Insights Some sins in physics didactics - comments

[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.

 

[vanhees71]

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.

With mind-boggling I don't mean mathematically difficult. For me the math to learn for E+M or GR was more of a challenge than to learn that of QM 1. There I had (and sometimes still have) more problems with the physics intuition, and a first step was to unlearh "old quantum theory". :-)

 

[vanhees71]

From another thread Wick theorem in "QFT for the Gifted Amateur"
Hmmm, there is a "true many-body theory" in quantum mechanics?

Well, I just meant that there you really deal with (very) many particles, while in HEP you usually deal with just a few in scattering processes. Of course there's the fascinating case, where both comes together in relativistic many-body theory, as I need it in my research.

 

[atyy]

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.

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.

Alternatively, that is what one means by the "particle nature" of the Hilbert space. I mean, one can formulate things rigourously, but I deliberately was trying to avoid that to get the intuitive idea across first.

As a Bourbakist, you should know that the most important concepts are true by definition :)

(If you want the rigourous view, you can probably say first we decide we have a system of N particles, where N is an integer and that integer nature implies discrete entities which is what we mean by particle, then from there we decide we have observables and canonical commutation relations for the N particles which in the framework of quantum mechanics via the Stone-von Neumann theorem picks the group representation which picks the Hilbert space representation ... but the intuition should come first - we do mean something well-defined enough when we talk about the Schroedinger equation for N particles, ie. a wave equation for particles ... anyway, I glad you are happy with Demystifier's explanation, since what he says as an acceptable interpretation of my words.)

 

[atyy]

Well, I just meant that there you really deal with (very) many particles, while in HEP you usually deal with just a few in scattering processes. Of course there's the fascinating case, where both comes together in relativistic many-body theory, as I need it in my research.

Yes, yes, just teasing you there for talking about particles after you said there are no particles, only quanta :)

With mind-boggling I don't mean mathematically difficult. For me the math to learn for E+M or GR was more of a challenge than to learn that of QM 1. There I had (and sometimes still have) more problems with the physics intuition, and a first step was to unlearh "old quantum theory". :-)

Yes, it's unfortunate :) that your teacher only told you it was nonsense after you had learned it, and before you started learning the proper quantum formalism.

My teacher told me old quantum theory is nonsense before teaching it, so there was nothing to unlearn. I do agree that old quantum theory should not be taught in a way in which it has to be unlearnt.

 

[vanhees71]

You have a point. I'm preaching water and drinking wine in still talking about "particles", but of course everybody talks about particles. Physicists of course understand particles usually in the right way as being described by quantum (field) theory and not as microscopic bullet-like classical entities.

 

[Demystifier]

Yes, I do know that she is a biologist.

She? I didn't know atyy is "she".
Now I like atyy even more.

 

[Demystifier]

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.

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

I think atyy is to physics and biology what second-order logic is to set theory and logic. For those who have no idea what that means, I want to say that atyy is a physicist in a sheep's clothing (just like, according to Quine, second-order logic is set theory in sheep's clothing), where "sheep" stands for either biology or logic.

(I hope that at least martinbn will appreciate the abstract-nonsense structure-preserving mapping above.)

 

[martinbn]

She? I didn't know atyy is "she".
Now I like atyy even more.

Just a guess, I don't know for sure.

 

[ShayanJ]

Just a guess, I don't know for sure.

I think you went for the flowers in his avatar. I don't know for sure too, but statistics tell me he's a male!

 

[vanhees71]

Well, what surprises me more is that atyy is a biologist rather than a (quantum) physicist. I've never met a biologist which such deep knowledge about quantum theory. The gender of a scientist is, in my opinion, totally irrelevant concerning the science done by that person!

 

[martinbn]

I think you went for the flowers in his avatar. I don't know for sure too, but statistics tell me he's a male!

No, I didn't. Something else is the tell. I didn't mean to reviel personal information, I thought everyone knew since it is obvious.

 

[atyy]

Hmmm, not that that it matters, but I'm male.

 

[Demystifier]

Hmmm, not that that it matters, but I'm male.

So you are a male biologist having all attributes of a female physicist? (Just kidding!)

 

[vanhees71]

Argh! Could we come back to physics?

 

[Demystifier]

Argh! Could we come back to physics?

Of course! Sorry!

 

[atyy]

You have a point. I'm preaching water and drinking wine in still talking about "particles", but of course everybody talks about particles. Physicists of course understand particles usually in the right way as being described by quantum (field) theory and not as microscopic bullet-like classical entities.

Yes, there is no classical particle with definite position and momentum (sticking to non-relativistic QM). But what the quantum notion preserves from the classical mechanics when we use "particle" as opposed to "continuum" is that there is a discrete number of entities N, so that it makes sense to refer to a number operator whose eigenvalues are integers. Also, in the classical limit, the quantum particle does become the classical particle, so I think "particle" is a good choice of terminology for the quantum entity that we talk about.

 

[vanhees71]

Yes, "particle" still is a good choice. Sometimes, particularly here in the forum when discussing about these fundamental quantum questions, I use "quantum" to emphasize that I talk about a quantum system. Where, I'd never use the word "particle" is when I talk about photons. Here, "photon" is the right word but should be exclusively understood in the sense of relativistic quantum field theory. Somewhat problematic is that it is often used in the sense of "particle", bur for photons this is so wrong that it is "not even wrong" in Pauli's sense. So we are back at my initial motivation for starting my "Didactical Sins" series of postings here in the Insights section with this particular example.

I guess the next entries will be about the sin to use "non-covariant representations in relativity" and (closely related) "against `hidden momentum'". ;-).

 

[andresB]

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.

I can't sympathize with this.

Of course the Bohr description is not valid, but remembering my time as undergraduate student it never mislead me into believing the old quantum theory was the right thing because my teacher (and the textbook) specifically warned me about that.

Now, how would you make the jump from classical physics to quantum physics?

 

[andresB]

Another example would be the Einstein clock int he box mental experiment. It would be wrong to show only Borh solution to the problem and stop at it.

But there is no problem with teaching Borh solution (it will not take more than 20 minutes anyways) and then to show the modern one in terms on nonlocality.

 

[Greg Bernhardt]

http://arxiv.org/abs/1309.7070 is an interesting read

 

[A. Neumaier]

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

See the new thread https://www.physicsforums.com/threads/818386/

 

[samalkhaiat]

I am a late comer to this thread and have not read all the posts. So, I apologize in advance if people already said what I am about to say.
1) Students learn about the Bohr-Sommerfeld quantization rule (BSQR) from the old quantum theory. They can not afford to “unlearn” the BSQR, as it is still extremely useful when the problem is too complicated. We are still using BSQR to quantize time-dependent solutions in field theory, to tackle the bound-state problem in QFT, to quantize the electric charge in field theory with non semi-simple gauge group, etc.
2) We perform the classical limit on the dynamics and operator algebra, but not on the Hilbert space \mathcal{H}. Itself, the Hilbert space is not an algebra, so we can not assign any meaning to the limit \lim_{\hbar \to 0} \mathcal{H}, even though the classical limit of the operators algebra in \mathcal{H} is a Poisson-Lie algebra.
3) In QM, the canonical commutation relations are nothing but statement about the homogeneity of the physical 3-Space.
4) There is nothing “wavy” in the Schrodinger equation i \partial_{t} | \psi \rangle = H | \psi \rangle. Waves, such as sound and EM waves, carry energy and momentum, while probability-wave (i.e. the wave function \langle x | \psi \rangle) does not carry energy or momentum.
5) Concepts such as “Particle Hilbert Space” and “Wave Hilbert Space” have no precise mathematical meanings. If you argue for “Particle Hilbert Space”, one can provide a “better” argument for “Wave Hilbert Space”: Without the Superposition Principle (i.e. waves and interference), “Hilbert Space” can not be linear vector space. However, even this argument is meaningless.
6) The axioms of any physical theory consist of two parts. There is (A) the abstract mathematical part, and (B) the physical part which maps the abstract entities introduced in (A) onto observations. Take QM as an example. In (A) we introduce \mathcal{H}, subsets of \mathcal{H} representing pure states, the algebra of bounded operators \mathcal{B}(\mathcal{H}), etc. While in (B) a) we talk about state-preparation procedure, b) introduce the average-value of the measurements of the observable (represented by self-adjoint operator) in the pure state (represented by the one-dimensional projection operator | \psi \rangle \langle \psi |), c) postulate the possible outcomes of a measurement and the frequency with which they are obtained, etc.
7) The classical notions of particles and waves are neither defined nor derived from the axioms of QM. The axioms are meant to describe (quantum) systems which we (the unfortunate creatures) are unable to perceive by our senses.

 

[A. Neumaier]

... the axioms of QM. The axioms are meant to describe (quantum) systems which we (the unfortunate creatures) are unable to perceive by our senses.

We (the blessed creatures) see huge quantum systems such as the sun, whose light and heat is produced by quantum processes, as well as small quantum systems such as single photons. We smell and taste molecules described by small but complex quantum systems, and we touch solids and liquids, large quantum systems described by elasticity equations and fluid dynamics, whose characteristics are computed from quantum statistical mechanics.

All these are covered by the axioms of quantum mechanics. My favorite set of axioms is described here.

 

[atyy]

I am a late comer to this thread and have not read all the posts. So, I apologize in advance if people already said what I am about to say.
1) Students learn about the Bohr-Sommerfeld quantization rule (BSQR) from the old quantum theory. They can not afford to “unlearn” the BSQR, as it is still extremely useful when the problem is too complicated. We are still using BSQR to quantize time-dependent solutions in field theory, to tackle the bound-state problem in QFT, to quantize the electric charge in field theory with non semi-simple gauge group, etc.
2) We perform the classical limit on the dynamics and operator algebra, but not on the Hilbert space \mathcal{H}. Itself, the Hilbert space is not an algebra, so we can not assign any meaning to the limit \lim_{\hbar \to 0} \mathcal{H}, even though the classical limit of the operators algebra in \mathcal{H} is a Poisson-Lie algebra.
3) In QM, the canonical commutation relations are nothing but statement about the homogeneity of the physical 3-Space.
4) There is nothing “wavy” in the Schrodinger equation i \partial_{t} | \psi \rangle = H | \psi \rangle. Waves, such as sound and EM waves, carry energy and momentum, while probability-wave (i.e. the wave function \langle x | \psi \rangle) does not carry energy or momentum.
5) Concepts such as “Particle Hilbert Space” and “Wave Hilbert Space” have no precise mathematical meanings. If you argue for “Particle Hilbert Space”, one can provide a “better” argument for “Wave Hilbert Space”: Without the Superposition Principle (i.e. waves and interference), “Hilbert Space” can not be linear vector space. However, even this argument is meaningless.
6) The axioms of any physical theory consist of two parts. There is (A) the abstract mathematical part, and (B) the physical part which maps the abstract entities introduced in (A) onto observations. Take QM as an example. In (A) we introduce \mathcal{H}, subsets of \mathcal{H} representing pure states, the algebra of bounded operators \mathcal{B}(\mathcal{H}), etc. While in (B) a) we talk about state-preparation procedure, b) introduce the average-value of the measurements of the observable (represented by self-adjoint operator) in the pure state (represented by the one-dimensional projection operator | \psi \rangle \langle \psi |), c) postulate the possible outcomes of a measurement and the frequency with which they are obtained, etc.
7) The classical notions of particles and waves are neither defined nor derived from the axioms of QM. The axioms are meant to describe (quantum) systems which we (the unfortunate creatures) are unable to perceive by our senses.

My remarks were made in the context of wave-particle duality. Is this a misleading concept, or does it actually have a counterpart in proper quantum theory? My argument is that wave-particle duality does have a counterpart in proper quantum theory. There are no classical particles, only quantum particles. Unlike classical particles, quantum particles do not have simultaneously well-defined position and momentum. We call these quantum things particles for two reasons. First the particle number is an integer that we use to write down the commutation relations, and their representations, so that the proper theory does contain concepts such as the wave function of one particle or the wave function of two particles. Because particle number is an integer, it refers to discrete entities, just as classical particles are discrete entities. We call the quantum entity a particle, because in the classical limit, say in the path integral picture, the path of this thing is the path of the classical particle. So that justifies the particle aspect of quantum theory. I have already used the wave aspect in referring to the classical limit, but to use it again, the Schroedinger equation is a wave equation, so that justifies the wave aspect of quantum theory. So wave-particle duality does have its place in proper quantum mechanics, even though it is a loose concept of old quantum theory.

 

[samalkhaiat]

We (the blessed creatures) see huge quantum systems such as the sun

Yes, thanks for its hugeness. So, why can't we explain the sun orbital motion using QM?

as well as small quantum systems such as single photons.

Did we? What does a single photon look-like? Is it rounded like football?
A glass full of liquid Helium is very much a quantum system, but to our senses it is no more that a glass full of very cold liquid .
Dear sir, my post contains no inaccurate or confusing statement, and by the piece you quoted I meant the following: We evolved to sense the macroscopic (classical) world and invented language to describe what we see, hear, feel, smell and taste. We are unfortunate because we cannot form a mental picture for the electron but, thanks to mathematics, we can live with that misfortune.

 

[samalkhaiat]

My remarks were made in the context of wave-particle duality. Is this a misleading concept, or does it actually have a counterpart in proper quantum theory? My argument is that wave-particle duality does have a counterpart in proper quantum theory. There are no classical particles, only quantum particles. Unlike classical particles, quantum particles do not have simultaneously well-defined position and momentum. We call these quantum things particles for two reasons. First the particle number is an integer that we use to write down the commutation relations, and their representations, so that the proper theory does contain concepts such as the wave function of one particle or the wave function of two particles. Because particle number is an integer, it refers to discrete entities, just as classical particles are discrete entities. We call the quantum entity a particle, because in the classical limit, say in the path integral picture, the path of this thing is the path of the classical particle. So that justifies the particle aspect of quantum theory. I have already used the wave aspect in referring to the classical limit, but to use it again, the Schroedinger equation is a wave equation, so that justifies the wave aspect of quantum theory. So wave-particle duality does have its place in proper quantum mechanics, even though it is a loose concept of old quantum theory.

I never talk about the wave-particle duality. I only talk about things that I can describe by equations.
I saw inaccurate statements was made “particle nature of Hilbert space”, “position is particle and momentum is wave” so I responded to those by reminding you that the mathematical formalism of QM does not describe the world in terms of waves and particles; it is only our observations of the world that may be described in those terms. You also made another incorrect statement about the commutation relations in QM, so I responded by stating something that can be proved rigorously:
In QM, the commutation relations follow from the homogeneity of the 3-space.

 

[atyy]

I never talk about the wave-particle duality. I only talk about things that I can describe by equations.
I saw inaccurate statements was made “particle nature of Hilbert space”, “position is particle and momentum is wave” so I responded to those by reminding you that the mathematical formalism of QM does not describe the world in terms of waves and particles; it is only our observations of the world that may be described in those terms. You also made another incorrect statement about the commutation relations in QM, so I responded by stating something that can be proved rigorously:
In QM, the commutation relations follow from the homogeneity of the 3-space.

The first two are correct. What was wrong with my remark out the commutation relations?

 

[samalkhaiat]

What is it "correct" about "position is particle and momentum is wave"? What "particle nature" does an abstract complex vector space can possibly have? The statement "wave-particle duality is formalised by commutation relations" can not be proven. The statement that I made about the commutation relations can be proven

 

[atyy]

What is it "correct" about "position is particle and momentum is wave"? What "particle nature" does an abstract complex vector space can possibly have? The statement "wave-particle duality is formalised by commutation relations" can not be proven. The statement that I made about the commutation relations can be proven

Yes, the language is not standard, but I hope to convince you it can be correct. The idea is that "wave-particle duality" which is a vague heuristic in old quantum theory is still worth teaching, because there are several things in the proper theory which can be seen as formalizations of the heuristic.

So by "position is particle and momentum is wave", I just mean that in the position basis, the position eigenfunction is localized like a particle, while the momentum eigenfunction is a sinusoidal wave. Since this is captured by the commutation relation between the position and momentum operators, this is one way in which wave-particle duality is formalized.

Another formalization is that in non-relativistic quantum mechanics, the Hilbert space is constructed by thinking about discrete entities called particles. For example, the Hilbert space for two particles is constructed as the tensor product of the one particle spaces. Or in quantum field theory in the second quantized language, the Fock space is again constructed by thinking about discrete entities that are called particles. Then the notion of wave enters in that the Schroedinger equation in non-relativistic quantum mechanics, or the equation of motion for the operators in the Heisenberg picture of quantum field theory in the second quantized language is a wave equation. So we have both particle and wave aspects in the construction of the theory. The important point is that these are not classical particles, but quantum particles which do not have trajectories except in appropriate limits.

 

[atyy]

So, why can't we explain the sun orbital motion using QM?

Although I have never seen a calculation done, I believe in principle we can explain the sun's orbital motion using QM. The main difficulty is that QM is a statistical theory, so ideally we would like to have multiple independent preparations so that frequentist reasoning becomes easy. However, we have only one sun on which we can make sequential observations. In this case, what one would like is that the probability for the observed trajectory is sharply peaked around the classical trajectory. So what one does is measure some observable that corresponds to a rough estimate of position, which collapses the wave function, and then one makes another measurement that corresponds to a rough estimate of position. We do this repeatedly, and we should get a probability distribution over observed trajectories. That distribution should be sharply peaked around to observed orbit of the sun.