting discussion my article against teaching "old quantum theory", in particular the pseudo-explanation of the photoelectric effect as an evidence for photons. I'm quite busy at the moment.

Just a remark: Of course, it's subjective, which "wrong" models one should teach and which you shouldn't. That's the (sometimes hard) decision to make for any who teaches science at any levels of sophistication. I personally think, one should not teach "old quantum theory", not because it's "wrong" but it leads to wrong qualitative ideas about the beavior of matter at the micrscopic level. E.g., the Bohr-Sommerfeld model contradicts well-known facts about the hydrogen atom, even known by chemists at the days when Bohr created it (e.g., it's pretty clear that the hydrogen atom as a whole is not analogous to a little disk but rather a little sphere, if you want to have a classical geometrical picture at all). The reason for, why I wouldn't teach old quantum theory (and also not first-quantized relativistic quantum mechanics) is that it leads to the dilemma that first the students have to learn these historical wrong theories and then, when it comes to "modern quantum theory", have to explicitly taught to unlearn it again. So it's a waste of time, which you need to grasp the mind-boggling discoveries of modern quantum theory. It's not so much the math of QT but the intuition you have to get by solving a lot of real-world problems. Planck once has famously said that the new "truths" in science are not estabilished by converting the critiques against the old ones but because they die out. In this sense it's good to help to kill "old models" by not teaching them anymore.

Another thing are "wrong" models which still are of importance and which are valid within a certain range of applicability. One could say all physics is about is to find the fundamental rules of nature at some level of understanding and discovery and then find their limits of applicability ;-)). E.g., one has to understand classical (non-relativistic as well as relativistic) physics (point and continuum mechanics, E+M with optics, thermodynamics, gravity), because without it there's no chance to understand quantum theory, which we believe is comprehensive (except for the lack of a full understanding of gravity), but this also only means we don't know its limits of application yet or whether there are any such limits or not (imho it's likely that there are, but that's a personal belief).

As for the question, why there's (sometimes) a "delay" in the propagation of electromagnetic waves through a medium, classical dispersion theory in the various types of media is a fascinating topic and for sure should be taught in the advanced E+M courses. You get, e.g., the phenomenology of wave propagation in dielectric insulating media right by making the very simple assumption that a (weak) electromagnetic fields distort the electrons in the medium a bit from the equilibrium positions, which leads to a back reaction that can be described effectively by a harmonic-oscillator and a friction force. You get a good intuitive picture, which is not entirely wrong even when seen from the quantum-theoretical point of view. The classical theory is best explained in Sommerfeld's textbook on theoretical physics vol. IV. There's also a pretty good chapter in the Feynman Lectures, but I've to look up at the details of the mentioned intuitive explanation in that book. Of course, a full understanding needs the application of quantum theory, and you can get pretty far by working out the very simple first-order perturbation theory for transitions between bound states. You can also get quantitative predictions for the resonance frequencies and the oscillator strengthts in the classical model. A full relativistic QED treatment is possible (and necessary), e.g., for relativistic plasmas (as the quark-gluon plasma created in ultrarelativistic heavy-ion collisions), where you have to evaluate the photon self-energy to find the "index of refraction".

In any case you learn, that you have to refine your idea of "the wave gets delayed". The question is what you mean by this, in other words, what you consider as the signal-propagation speed. That's not easy. There is first of all the phase velocity, which usually gets smaller than the vacuum speed of light by a factor of $1/n$, $n$ is the index of refraction. Nevertheless $\mathrm{Re} n$ (usually a complex number) does not need to be $>1$, and the phase velocity can get larger than $c$. Another measure is the group velocity, which (when applicable at all!) describes the speed of the center of a wave packet through the medium. Usually it's also smaller than $c$ although in regions of the em. wave's frequency close to a resonance frequency of the material, that's not true anymore and it looses its meaning, because the underlying approximation (saddle-point approximation of the Fourier integral from the frequency to the time domain) is not applicable anymore (anomalous dispersion). The only speed which has to obey the speed limit is the "front velocity", which describes the speed of the wave front. In the usual models it turns out to be the vacuum speed of light, as was found famously by Sommerfeld as an answer to a question by W. Wien concerning the compatibility with the known fact that the phase and group velocities in the region of anomalous dispersion can get larger than $c$ with the then very new Special Theory of Relativity (1907). This was further worked out in great detail by Sommerfeld and Brillouin in two famous papers in "Annalen der Physik", which are among my favorite papers on classical theoretical physics.

 

[martinbn]

Well, do you at least agree with terminology like the Schroedinger equation for 1 particle, or the Schroedinger equation for two particles?

It is not a question whether I agree with terminology or not. The problem is that I don't understand your comment and was asking for a clarification. One of the things you said was that the Hilbert space has particle nature. I still don't know what you meant. It seems that you imply that a particular basis of the Hilbert space gives its nature. But that cannot be, that's why I am confused. The space has various bases (infinitely many) choosing a basis doesn't change the space nor its nature.

 

[ShayanJ]

It is not a question whether I agree with terminology or not. The problem is that I don't understand your comment and was asking for a clarification. One of the things you said was that the Hilbert space has particle nature. I still don't know what you meant. It seems that you imply that a particular basis of the Hilbert space gives its nature. But that cannot be, that's why I am confused. The space has various bases (infinitely many) choosing a basis doesn't change the space nor its nature.

I think you're looking at what atyy said too mathematically,which isn't strange, you're a mathematician!
You're right that there is nothing "particlish" about Hilbert spaces. In fact, mathematically, what atyy says is meaningless which is the source of the fact that you don't understand him. But I, as a physics student, understand what he means and actually think he's right. The point is, the mathematics used in a theory is a bit different from the mathematical formulation of that theory. The mathematical formulation of a theory has some interpretations attached to it. I mean how you relate the mathematical concepts to the physical concepts. What atyy is saying, is that in QM, we acknowledge the existence of particles and give them physical meaning. So in our mathematical formulation, we relate some concepts of the mathematics used in our theory, to particles. We give each particle its own wavefunction and define operators to act on only one of the particles. Of course we can have non-separable operators(I guess!) but we start with thinking in terms of individual particle. So I should say what atyy said doesn't concern Hilbert spaces, but how we relate physical concepts to Hilbert spaces.I hope this clarifies the issue.

 

[atyy]

It is not a question whether I agree with terminology or not. The problem is that I don't understand your comment and was asking for a clarification. One of the things you said was that the Hilbert space has particle nature. I still don't know what you meant. It seems that you imply that a particular basis of the Hilbert space gives its nature. But that cannot be, that's why I am confused. The space has various bases (infinitely many) choosing a basis doesn't change the space nor its nature.

Would you agree that the wave functions for the 1 particle and 2 particle Schroedinger equations belong to different Hilbert spaces?

 

[TrickyDicky]

Would you agree that the wave functions for the 1 particle and 2 particle Schroedinger equations belong to different Hilbert spaces?

I think that the source of the confusion(as shyan points out) is that mathematically those Hilbert spaces are isomorphic, so they are not different Hilbert spaces. But in QM they are different by particularising a basis, and this is inherent to quantization itself. It just shows one way in which QM is not mathematically well defined.

 

[atyy]

I think that the source of the confusion(as shyan points out) is that mathematically those Hilbert spaces are isomorphic, so they are not different Hilbert spaces. But in QM they are different by particularising a basis, and this is inherent to quantization itself. It just shows one way in which QM is not mathematically well defined.

Yes, they are. But if you say ψ(x) is the wave function for two particles - as is certainly permitted by the isomorphism between Hilbert spaces, then the commutation relations for the positions and momenta of the particles will not be the canonical commutation relations. This is why we do say that ψ(x) is the wave function for 1 particle, and ψ(x,y) is the wave function for two particles.

 

[TrickyDicky]

Yes, they are. But if you say ψ(x) is the wave function for two particles - as is certainly permitted by the isomorphism between Hilbert spaces, then the commutation relations for the positions and momenta of the particles will not be the canonical commutation relations. This is why we do say that ψ(x) is the wave function for 1 particle, and ψ(x,y) is the wave function for two particles.

Exactly.

 

[TrickyDicky]

It's just that this distinction cannot be accommodated by the Hilbert space model, therefore ambiguities arise that lead to all the well known interpretational problems(factorization, entanglement, Schrodinger's cat, ...).
No wonder mathematicians feel confused about what Hilbert spaces have to do with particles.