Why Is Quantum Mechanics So Difficult? - Comments

[Demystifier]

There is meaning we create and belief there is a meaning beyond our own creation, which are you referring to?

I am talking about human creation. Physics, as a scientific discipline, is created by humans. Physics is a human way to describe and predict what we see. The "true reality" (whatever that means) may be entirely different.

 

[Demystifier]

Check last sentence above.

Thanks, I've corrected the error.

 

[houlahound]

I am talking about human creation. Physics, as a scientific discipline, is created by humans. Physics is a human way to describe and predict what we see. The "true reality" (whatever that means) may be entirely different.

I have a different meaning of "meaning".

 

[Demystifier]

I have a different meaning of "meaning".

Like the ultimate meaning of life according to some dogmatic religion? I don't need that kind of meaning in physics.

 

[houlahound]

No

 

[Demystifier]

No

Then what do you mean by "meaning"?

 

[houlahound]

This sort of thing;

https://en.m.wikipedia.org/wiki/Teleology

 

[Demystifier]

This sort of thing;
https://en.m.wikipedia.org/wiki/Teleology

In physics, this can be achieved by specifying initial conditions in the future.

 

[houlahound]

What about predicting the result after something happened?

 

[Demystifier]

What about predicting the result after something happened?

If "initial" condition is known at some arbitrary time $t_0$, the differential equation determines behavior for both $t>t_0$ and $t<t_0$ (if this is what you ask).

 

[vanhees71]

So, in your opinion, could Bohmian mechanics have some value, even if it is not science? After all, it has some non-trivial mathematical structure. In addition, similarly to ethics if you wish, it offers some meaning of QM for those human physicists who, for some personal reasons, need some meaning in physics for internal motivation. (After all, if physics does not have any meaning for you, then why do you do it?)

Or let me put it this way. Even if BM is not science, it is certainly a non-trivial intellectual discipline. So how should we classify it? Philosophy? Philosophy of science? Isn't philosophy of science a part of science as much as it is a part of philosophy?

Well, in the case of BM I don't see any value. Maybe it's interesting as mathematics. Physics and the natural sciences do not provide "meaning" to anything. It is a method to learn in an objective and quantitative way about how nature is (or let's say more carefully about the objectively by human beings knowable part). Philosophy of science (or metaphysics) is part of philosophy and not science. It's providing amazingly little to the progress of science itself. It can, however, be valuable to subsume science into a bigger view of human knowledge and to systematially analyze its role in the progress of culture in general. It can also provide heuristical lines of thought in model/theory building, as shows the example of Einstein. His case is paradigmatic for the good and the bad of the use of philosophy of science in science itself: On the one hand Einstein's strong believe in a deterministic world view for sure helped him to very clearly resolve some of the outstanding puzzles of physics of his time starting from the incompatibility of Maxwell electrodynamics with the Newtonian space-time picture (ingeniously solving it by adapting the space-time picture and mechanics to the findings included in Maxwell's equations rather than the other way around as was the approach by the other physicists of his time, including Hertz, Lorentz, and Poincare) as well as early quantum theory (which however he found dissatisfying from the very beginning). On the other hand, this strong philosophical (if not quasi-religious) believes also hindered Einstein to participate in the development of modern quantum (field) theory for the last 20-30 years of his life (most probably more to the disadvantage for science than for himself).

 

[Demystifier]

It can also provide heuristical lines of thought in model/theory building, as shows the example of Einstein.

Yes, Einstein is a good example. But do you think that any big step in science, any change of paradigm, is possible without some philosophy as a heuristic line of thought for theory building? Newton, Darwin, Einstein, Bohr, could any of them make their main achievements without any philosophy?

 

[stevendaryl]

Yes, Einstein is a good example. But do you think any big step any science, any change of paradigm, is possible without some philosophy as a heuristic line of thought for theory building? Newton, Darwin, Einstein, Bohr, could any of them make their main achievements without any philosophy?

This is the question of what science is about. Starting with Popper (or earlier, with the logical positivists), many people think that science is about making better and better predictions. I actually don't think that is the goal of science. I think the goal of science is understanding the universe, and that the focus on predictions is an attempt to keep science grounded, so that it doesn't drift too far into pure speculation. Falsifiable predictions isn't the goal of the scientist, but is a measure of his progress.

When Einstein worked on Special Relativity, his goal wasn't to make new predictions, but to reconcile Newtonian mechanics with electrodynamics (which turned out to require a modification of the first). When he worked on General Relativity, his goal wasn't to make new predictions, but to reconcile Special Relativity with Newtonian gravity (which turned out to require a modification of both). In both cases, there were falsifiable predictions, but that wasn't the goal. To me, it's like exams in school. They are an important check on a student's understanding, but it's a mistake to think that doing well on exams is the purpose of learning.

 

[Demystifier]

Falsifiable predictions isn't the goal of the scientist, but is a measure of his progress.

Exactly!

I have one falsifiable prediction: The collision of two pink elephants in liquid helium produces 7 red frogs. Nobody so far has made that experiment, but it can be done with present technology. Is that science?

 

[vanhees71]

Yes, Einstein is a good example. But do you think that any big step in science, any change of paradigm, is possible without some philosophy as a heuristic line of thought for theory building? Newton, Darwin, Einstein, Bohr, could any of them make their main achievements without any philosophy?

In the model building itself, I think it's an important aspect, but I don't know of any example in the history of physics that progress has been made without solid foundation on empirical facts. That's why I think physical theories are discoveries and not inventions (as are mathematical structures like groups, rings, fields, vector spaces, calculus,...). Take "modern physics". It's almost entirely based on observations contradicting previous theories. The key point from the 19th century was the discovery (!) of the modern (and still valid) form of the theory of the electromagnetic interactions by Faraday (experiment) and Maxwell (theory). Note that Maxwell's heuristics is entirely wrong, while his theory has had a solid foundation in Faraday's experiments and thus still holds in modern form today. Maxwell himself later abandoned the wrong mechanical heuristics by the way. The discovery of quantum theory is also triggered in major flaws of the classical picture. Here also the theory of light (i.e., Maxwell electrodynamics) plays some role (photoelectric effect) but even more importantly the flaw of classical thermodynamics and statistical physics in explaining the stability of macroscopic matter. Last but not least it was Planck's derivation of his previously empirically found formula for the black-body spectrum based on highly accurate measurements by Kurlbaum et al at the Reichsanstalt (BTW with the main motivation to define a reproducible and reliable measure for the luminosity of all kinds of lightning, among them the then pretty new electric light bulbs :-)).

Of course there is also other input which I'd summarize as "intuition" of the physicists going into model building. E.g., the "old quantum theory", particularly concerning the Bohr-Sommerfeld model of atomic structure and spectral lines (again electromagnetism!) was considered (even by Bohr and Sommerfeld themselves) as a pretty poor picture with all its ad-hoc assumptions just to tweak the theory to all the accurate measurements of atomic spectra, the wrong counting of multiplicity of spectral lines, the socalled anomalous Zeeman effect etc. Here the main motivation to develop a more consistent theory was mostly on esthetical grounds, i.e., to find a more convincing or if you wish "more beautiful" theory. Best known is Heisenberg's heuristics of his famous Helgoland paper: He wanted to include only "observable quantities" into his theory and invented what Born has recognized as matrix calculus and worked it out together with Heisenberg and Jordan to matrix mechanics. Schrödinger's heuristics was very mathematical and strongly influenced by de Broglie's thesis, which itself can be seen also as an attempt to unify the disturbing heuristics of old quantum theory to a new basic principle, "wave-particle dualism", which is of course also flawed, as the development of modern QT shows. Schrödinger based his "wave mechanics" entirely on the analogy between ray optics and wave optics (the former being the leading-order eikonal approximation of the latter) on the one hand and point particle mechanics in terms of the Hamilton-Jacoby partial differential equations, taking the Hamilton-Jacobi PDE as the eikonal approximation of his "matter waves". Of course, he had a completely wrong intuition about the meaning of his waves, which he never got completely convinced until the end of his life.

So to discover a successful model or even a paradigm shifting general theory (which is a very rare thing in the history of science; in my opinion there are only two: the first is the advent of the field-point of view, based on observations by Faraday (finally leading to the relativistic space-time model; in full form achieved by Einstein with his General Relativity Theory) and quantum theory, which was initially mainly enforced on the usually rather conservative physicists by overwhelming empirical evidence) there is a very strong need for a solid foundation in empirical evidence, usually consisting of high accuracy measurements of clearly defined experimentally setup situations, but as well some intuition of ingenious physicists, including a lot of (often even wrong) heuristics. New models and theories in turn lead to new experimental setups as ingenious as the theories. For the longest time scientific progress is made in testing and investigating the predictions of models at higher and higher accuracy and ever new ways (again a lot of intuition is needed to invent the proper technology to perform the pertinent measurements), not so often in paradigm-shifting "revolutions" in theory. That's why I think Kuhn's famous work on these revolutions is flawed in the sense that it describes very rare special events in the history of sciences not the usual slow progress eventually leading to them. An example are gravitational waves. Predicted, with many doubts, by Einstein in 1916 almost precisely 100 years later they could be observed for the first time with an instrument taking the collaborative effort of many physicists around the world for decades. Who knows, what will come out of this completely new "window" (which can be understood in a quite literal sense) to observing the universe!

 

[Demystifier]

Falsifiable predictions isn't the goal of the scientist, but is a measure of his progress.

I would like to add that it is not the only measure of progress. For instance, reformulation of the theory such that calculations become simpler (e.g. matrix mechanics vs wave mechanics) is also a progress. Or a reformulation of the theory such that abstract mathematical objects can more easily be visualized (the same example) is also a progress. Or a reformulation of the theory such that it becomes more intuitive (which is a matter of interpretation) is also a progress.

Of course, different individuals do not need to agree which method of calculation is simpler, which mathematical object can more easily be visualized, or which interpretation is more intuitive. That's why different methods of calculation, different mathematical formulations, and different interpretations simultaneously exist. It is subjective and depends on someone's taste, but it does mean that it is not a part of science.

 

[A. Neumaier]

physical theories are discoveries and not inventions (as are mathematical structures like groups, rings, fields, vector spaces, calculus,...).

The mathematical structures you mention are also discoveries and not inventions. You can invent a set of axioms by listing some properties and give the structures that have these properties a name - but nobody would be interested in them. Even if new it is dead wood, not more interesting than listing new truths of the form a+b=c where a,b are huge numerically given numbers whose sums were never computed before by anyone.

On the other hand, one discovers that certain very special structures (like groups, rings, fields, vector spaces, calculus,...) have interesting properties that help to organize prior knowledge in a more powerful way and thereby give rise to progress, including new and interesting quests - just as with progress in physics.

 

[Demystifier]

So to discover a successful model or even a paradigm shifting general theory (which is a very rare thing in the history of science; in my opinion there are only two: ... the advent of the field-point of view ... and quantum theory

You forgot the biggest paradigm shift, that physics should be formulated in terms of equations (Newton).

Of course, there are also many "smaller" paradigm shifts. For example, that QFT does not need to be renormalizable in order to make sense, that there may be more than 3 spatial dimensions, ... All they require a bit of philosophy.

 

[vanhees71]

The mathematical structures you mention are also discoveries and not inventions. You can invent a set of axioms by listing some properties and give the structures that have these properties a name - but nobody would be interested in them. Even if new it is dead wood, not more interesting than listing new truths of the form a+b=c where a,b are huge numerically given numbers whose sums were never computed before by anyone.

On the other hand, one discovers that certain very special structures (like groups, rings, fields, vector spaces, calculus,...) have interesting properties that help to organize prior knowledge in a more powerful way and thereby give rise to progress, including new and interesting quests - just as with progress in physics.

Well, but mathematical strucures are pure inventions of the human mind, only restricted for being consistent within the usual logics. Whether they are interesting is another question, while physical theories (usually formulated using well-established mathematical structures or inventing new ones as needed, which are then often happily adopted by the mathematicians like functional analysis (Dirac's $\delta$ function made rigorous) or fiber bundles (setting gauge theories on a rigorous mathematical footing)), are usually discovered by thinking about observed phenomena.

 

[A. Neumaier]

Well, but mathematical structures are pure inventions of the human mind, only restricted for being consistent within the usual logics.

Well, physical structures are also pure inventions of the human mind, only restricted for being consistent with experiment within their domain of validity. Whether they are interesting is another question. As in mathematics, it is only the interesting, most widely applicable ones that are preserved and taught; the remaining ones, though consistent with experiment within their domain of validity, fall into oblivion.

while physical theories are usually discovered by thinking about observed phenomena.

In each discipline, the relevant results are usually discovered by thinking about its subject matter. If I look into a typical issue of ''Nuclear Physics B'' or ''Classical and Quantum Gravity'' I find lots of theory obtained primarily by thinking about other theory, not about observed phenomena.

 

[vanhees71]

Well, I was talking about discovering new theories, not work on established ones, which of course makes the most of what's done by theoretical physicists in their daily work.

 

[Demystifier]

If I look into a typical issue of ''Nuclear Physics B'' or ''Classical and Quantum Gravity'' I find lots of theory obtained primarily by thinking about other theory, not about observed phenomena.

Excellent point!

 

[Demystifier]

Well, I was talking about discovering new theories, not work on established ones, which of course makes the most of what's done by theoretical physicists in their daily work.

Is SUSY discovered or invented? If it is invented, would you be still claiming so if it happened that LHC confirmed SUSY in experiments?

 

[vanhees71]

Hm, isn't this purely semantic again. In a sense SUSY seems to be triggered by the famous Coleman-Mandula no-go theorem, and some physicists tend to dislike no-go theorems and event more general mathematical structures to circumvent it. In this sense the introduction of a $\mathbb{Z}_2$ graded algebra into the symmetry structure of Lagrangians is an invention (interesting would be to know, whether mathematicians had thought of such a struture before; I guess so since if you have Grassmann numbers, it's not far to come to the idea to formulate such structures). It stays of course invented even if LHC confirms SUSY, because that's the way it was found historically. The only point is, how likely it is that it is really found. It's very rare that from such a pure invention without being triggered by some necessity from observations something turns out to be finally true, although the prediction of the Higgs boson is close to something like it. It has been introduced to get a consistent gauge theory with massive gauge bosons to describe the weak interactions within a renormalizable QFT, and nobody could find another way to do this (except for the Abelian case, which however is not describing QFD in accordance with experiments). The idea to use the apparent spontaneous breaking of a local gauge theory was, however, in some sense a mathematical invention (by condensed-matter physicists by the way, mostly Andersen, in the context of superconductivity) like SUSY. On the other hand the application to the ew. interaction was very much triggered by phenomenology (even in form of Fermi's theory of $\beta$ decay and its variants after parity violation was discovered and in this context the correct (V-A)-realization of this breaking was found based on these observations).

So it's tricky to ask, what's invented and what's discovered in theoretical physics, but it's fortunately completely irrelevant to the progress of science either :-)).

 

[Demystifier]

So it's tricky to ask, what's invented and what's discovered in theoretical physics, but it's fortunately completely irrelevant to the progress of science either :-)).

Well, it's not completely irrelevant. If one says it's invented and not discovered, then it's close to saying that it's mathematics and not physics, which can influence someone's decision on whether such research should be financed or not, which may have a high impact on progress of science.

Or suppose I say (which I often do) that Bohmian mechanics is not an alternative interpretation, but an alternative mathematical formulation. That's just semantics, but such a change of words can have a high impact when someone needs funds for research on the Bohmian approach.

 

[fluidistic]

Discussing the PF Insight : My experience with my 2 undergraduate QM courses has been rather strange in that, to me, the math part was actually the most challenging part. I did not understand most of it, I believe. For instance in the 1st course we spent some classes to reach the spectral theorem and how it relates to operators in QM. There were many homework problems involving pure math proofs. It seemed to me like a deep linear algebra course (I had already took a linear algebra course but QM really digged much deeper. For example I had never dealt with unbounded operators and some theorems that were valid for finite dimension vector spaces were suddenly not valid for infinite dimensional ones. The ones that appear in QM!)
In my second course what really got me ENTIRELY lost was the spherical tensors part. I did not grasp a single bit out of it.

Whilst on the "physics" part of QM that challenged my intuition most, or got me think "wow... that's freaking weird" was :
1) That in a central potential, the wavefunction of the electron is not restriced to a plane. That's entirely different than in classical mechanics.
2)The wavefunction of the ground level of the SHO for 1 particle, implies that there's a greater probability to find the particle at the center rather than on the tails of the quadratic potential. That is opposite to classical mechanics.

There are a few others, dealing for example with perturbations that demolished my intuition again and again. But it was not what made the course hard.

 

[lavinia]

I am just starting out in quantum mechanics. The approach has been to talk about Hermitian operators as obervables acting on a complex vector space of states.Mathematically this is simple stuff, linear algebra. The Shroedinger equation makes it a little more difficult as does the use of symmetries to find degeneracies. But net net this is easy mathematics. Basic Quantum Mechanics is not difficult because of the mathematics.

Classical phase space is just a manifold of points. This is a straight forward generalization of our visual picture of the space around us. Quantum Mechanical phase space is a complex vector space, and points in the phase space can be linearly combined to make new states. I think for anybody this is difficult to intuit. Also the whole deal with eigen states as the results of observations. We end up with a mathematical formalism that is interpreted in terms of weird experiments. That is hard.

In Mathematics unintuitive ideas come to life through examples and applications to other areas of mathematics. That is how I want to learn Quantum Mechanics. In this context the historical approach has got to be enlightening. I don't agree that it should not be taught. I want to know it.

Another conceptual problem with Quantum Mechanics is that it is taught deductively from a set of Axioms (as is Special Relativity). These Axioms can seem like pure formalism and the exposition of the subject a syllogism. That can not be a good way to teach. That is not the way mathematics is taught nor is it the way mathematical ideas are discovered. This again urges the historical approach at least to gain insight into why these Axioms are necessary and how they were thought of.

I am a little surprised by learning Physics. It is a strange experience. It seems that Physical theories(unlike most mathematical theories which is what I am familiar with) are axiomatic systems from which physical phenomena are revealed through pure deduction. This again argues for understanding why these axioms are assumed.

 

[rootone]

I am a little surprised by learning Physics. It is a strange experience. It seems that Physical theories(unlike most mathematical theories which is what I am familiar with) are axiomatic systems from which physical phenomena are revealed through pure deduction. This again argues for understanding why these axioms are assumed.

Physical theories are not axiomatic like mathematical proof is.
They are models (of which math is an essential part), which fit with observations.
If the model predicts something that can be tested, it's a good theory.

 

[Stephen Tashi]

I am a little surprised by learning Physics. It is a strange experience. It seems that Physical theories(unlike most mathematical theories which is what I am familiar with) are axiomatic systems from which physical phenomena are revealed through pure deduction. This again argues for understanding why these axioms are assumed.

I don't think a theory deals with physical probability can be deduced from the standard (Kolmogorov) axioms for probability. Those axioms lack assumptions having to do with a "probable" event becoming an "actual event". I see nothing in the Kolmogorov axioms that even asserts it is possible to take a random sample - i.e. to have a "realization" of random variable. It seems to me that in order to axiomatize physical probability, one would have to introduce the notion of time into the axioms, so that an event could have a probability at one time and then become (or fail to become) actual at a later time.

 

[A. Neumaier]

Quantum Mechanical phase space is a complex vector space

No, it is a complex projective space formed by the rays in the Hilbert space. Geometrically it is again a symplectic manifold (though often infinite-dimensional). Note that rays cannot be superimposed, only state vectors.

You might like my online book Classical and Quantum Mechanics via Lie algebras, where I present quantum mechanics in a mathematician-friendly way.

 

[A. Neumaier]

why these axioms are assumed.

Because they allow a concise description of models of real physical systems. Just like in mathematics axioms for group theory or for vector spaces are assumed because they allow a concise description of symmetries and linear algebra.

In both cases, the experience with the concrete stuff comes first and gives later rise to a formal, abstract edifice in which the experience gained is transformed into powerful tools that allow to tackle more complex problems than what one can do when staying on the concrete level.

 

[vanhees71]

I am just starting out in quantum mechanics. The approach has been to talk about Hermitian operators as obervables acting on a complex vector space of states.Mathematically this is simple stuff, linear algebra. The Shroedinger equation makes it a little more difficult as does the use of symmetries to find degeneracies. But net net this is easy mathematics. Basic Quantum Mechanics is not difficult because of the mathematics.

Classical phase space is just a manifold of points. This is a straight forward generalization of our visual picture of the space around us. Quantum Mechanical phase space is a complex vector space, and points in the phase space can be linearly combined to make new states. I think for anybody this is difficult to intuit. Also the whole deal with eigen states as the results of observations. We end up with a mathematical formalism that is interpreted in terms of weird experiments. That is hard.

In Mathematics unintuitive ideas come to life through examples and applications to other areas of mathematics. That is how I want to learn Quantum Mechanics. In this context the historical approach has got to be enlightening. I don't agree that it should not be taught. I want to know it.

Another conceptual problem with Quantum Mechanics is that it is taught deductively from a set of Axioms (as is Special Relativity). These Axioms can seem like pure formalism and the exposition of the subject a syllogism. That can not be a good way to teach. That is not the way mathematics is taught nor is it the way mathematical ideas are discovered. This again urges the historical approach at least to gain insight into why these Axioms are necessary and how they were thought of.

I am a little surprised by learning Physics. It is a strange experience. It seems that Physical theories(unlike most mathematical theories which is what I am familiar with) are axiomatic systems from which physical phenomena are revealed through pure deduction. This again argues for understanding why these axioms are assumed.

I have the opposite impression. Modern mathematics textbooks are very often in an awful "Bourbaki style". It's all well ordered in terms of axioms, definitions, lemmas, and theorems with proofs, but no intuition whatsoever. I think in math the intuition comes from the applications, and physics is of course full of applications of interesting math issues (from analytical geometry via calculus to group theory and topology nearly everything interesting in math finds applications in physics).

Concerning the historic approach, I'm a bit unsure. From my own experience, I don't like it at all. In high school we learned "old quantum mechanics" first. This is physics of a transition era from classical ways of thought, which were proven wrong by observations at the time (e.g., the very familiar stability of matter is a complete enigma to classical physics as soon as you know that atoms consist of a positive charged particle (now called atomic nucleus) surrounded by negatively chargend electrons, as was revealed by the famous gold-foil experiment by Rutherford. The collisions of $\alpha$ particles (He nuclei) were perfectly described by classical scattering theory in the Coulomb field (that's why the corresponding cross section is named after Rutherford). So far so good, but then you had to understand the bound state making up an atom itself, and this caused a lot of trouble since on the one hand Bohr had just to solve the classical motion of a charged particle running around the nucleus under the influence of the mutual electrostatic interaction and assuming the quantization of the action, which was an ingeneous ad-hoc extrapolation of Planck's treatment of black-body radiation. Lateron the model was refined by Sommerfeld. However it was contradicting classical electromagnetics according to which there should be some bremsstrahlung and in a very short time the electrons should crash into the nucleus, i.e., the atom should be unstable. Instead of the bremsstrahlun, however, what's observed are clean pretty narrow spectral lines whose frequency was given by the distance between the energy levels known from the Bohr-Sommerfeld model, which is totally ununderstandable from classical physics. Last but not least the chemists knew that hydrogen atoms are spherical and not little flat disks.

Learning the old quantum mechanics first, cements very wrong pictures in the students's mind whic h have to be unlearnt and corrected when you want them to understand modern quantum theory. As you rightly realized the problem is indeed not the math. Taking the wave-mechanics approach for a long time of the QM 1 lecture you deal just with a scalar field. Compared to electromagnetics, where you deal with a lot of scalar and vector fields which are coupled via Maxwell's equations, it's a piece of cake to deal with a single pseudoparabolic pde, known as the Schrödinger equation. The problem is that you have to build the entire intuition behind this math via this math itself. There is no correct intuitive picture from our classical experience, which ironically becomes a pretty tough question to answer, why it is right (the answer is decoherence, and the course-graining concept of quantum statistical physics, and after you swallowed this many loose ends of classical statistics like the Ehrenfest paradoxon the question of the absolute measure of entropy, etc. are solved either).

On the other hand, you are right in saying that learning about the historical development of theories can help a lot to gain intuition, and particularly for QM 1 I'd not know how to teach it without a (however brief) historical introduction about how the idea to have the wave-mechanics approach or, in my opinion even better, the representation free Hilbert space formulation a la Dirac. I've never taught QM 1, but I guess, I'd give a brief introduction about the historical development, starting with Planck's black-body law, then talking about "wave-particle dualism" (sparing out however, the completely misleading picture of photons being "particles" in any classical sense, and the photoelectric effect does NOT prove the necessity of photons, i.e., the necessity of quantizing the em. field, which is anyway not discussed in QM 1) and finally getting via the wave-mechanics heuristics a la Schrödinger as quickly as possible over to Dirac's very neat mathematical heuristics to the Hilbert-space $C^*$-algebra approach, where first the commutation relations can be reduced to the Heisenberg algebra of position and momentum operators, i.e., a canonical quantization approach.

I've once taught QM 2, where I emphasized the role of symmetries, which is in my opinion the right intuition for modern physics anyway. The $C^*$ algebra of non-relativistic QM is then very clearly deduced from the ray-representation theory of the Galileo group, which then helps very much to understand, by the notion of mass is so different in Galilean physics (central charge of the Galileo algebra) compared to Minkowskian physics (Casimir operator of the proper orthochronous Poincare group). Given the evaluations of this lecture by the students, this approach seems to be not too bad since they liked it, although they argued that one better keeps the discussion of the Galilean part shorter and rather go farther in the relativstic part, whic h they found much more interesting. Of course, I did not teach the wrong wave-mechanics interpretation of relativsitic QT a la Bjorken-Drell vol. I. This is really too much sacrifice to the historical approach, since once the Dirac formalism is motivated in QM 1, there is no reason, not to start relativistic QT right away as a local quantum field theory.

I think the best way in a good balance between the historical and the deductive approach to teach theoretical physics can be found in Weinberg's textbooks, where he usually has a historical introduction but then develops the theory itself in the deductive way emphasizing the mathematical structure and its meaning in context of the physics. I can only recommend to read his "Lectures on Quantum Mechanics" and of course "Quantum Theory of Fields".

 

[vanhees71]

Because they allow a concise description of models of real physical systems. Just like in mathematics axioms for group theory or for vector spaces are assumed because they allow a concise description of symmetries and linear algebra.

In both cases, the experience with the concrete stuff comes first and gives later rise to a formal, abstract edifice in which the experience gained is transformed into powerful tools that allow to tackle more complex problems than what one can do when staying on the concrete level.

It should, however, be clear that in physics the axioms are not the starting point but the goal of theory building. In math you start with the axioms, because it defines your "universe". In physics the "universe" is imposed on us by nature, and we have to figure out the "axioms" for describing it. It's amazing enought, how far you come with this axiomatic approach to physics, although admittedly the most successful theory we have has no rigid foundation in math, relativistic QFT and the Standard Model of HEP.

 

[A. Neumaier]

In math you start with the axioms, because it defines your "universe".

Only in a textbook treatment, just like in theoretical physics. In a historical context, real numbers were used 1000 years before their axiomatization, matrices and quadratic forms were used 150 years before the axiomatic concept of a vecor space appeared, transformation groups were studied over 100 years before the notion of an abstract group was coined, probability 150 years before Kolmogorov's axioms, etc.. The appropriate axiomatic setting of a subject matter arises only after enough experience has accumulated, both in mathematics and in theoretical physics.

The goal of a theory, both in mathematics and physics, is to fully account for the corresponding part of the domain of discourse the theory covers, and to do so in an efficient way that is easy to teach and provides quickly all relevant tools and results.

The axioms are a device serving the concise introduction of the concepts to be used later, again both in mathematics and in theoretical physics. Except that the mathematician's axioms cover the full conceptual content while physical axioms usually have an exemplary nature and are not sufficient to give a solid basi of the theory. Born's rule (which applies only to certain highly idealized experiments) is the best known example of such a caricature axiom, supposedly defining the formal meaning of a measurement.

 

[bhobba]

I have the opposite impression. Modern mathematics textbooks are very often in an awful "Bourbaki style". It's all well ordered in terms of axioms, definitions, lemmas, and theorems with proofs, but no intuition whatsoever. I think in math the intuition comes from the applications, and physics is of course full of applications of interesting math issues (from analytical geometry via calculus to group theory and topology nearly everything interesting in math finds applications in physics).
the absolute measure of entropy, etc. are solved either).

How true that is.

But it requires its own thread, not in the QM sub-forum but in one of the math ones.

It should look at the history of Rigged Hilbert spaces and its now central role in White Noise Theory that has far reaching applications to QM.

I've once taught QM 2, where I emphasized the role of symmetries, which is in my opinion the right intuition for modern physics anyway.

Again -

To start this journey, and its the most important insight of 20th century physics IMHO, see Landau who in his terse no BS style gets to the heart of the matter:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20

I could explain it, but IMHO its so important the OP, and anyone reading this, needs to nut it out for themselves, which is the only way for true understanding. Its best done in the classical mechanics section. Once that is understood then rest falls rather quickly into place via Noether's beautiful theorem:
https://www.amazon.com/dp/0801896940/?tag=pfamazon01-20

It left Einstein basically - well his own words are best
http://cwp.library.ucla.edu/articles/noether.asg/noether.html
https://www.washingtonpost.com/news...in-called-her-a-creative-mathematical-genius/
http://www.math.wichita.edu/history/women/noether.html
'Einstein also wrote in a letter to Professor David Hilbert that Emmy Noether display "penetrating mathematical thinking."

Its the type of thing that makes me laugh about philosophy. I never ever see philosophy articles about it, they get into all sorts of irrelevant tangents that skirt around the real issue, but it's central, vital, and one of the truly great insights of modern physics. So much so I believe, and this is simply conjecture on my part, some truly great and strikingly simple symmetry, lies at the heart of all physics. But unraveling that first requires figuring out what it is a symmetry in. In classical physics we know its symmetries in the Lagrangian which depends on QM. In QFT we know its symmetries again in Field Equations, but they can also be expressed in terms of Lagrangians. That however is not explainable in terms of QM because QFT is what explains QM. To me this is the deep deep mystery whose solution will illuminate the fundamental law at the foundation of all of physics ie the TOE.

Just an opinion of course - only time and further research will solve it.

Thanks
Bill