Insights Some sins in physics didactics - comments

[Greg Bernhardt]

I would also like to propose some sins in physics didactics:

- The Michelson-Morley experiment is taught to be a proof that aether does not exist. Nevertheless, this experiment by itself does not prove it. The experiment does not exclude the possibility that the aether is dragged by the Earth. (Of course, there are other experiments that exclude this possibility, but not the Michelson-Morley one.)

- The spinorial transformation of the Dirac wave function is often taught as being derived from the Dirac equation. However, the Dirac equation does not really imply the spinorial transformation. The Dirac equation allows also a (physically equivalent) alternative, according to which the wave function transforms as a scalar:
http://lanl.arxiv.org/abs/1309.7070 [Eur. J. Phys. 35, 035003 (2014)]

- In 1930 Einstein proposed the photon-in-the-box thought experiment, which was supposed to demonstrate an inconsistency of the time-energy uncertainty relations. The Bohr's resolution of the problem, based on adopting some principles of general relativity, is often taught to be the correct way to save consistency of the time-energy uncertainty relations. But it is not. The correct resolution of the photon-in-the-box paradox, similarly to the latter EPR paradox, is the non-local nature of quantum correlations:
http://lanl.arxiv.org/abs/1203.1139 [Eur. J. Phys. 33 (2012) 1089-1097]

Maybe a follow up entry? :)

 

[ZapperZ]

I would also like to propose some sins in physics didactics:

- The Michelson-Morley experiment is taught to be a proof that aether does not exist. Nevertheless, this experiment by itself does not prove it. The experiment does not exclude the possibility that the aether is dragged by the Earth. (Of course, there are other experiments that exclude this possibility, but not the Michelson-Morley one.)

I disagree. That's like saying "OK, so you found that there's no unicorn. But that was because you were looking for 4-legged unicorns. What if there are 2-legged unicorns?"

The MM-experiment was specifically testing a particular characteristic of light, and based on what was described at that time, it tested it perfectly well. Besides, if you bring the same setup to the ISS, the MM-experiment is equally up to the challenge to even test the ether drag. So the experiment in itself is adequate.

Zz.

 

[Ken G]

This article is suggesting that the photo-electric effect doesn't actually prove anything about the quantization of the electromagnetic field; the quantization of energy levels of matter is sufficient to explain it. So does ANYTHING prove the quantization of the E&M field? I guess not, because Feynman's "absorber theory" reformulates QED so that there are no additional degrees of freedom in the E&M field.

On the other hand, it seems strange to treat matter (fermions) completely different than gauge particles, when their physics is so similar.

I think a big issue is the question, what is the "quantum" in "quantum mechanics?" We might say it's first quantization, and then the quantum in "quantum field theory" is second quantization. But first quantization doesn't give us photons, it just gives us the analysis vanhees71 gave. So his remarks can be interpreted as suggesting that we separate what experiments support the theory of first quantization from the experiments that support second quantization, and not simply follow the historical path there. I think we must agree that had Bohr come up with his model of the atom before Einstein did the photoelectric effect experiment, then that experiment is just a way to generalize the concepts of first quantization to other regimes. There might not be any hint that second quantization is needed, so if we teach it the historical way, we are promoting misconceptions about the differences between these two brands of "quanta".

 

[rude man]

" ... the introduction of a velocity-dependent mass in special relativity, which is a relic from the very early years after Einstein’s ground-breaking paper of 1905. "

The statement is incorrect. See below.

I have never liked the elimination of rest mass as a separate parameter. It changes several formulae that were accurate before this change, not the least being E = mc^2 for a moving particle.

If it was good enough for Richard Feynman it's good enough for me. Reminder: the milennial edition of "The Feynman Lectures on Physics" was issued just a year or two ago. It includes significant revised material from earlier editions but the use of rest mass as a separate parameter was retained. And wisely so IMO.

 

[ZapperZ]

" ... the introduction of a velocity-dependent mass in special relativity, which is a relic from the very early years after Einstein’s ground-breaking paper of 1905. "

The statement is incorrect. See below.

I have never liked the elimination of rest mass as a separate parameter. It changes several formulae that were accurate before this change, not the least being E = mc^2 for a moving particle.

If it was good enough for Richard Feynman it's good enough for me. Reminder: the milennial edition of "The Feynman Lectures on Physics" was issued just a year or two ago. It includes significant revised material from earlier editions but the use of rest mass as a separate parameter was retained. And wisely so IMO.

I don't what's "incorrect" about that. In fact, check out one of my earlier posting about this:

https://www.physicsforums.com/threads/relativistic-mass.642188/#post-4106101

Note that even Einstein later on stopped using it.

Zz.

 

[atyy]

I don't know. It looks like it is employing the dipole transition matrix for each transition due to photon absorption. That smells very much like it already assumes the photon model.

I think that although the dipole approximation is used, it is a treatment in which the EM field is not quantized. The Hamiltonian they use for the two-photon process is http://cua.mit.edu/8.421_S06/Chapter9.pdf (Eq 9.3), which looks to me of the same form as http://cua.mit.edu/8.421_S06/Chapter7.pdf (Eq 7.31), which has a classical EM field. In their notation if the EM field is quantized, I would expect to see an expression more like their Eq 7.46.

BTW, here is a reference that I have on an example of multiphoton photoemission. Look at Eq. 1 and how it manifests itself as the slope of the charge with light intensity.

http://qmlab.ubc.ca/ARPES/PUBLICATIONS/Articles/multiphoton.pdf

That is interesting. I had to look up the Fowler-Dubridge theory they mention, on which the BSB theory is based. It basically assumes the E=hf from old quantum theory like Planck and Einstein.

 

[atyy]

Of course, you can treat the photoeffect also with single photons. For that you have to quantize the electromagnetic field. The only difference at this order is that for the excited bound states there's a transition probability from an excited (bound) state to a lower state under emission of (one or more) photons, spontaneous emission, and that's why Planck's Law shows the necessity for the quantization of the em. field, as Einstein has figured out in 1917 from another semiclassical argument within old quantum theory. To get the correct radiation law, he had to assume spontaneous emission, and that was later explained by Dirac when introducing the formalism for non-conserved "particle numbers" in terms of creation and annihilation operators.

Probably the strongest argument for teaching the "old quantum theory" view of E = hf and the photoelectric effect using E = hf is that the photoelectric effect is still how we detect single photons!

 

[stevendaryl]

- The spinorial transformation of the Dirac wave function is often taught as being derived from the Dirac equation. However, the Dirac equation does not really imply the spinorial transformation. The Dirac equation allows also a (physically equivalent) alternative, according to which the wave function transforms as a scalar:
http://lanl.arxiv.org/abs/1309.7070 [Eur. J. Phys. 35, 035003 (2014)]

I seem to recall a somewhat heated discussion the last time this was brought up, but I don't remember what the objections were.

It seems that there are maybe three approaches to understanding the gamma-matrices in the Dirac equation:

1. They are just four constant matrices, and the index does not imply that they form a vector.
2. They are matrix-valued components of a 4-vector.
3. Each gamma matrix is a vector. The index \mu in \gamma_\mu indicates which vector, rather than which component.

I'm not 100% sure whether the third approach is well-worked-out, but it is the approach taken by Hestenes in his "geometric algebra", which is inspired by Clifford algebras. In a Clifford algebra, the anticommutation relation

e_\mu e_\nu + e_\nu e_\mu = 2 g_{\mu \nu}

is supposed to hold for basis vectors e_\mu; the \mu indicates which basis vector, rather than which component.

 

[carllooper]

Explanations in physics, (be they old or new explanations), are for many people themselves in need of an explanation.

Put an equation in front of many people and it won't explain anything.

There is a case for pointing out to students particular experiments (observations) which can act as inspiration for particular explanations, regardless of whether such explanations are deemed today as correct or otherwise.

Now we can not prove that any particular experiment has historically inspired any particular explanation or understanding. But nor is that the goal. The goal is to identify those experiments which could have inspired a theory, or could re-inspire that same theory ... re-inspire the very understanding we might be otherwise entertaining and expressing in an otherwise difficult equation.

We can explain an explanation in this way.

C

 

[rude man]

" ... the introduction of a velocity-dependent mass in special relativity, which is a relic from the very early years after Einstein’s ground-breaking paper of 1905. "

The statement is incorrect. See below.

I have never liked the elimination of rest mass as a separate parameter. It changes several formulae that were accurate before this change, not the least being E = mc^2 for a moving particle.

If it was good enough for Richard Feynman it's good enough for me. Reminder: the milennial edition of "The Feynman Lectures on Physics" was issued just a year or two ago. It includes significant revised material from earlier editions but the use of rest mass as a separate parameter was retained. And wisely so IMO.

I don't what's "incorrect" about that. In fact, check out one of my earlier posting about this:

https://www.physicsforums.com/threads/relativistic-mass.642188/#post-4106101

Note that even Einstein later on stopped using it.

Zz.

Well, you said it was a relic from the early years of 1905. Feynman taught the course in question at Caltech in the '60's.

I'm aware Einstein later changed his mind but Feynman certainly did not.

 

[rude man]

Well, you said it was a relic from the early years of 1905. Feynman taught the course in question at Caltech in the '60's.

I'm aware Einstein later changed his mind but Feynman certainly did not.

 

[ZapperZ]

Well, you said it was a relic from the early years of 1905. Feynman taught the course in question at Caltech in the '60's.

I'm aware Einstein later changed his mind but Feynman certainly did not.

I didn't say anything about a relic.

So how do you decide who to listen to? The one better looking and with less messy hair?

Zz.

 

[carllooper]

The idea that previous explanations for something are "relics" is typical of the belief that the past is no longer relevant - that contemporary theory is the only theory that should be entertained. As if theory as a whole should be whatever is currently fashionable - that anything older than this morning should be put out with the rubbish.

I've read criticisms of some theories (even on this forum) where the critique is literally no more than: "that's old fashioned".

Well, Einstein's Relativity Theory is old fashioned. It's more than a 100 years old.

The age of a theory has no bearing whatsoever on it's value. If a theory is wanting it won't be necessarily due to it's age. And there are plenty of freshly minted theories which could be framed as wanting.

Another critical angle is this notion of "correctness" or "truth value" - that the value of a theory is in terms of how correct or true it is.

No theory is correct. No theory is true.

Theories are particular ways of understanding the way in which nature works. Nature herself doesn't care. She behaves the way she behaves regardless of whatever theory we might develop. How we understand her behaviour is an entirely different thing - be it on a simple approximate level or in enormous detail. The value of a theory is to be found in what it might allow - on various levels - technology being the most powerful driver of theories (of the fashionable variety) but by no means the only driver.

The history of a theory is important in how a theory is to be understood. The genesis of Einstein's Theory of Relativity didn't just spring out of thin air. It has a context in which ideas such as the aether fed into such, and the Michelson-Morley experiment, and so on, each of which help to understand Einstein's theory and why it emerged at that time and why it is the way it is.

Getting in the way of this are myths about personal genius (Einstein as a genius). They distract from understanding why theories (fashionable or otherwise) are the way they are. When cleansed of all historical context they appear either silly or genius, neither of which are true.

C

 

[bhobba]

I know first hand in answering questions on this forum the many misconceptions people have because of popularisations and beginner texts.

Feynman was aware of it and, if I recall correctly, has a section devoted to it somewhere in his lectures. He laments you can't always tell the students the truth from the start, but doesn't know any other way of resolving the issue - they simply do not have the background for the complete story.

I see no reason we can't keep doing the same thing, but simply, like Feynman, have the occasional lecture explaining some of the stuff they are learning will need to be unlearned later, that's simply the way physics is, nothing much can be done about it, but just be aware that's the case.

That doesn't mean of course we shouldn't look at physics curriculum to ensure students need to unlearn as little as possible.

Thanks
Bill

 

[atyy]

No, of course one should never teach anything which may require unlearning later. That does not mean one should not teach old quantum theory.

 

[bhobba]

I'm aware Einstein later changed his mind but Feynman certainly did not.

I find that Feynman didn't understand the issues of relativistic mass a little difficult to fathom. What happens when you apply a force in the direction of motion? What mass do you use then? And such being the case how does that gell with the usual concept of mass being a scalar?

Thanks
Bill

 

[atyy]

I find that Feynman didn't understand the issues of relativistic mass a little difficult to fathom. What happens when you apply a force at right angles to the direction of motion? What mass do you use then? And such being the case how does that gell with the usual concept of mass being a scalar?

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

 

[vanhees71]

This article is suggesting that the photo-electric effect doesn't actually prove anything about the quantization of the electromagnetic field; the quantization of energy levels of matter is sufficient to explain it. So does ANYTHING prove the quantization of the E&M field? I guess not, because Feynman's "absorber theory" reformulates QED so that there are no additional degrees of freedom in the E&M field.

On the other hand, it seems strange to treat matter (fermions) completely different than gauge particles, when their physics is so similar.

Yes, as stressed in the article and many times in this discussion, e.g., the Planck Black Body radiation law proves the quantization of the electromagnetic field, because you need spontaneous emission to derive it from kinetics, as was found by Einstein already in 1917, but there he had to introduce spontaneous emission ad hoc, while in QFT it's derived from the bosonic nature of the em. field (symmetry under exchange of identical bosonic quanta). The analogue for fermions is Pauli blocking, which was introduced by Pauli (as the name correctly suggest) in an ad hoc way also before the discovery of modern quantum theory and is nowadays implied by the fermion many-body space (antisymmetry under exchange of identical fermionic quanta).

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").

 

[bhobba]

No, of course one should never teach anything which may require unlearning later. That does not mean one should not teach old quantum theory.

Ok - then Feynman's QED - The Strange Story of Light And Matter needs to be banned eg its explanation of why light moves slower in glass is wrong:
https://www.physicsforums.com/threads/do-photons-move-slower-in-a-solid-medium.511177/

Should one expose beginning students to Zappers correct explanation and forget the intuitive incorrect one? Would beginning students even understand what Zapper said?

Like I said - Feynman was a teacher of some renown, and grappled with the issue. He decided students, correctly IMHO, need to be eased into the correct understanding.

Thanks
Bill

 

[atyy]

Ok - then Feynman's QED - The Strange Story of Light And Matter needs to be banned eg its explanation of why light moves slower in glass is wrong:
https://www.physicsforums.com/threads/do-photons-move-slower-in-a-solid-medium.511177/

Should one expose beginning students to Zappers correct explanation and forget the intuitive incorrect one? Would beginning students even understand what Zapper said?

Like I said - Feynman was a teacher of some renown, and grappled with the issue. He decided students, correctly IMHO, need to be eased into the correct understanding.

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.

 

[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.