Insights Some sins in physics didactics - comments

[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.

 

[atyy]

Well, it's interesting to try to discuss a rigrourous version later. But the basic idea is that in physics speak, ψ(x) is the wave function for one particle, but it is not the wave function for 2 particles, and ψ(x,y) is the wave function for two particles, but it is not the wave function for one particle.

If we can at least agree that this is meaningful, then it is obvious that the Schroedinger equation for 1 particle is correctly named and it is different from the Schroedinger equation for 2 particles. The Schroedinger equation is obviously a wave equation, and which Schroedinger equation we use is specified by the number of particles. So the Schroedinger equation for N particles is a formlization of the heuristic concept of wave-particle duality.

We should at least agree on this idea before discussing what conditions we need to add to make it rigrourous. It is clear that the isomorphism of Hilbert spaces is an objection that can be overcome by adding some conditions if one is interested in rigour, since by the isomorphism, the single particle Hilbert space is also the Hilbert space of Yang-Mills and the Hilbert space of quantum gravity.

 

[TrickyDicky]

Hmm, isn't this issue what demanded going to relativistic QFT to begin with(and its own issues with rigour).
Strictly speaking the Schrodinger equation is a "one particle" equation. You add more particles and all hell breaks loose, you have to account quantum mechanically with possible interactions between them also, or simply go for the semiclassical approximation if it works, but then the model is not purely quantum...

 

[atyy]

The Schroedinger equation for N particles is perfectly fine, as long as N is finite.

 

[TrickyDicky]

The Schroedinger equation for N particles is perfectly fine, as long as N is finite.

You mean "fine" mathematically or physically?
Mathematically is fine if you don't distinguish one particle from N particles, that's the $L^2$ isomorphism. Wich leads to martinbn questions.
Physically is fine of course, think of condensed matter physics. Then again there is no pretense whatsoever of mathematical rigour(or even physical, being a nonrelativistic approximation) in the sense we are discussing about Hilbert spaces in condensed matter physics.

 

[vanhees71]

Well, you can as well argue that nonrelativistic quantum theory is a quantized field theory, i.e., the corresponding Hilbert space shows that it describes fields. Just read the Schrödinger equation as a field equation, write it in terms of an action functional and then quantize it in some formalism (canonical operator quantization or path-integral quantization or whatever you prefer). Then you realize, there are conserved currents, which can be interpreted as conservation laws for particle numbers for many interactions occurring when describing real-world situations. This implies that you can formulate everything in the subspace of a fixed particle number, and that's equivalent to the "first-quantization formalism" for identical bosons or fermions.

There's no way to a priori say, you describe particles or fields. You describe quanta, and that's what it is. There are some aspects which you'd consider as "particle like" and some that are "wave like". It simply depends on the observables you look at, but there's no "wave-particle duality" but a consistent probabilistic theory called quantum theory that precisely describes these particle or wave-like aspects (or some aspects that are neither nor such as, e.g., entanglement and the corresponding strong non-classical correlations).

 

[atyy]

Well, you can as well argue that nonrelativistic quantum theory is a quantized field theory, i.e., the corresponding Hilbert space shows that it describes fields. Just read the Schrödinger equation as a field equation, write it in terms of an action functional and then quantize it in some formalism (canonical operator quantization or path-integral quantization or whatever you prefer). Then you realize, there are conserved currents, which can be interpreted as conservation laws for particle numbers for many interactions occurring when describing real-world situations. This implies that you can formulate everything in the subspace of a fixed particle number, and that's equivalent to the "first-quantization formalism" for identical bosons or fermions.

There's no way to a priori say, you describe particles or fields. You describe quanta, and that's what it is. There are some aspects which you'd consider as "particle like" and some that are "wave like". It simply depends on the observables you look at, but there's no "wave-particle duality" but a consistent probabilistic theory called quantum theory that precisely describes these particle or wave-like aspects (or some aspects that are neither nor such as, e.g., entanglement and the corresponding strong non-classical correlations).

Alternatively, that is what we mean by wave-particle duality! Changing the name from particle to "quanta" is just a game, when everyone calls them "particles" and uses terms like "1 particle subspace". Also the equation of motion is a wave equation.

To be consistent, you should say "particle physics" is a myth, and the "Schroedinger equation for N particles" is a myth, since there are no particles, only quanta.

Here is another myth: http://pdg.lbl.gov/.

 

[carllooper]

The term "particle" has become shorthand for both wave-like and Newtonian particle-like behaviour. One uses the term "particles", but the Newtonian particle aspect of which is limited to the point of absorption (and emission). The propagation of the particle (so called) through a vacuum is otherwise modeled as a wave (using the wave function). This obviously goes against what we would otherwise intuit from a particle-like detection. We'd otherwise intuit a particle-like object (a ray of light so to speak) as that which created a particle-like detection (had we been born a 100 years ago or otherwise a newbie to this sort of thing). If we opt for a wave model it's purely because, in addition to the particle-like detections (that we can clearly see), there are also wave-like aspects to the detections as well - not immediately obvious given just a few detections. For on the one hand we can clearly see each of the individual detections (absorptions) which we can clearly characterise in terms of a point like descriptor, eg. we can assign each detection a precise point in space and time. But on the other hand, (once we remove our blinders, or our fetish for localisable phenomena) we can also clearly see the distribution of said point-like detections (the pattern they form). But we can't describe this pattern in terms of a "rays of light" model. What we can do is characterise this pattern in terms of a wave function model. Now while we can clearly see (in the sensory sense) both phenomena (ie. each individual detection and their ensemble distribution), we nevertheless have difficulties reconciling such clear information in terms of a model that would be internally consistent (ie. a purely mathematical model).

Now all of this is really "newbie" stuff - but that is what history provides - it provides a perfect context in which newcomers can come face to face with the same problems and the same possible answers that faced, and occurred to, Einstein and Bohr (to name but two). For they too were newbies. They were working from scratch (in terms of creating a viable quantum theory). What is at issue is not whether the models they created were (or are) correct (that is of course something to investigate in due course), but why these models were created in the first place: what is the actual problem that such models were (or are) hoping to solve?

Historical models (and the experiments that inspired them) provide a way to understand the problem.

More complex solutions (or models) become easier to understand once you grasp the problem (so called) behind such solutions.

C

 

[atyy]

From another thread Wick theorem in "QFT for the Gifted Amateur"

On the other hand, condensed-matter theory usually uses QFT as a true many-body theory, i.e., you look at systems which contain many particles and not like in relativistic vacuum QFT as used in high-energy particle physics, with one or two particles in the initial state and a few particles in the final state, where you calculate cross sections and the like.

Hmmm, there is a "true many-body theory" in quantum mechanics?

 

[atyy]

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.

Well, ok it's subjective. We certainly both agree that one should not teach that the photoelectric effect "proves" the existence of photons, and I'm happy to let the teacher choose his syllabus. But hopefully that you agree it's subjective means that it is also fine to teach old quantum theory, provided it can be taught in a way that is not misleading.

For example, whereas you prefer you prefer to say there is no wave-particle duality because QM is a consistent theory, I prefer to say wave-particle duality is a vague historical heuristic which is implemented in QM as a consistent theory.

But a point of disagreement is that you stress that QM is "mind-boggling". I think that is a myth. QM is almost 100 years old now, and I don't think it should be taught as any more mind-boggling than classical physics. In fact, I personally find classical physics much more mind-boggling - rolling motion is really difficult, and I always have to look up the Maxwell relations in thermodynamics. QM does have the measurement problem, but most of what people consider mind-boggling comes after one has chosen the apparatus and system, ie. operators and Hilbert space, whereas the measurement problem comes before that.

 

[phion]

Wow.

 

[Demystifier]

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.

Note that atyy is also not a physicist, but a biologist. Even though he knows about physics more than many physicists, it is really funny to see how a mathematician and a biologist talk about quantum physics without being able to understand each other, essentially because they come from communities (biologists vs mathematicians) with very different standards of precision in scientific talk.

Anyway, Hilbert space, as such, does not have a "particle nature". But wave functions $\psi(x1,...,xn)$, which can be thought of as vectors in the Hilbert space, represent the probability amplitude that n particles have positions x1,...,xn. More precisely, the probability density is
$|\psi(x1,...,xn)|^2$. Is that precise enough?

 

[atyy]

Note that atyy is also not a physicist, but a biologist. Even though he knows about physics more than many physicists, it is really funny to see how a mathematician and a biologist talk about quantum physics without being able to understand each other, essentially because they come from communities (biologists vs mathematicians) with very different standards of precision in scientific talk.

Lowly biologists are the most precise. Physicists* are imprecise because they have the measurement problem, and to place the cut they have to use the intuitive, non-rigourous language of biologists. Mathematicians are imprecise because to even define ZFC, one needs the metalanguage, which is again basically the intuitive, non-rigrourous language of biologists.

*Bohmians excluded

 

[ShayanJ]

Even though he knows about physics more than many physicists

It really scares me to think about how much biology he knows!

 

[atyy]

It really scares me to think about how much biology he knows!

I measure electrical signals so I only need to know 4 equations (usually less than that, but knowing about electromagnetic waves is useful for getting rid of noise).

 

[martinbn]

Note that atyy is also not a physicist, but a biologist. Even though he knows about physics more than many physicists, it is really funny to see how a mathematician and a biologist talk about quantum physics without being able to understand each other, essentially because they come from communities (biologists vs mathematicians) with very different standards of precision in scientific talk.

Yes, I do know that she is a biologist. The miscommunication is interesting, but I have that with physicists as well. It may not be related to me being a mathematician, but a bourbakist. Once I talked to a student of Arnold, and it took a good part of an hour of interrogations before I forced him to formulate something about ergodic theory in a way that I was happy about.

Anyway, Hilbert space, as such, does not have a "particle nature". But wave functions $\psi(x1,...,xn)$, which can be thought of as vectors in the Hilbert space, represent the probability amplitude that n particles have positions x1,...,xn. More precisely, the probability density is
$|\psi(x1,...,xn)|^2$. Is that precise enough?

This is something I know and I understand and your first sentence clarifies my confusion. But then why couldn't atyy simply say that Hilbert spaces have no particle nature and explain what she meant! It would have saved us quite a few posts.