Why Is Quantum Mechanics So Difficult? - Comments

[Stephen Tashi]

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.

Perhaps an even more fundamental concept than symmetry is that we agree to call different things the same - or is calling different things the same thing the essential idea of symmetry?

The idea of repeating the "same" experiment is, at face value, self-contradictory because if experiment #2 was precisely a repeat of experiment #1 then it wouldn't be a different experiment, it would just be another label for experiment #1. So when an experiment is repeated it only certain aspects of it are repeated. The "unessential" aspects of the experiment tend to be ignored, but if they didn't exist then we wouldn't have a repeated experiment. Any particular unessential aspect of an experiment (e.g. what color t-shirt the lab technician wore) is not critical, but it is critical that there be some unessential aspect that distinguishes two repeated experiments.

The concept of physical probability involves the convention that we will define "an event" in a way that actually denotes a collection of different events. The mathematical model of repeated independent trials as some sort of tensoring together of copies of the same sample space doesn't quite capture the requirement that a "repetition" of an experiment requires that something be different when an experiment is repeated.

 

[lavinia]

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.

Thanks but the math isn't what worries me. Leonard Susskind and Richard Feynman teach it explicitly as a complex vector space. That is what I am learning from. Your book sounds too mathematical for a first trip around the block.

My point was that linear combinations of states to get new states is difficult to intuit.

 

[A. Neumaier]

that linear combinations of states to get new states is difficult to intuit.

Linear combinations of state vectors give new state vectors, which is as it should be in a vector space, no problem for the intuition.

But state vectors are not states - physical states are normalized state vectors determined only up to a phase - i.e., rays in the Hilbert space, or points in the projective space.

It makes no sense to take linear combinations of states. Thus what is a difficulty for your intuition is based on a misunderstanding.

 

[A. Neumaier]

or is calling different things the same thing the essential idea of symmetry?

Not quite. It is the essential idea of equivalence relations.

Without calling different things the same we cannot form a single concept...

 

[bhobba]

Perhaps an even more fundamental concept than symmetry is that we agree to call different things the same - or is calling different things the same thing the essential idea of symmetry?

Yes - good one.

Its actually interconnected.

Take the fundamental symmetry of GR - no prior geometry. The interpretation is since there is no prior geometry all geometries are equivalent. If they are equivalent then they are dynamical ie geometry itself is dynamical. Physically it means geometry itself has its own Lagrangian and the striking thing is that all by itself basically leads to GR:
http://www.if.nu.ac.th/sites/default/files/bin/BS_chakkrit.pdf

Bottom line is this - symmetry requires a bit of interpretation on our part. IMHO finding that interpretation is the key goal of science.

In QM the thing it's symmetrical in is the laws of QM which are really 2 as found in Ballentine (its really one, but that is another story). The symmetry is the Born rule must follow the Gaelian transformation - specifically the probabilities must not depend on FOR. Its very intuitive - so intuitive you do not even think you are invoking the POR - but really you are. You find symmetry is a lot like that - its so magical it takes a bit of thought understanding just what your physical assumptions are.

I can do posts spelling them out but really its so critical you should nut it out for yourself like I did.

A good starting point is the physical assumption in the following almost magical derivation of Maxwell's equations:
http://cse.secs.oakland.edu/haskell/Special Relativity and Maxwells Equations.pdf

It took me a while to nut out what they were, and truth be told I can't remember them - but I felt really good when I finally realized what they were.

In my blurb about what attracts me to science its how we view science so its almost obvious - ie the physical assumptions are there but so cunningly 'hidden' you don't even realize they are there.

Its truly beauty incarnate - you feel so elevated - at least I do.

Thanks
Bill

 

[lavinia]

Linear combinations of state vectors give new state vectors, which is as it should be in a vector space, no problem for the intuition.

But state vectors are not states - physical states are normalized state vectors determined only up to a phase - i.e., rays in the Hilbert space, or points in the projective space.

It makes no sense to take linear combinations of states. Thus what is a difficulty for your intuition is based on a misunderstanding.

Yeah I know that and I am not sure why you are correcting this since it is a given on the first day of a Quantum Mechanics course that you have to normalize . Yeah normalize so you really have a projective space. But this is all obvious. The important thing is the difference in the nature of state space and that is intimately a consequence of linear combination. I suppose if you only want physically equivalent sates you would form the complex projective space of 2 planes in the Hilbert space.

 

[A. Neumaier]

The important thing is the difference in the nature of state space and that is intimately a consequence of linear combination.

No. Using vectors, matrices and functions is the natural way of describing any (mathematical or physical) system with a large number of degrees of freedom. For example, nonlinear manifolds are represented in terms of vectors when doing actual computations.

The classical phase space for a particle in an external field is also a vector space $R^6$ (or $C^3$ if you combine position and momentum to a complex position $z=q+i\kappa p$ with a suitable constant $\kappa$). And, unlike in the quantum case, one can form linear combinations of classical states.

Thus the problem with quantum mechnaics cannot lie in the use of vectors and their linear combinations. In the quantum case you just have many more states than classically, which is no surprise since it describes systems form a more microscopic (i.e., much more detailed) point of view.

What one must get used to is not the superpositions but the meaning attached to a pure quantum state, since this meaning has no classical analogue.

However, for mixed states (and almost all states in Nature are mixed when properly modeled), quantum mechanics is very similar to classical mechanics in all respects, as you could see from my book. (Note that the math in my book is no more difficult than the math you know already, but the intuition conveyed with it is quite different from what you can get from a textbook.)

Thus the difficulty is not intrinsic to quantum mechanics. It is created artificially by following the historically earlier road of Schroedinger rather than the later statistical road of von Neumann.

 

[Dr. Courtney]

I had a great high school chemistry course which served as the underpinnings for QM when I got there in college. My teacher was an old school former nun who made us memorize the periodic table and the orbitals for many of the atoms. We understood how the orbitals corresponded to the columns with the "filled shells" business and all of that.

Somewhere between my 2nd semester general physics course and my 3rd semester modern physics course, I realized that QM held the promise of explaining the "why" of the periodic table as well as explaining an awful lot about atomic and molecular spectra, which fascinated me since I was in elementary school.

The "electron cloud" and the probabilistic bit never bothered me. I learned to focus on the parts that could be measured in experiments and learned not to worry about bits which did not make predictions.

 

[vanhees71]

Yeah I know that and I am not sure why you are correcting this since it is a given on the first day of a Quantum Mechanics course that you have to normalize . Yeah normalize so you really have a projective space. But this is all obvious. The important thing is the difference in the nature of state space and that is intimately a consequence of linear combination. I suppose if you only want physically equivalent sates you would form the complex projective space of 2 planes in the Hilbert space.

Sure, the superposition principle is one very important feature of quantum theory, i.e., (except when there are superselection rules) any Hilbert space vector can be representant of a pure state (which indeed is only determined up to a multiplicative constant and thus is in fact rerpesented by the entire ray in Hilbert space or, equivalently by the corresponding projection operator of any normalized representant of that ray) and thus also the superpositions of any such vectors.

On the other hand it is very important to keep in mind that the true representants of the states are the statistical operators (or in case of pure states rays) since otherwise you'd have a hard time to define non-relativistic quantum theory: it's not the unitary reprsentations of the Galileo group but the ray representations, and thus you have more freedom, i.e., you can represent any central extension of the covering group, and thus you can introduce the mass as a non-trivial central charge of the Galilei algebra and use SU(2) to represent rotations. The former is the only way to make a physically sensible dynamics possible and the latter enables to describe half-integer spins, without which the description of the matter around us wouldn't be possible either.

 

[vanhees71]

I had a great high school chemistry course which served as the underpinnings for QM when I got there in college. My teacher was an old school former nun who made us memorize the periodic table and the orbitals for many of the atoms. We understood how the orbitals corresponded to the columns with the "filled shells" business and all of that.

Somewhere between my 2nd semester general physics course and my 3rd semester modern physics course, I realized that QM held the promise of explaining the "why" of the periodic table as well as explaining an awful lot about atomic and molecular spectra, which fascinated me since I was in elementary school.

The "electron cloud" and the probabilistic bit never bothered me. I learned to focus on the parts that could be measured in experiments and learned not to worry about bits which did not make predictions.

That's a very healthy approach. Too much philosophy hinders the understanding of science ;-)).

 

[ShayanJ]

That's a very healthy approach. Too much philosophy hinders the understanding of science ;-)).

I think this comment comes from a misunderstanding. Let me explain my point with an example:
There are physicists who work on lattice QCD simulations and other people who work on the problem of confinement. Some of the results from those simulations may be useful for developing models for confinement. But imagine there is a particular result that is still controversial among lattice QCD people. Should confinement people accept or reject that result? Should they participate in a serious discussion about that result although they don't know as much as lattice QCD people about the subject?
I think the answers to the above questions are clear. The confinement people just wait until the lattice QCD people come to a consensus and until then they just ignore that result. Of course, they may accept it or deny it or try to contribute to the discussion but because of the simple fact that they don't know enough about the subject, they may get confused or have uninformed ideas.

The field of "Foundations of QM" is just another field of research in physics with its own community of experts. The fact that people not in this community may get confused by the subject or have uninformed ideas doesn't mean that the subject is flawed, it just means that people should focus on what they know!

 

[Demystifier]

Too much philosophy hinders the understanding of science

Too much abstract formalism (Bourbaki) also hinders the understanding of science.
Too much equations without verbal explanations also hinders the understanding of science.
Too much verbal explanations without equations also hinders the understanding of science.
Too much of numerical computation also hinders the understanding of science.
Too much of general equations without putting numbers in also hinders the understanding of science.
Too much theory without experiments also hinders the understanding of science.
Too much experiments without theory also hinders the understanding of science.
Even too much science without looking at it from the outside (meta-science) hinders the understanding of science.

The point is to find a healthy dose of all that. And to realize that the correct dose depends on the individual.

 

[Demystifier]

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.

In their daily work, mathematicians don't start from axioms any more than physicists start from experimental facts. And you certainly know that physicists, in their daily work, often do not really start from experimental facts.

 

[Demystifier]

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.

Only if, by "modern" physics, you mean high-energy physics. But most physicists would not agree with such a definition of modern physics.

Indeed, you just confirmed my claim that high-energy spirit of QFT differs from condensed-matter spirit of QFT. Condensed-matter physicists rarely make such a strong emphasis on symmetry. They use symmetry in practice, but they rarely base their intuition on it.

When I was young (late high school and early college days), I thought that one of the keys for understanding the deepest secrets of nature is to understand the meaning of symbols such as SU(2) and SU(3). Needless to say, I don't think that anymore.

 

[vanhees71]

Well, I don't know condensed-matter physics as well as high-energy particle/nuclear physics, but I don't think that symmetries are less important in condensed-matter physics than in HEP. Also I think symmetries is a common thing for all subtopics of physics, and one should not specialize too early. As parts of the general theory course (and not lecture aiming at specialization) QM1 and QM 2 should provide the theoretical methodology for a broad range of "users", including experimental physicists. So, I think, that no matter in which field of research you specialize in your research, symmetries seems to be a good basis to understand theoretical physics (not only QT but also classical physics).

 

[A. Neumaier]

When I was young (late high school and early college days), I thought that one of the keys for understanding the deepest secrets of nature is to understand the meaning of symbols such as SU(2) and SU(3). Needless to say, I don't think that anymore.

Whereas the older I get the more I realize that properly understanding quantum mechanics means properly understanding unitary group representations. Whether spin and Stern-Gerlach experiments, or the Schroedinger equation, or the harmonic oscillator, or the spectrum of the hydrogen atom, or multiparticle scattering, or Hartree-Fock theory, or coupled cluster expansions, or coherent states, or the fractional Hall effect, or equilibrium statistical mechanics, or dissipative quantum mechanics, or free (condensed matter or relativistic) quantum fields, or QED and the standard model, or exactly solvable models, or nonperturbative quantum field theory, or conformal field theory, unitary group representations always give the best structural insights into what really matters. It is a great organizational principle.

 

[Demystifier]

Well, I don't know condensed-matter physics as well as high-energy particle/nuclear physics, but I don't think that symmetries are less important in condensed-matter physics than in HEP. Also I think symmetries is a common thing for all subtopics of physics, and one should not specialize too early. As parts of the general theory course (and not lecture aiming at specialization) QM1 and QM 2 should provide the theoretical methodology for a broad range of "users", including experimental physicists. So, I think, that no matter in which field of research you specialize in your research, symmetries seems to be a good basis to understand theoretical physics (not only QT but also classical physics).

Well, every theoretical physicist needs symmetry at the intuitive level, e.g. to develop instinct of using spherical coordinates whenever the spherical symmetry is obvious. But I don't think that use of advanced group theory is always necessary.

 

[Demystifier]

Whereas the older I get the more I realize that properly understanding quantum mechanics means properly understanding unitary group representations. Whether spin and Stern-Gerlach experiments, or the Schroedinger equation, or the harmonic oscillator, or the spectrum of the hydrogen atom, or multiparticle scattering, or Hartree-Fock theory, or coupled cluster expansions, or coherent states, or the fractional Hall effect, or equilibrium statistical mechanics, or dissipative quantum mechanics, or free (condensed matter or relativistic) quantum fields, or QED and the standard model, or exactly solvable models, or nonperturbative quantum field theory, or conformal field theory, unitary group representations always give the best structural insights into what really matters. It is a great organizational principle.

How can unitary group representations help to understand the measurement problem?

 

[atyy]

Indeed, you just confirmed my claim that high-energy spirit of QFT differs from condensed-matter spirit of QFT. Condensed-matter physicists rarely make such a strong emphasis on symmetry. They use symmetry in practice, but they rarely base their intuition on it.

I have heard one condensed matter physicist (Cenke Xu) explain it this way:

In HEP, the higher the energy, the more the symmetry.

In condensed matter, the lower the energy, the more the symmetry.

 

[Demystifier]

Speaking of symmetries, groups and representations, I was very disappointed with the new book by Zee
https://www.amazon.com/dp/0691162697/?tag=pfamazon01-20

In his previous books (Quantum Field Theory and Gravity) I have found a lot of new deep original insights, but that didn't happen with his last book on Group Theory. Perhaps it tells more about me than about the book, but I would certainly like to see what others think.

 

[atyy]

In their daily work, mathematicians don't start from axioms any more than physicists start from experimental facts. And you certainly know that physicists, in their daily work, often do not really start from experimental facts.

The great thing about Bourbaki is that they started from experimental facts, just like quantum mechanics! They said, well, we know what it means by two symbols on the page being the "same", even though it all probability, two different "ψ"s are almost certainly not the same down to the last atom.

 

[Demystifier]

I have heard one condensed matter physicist (Cenke Xu) explain it this way:

In HEP, the higher the energy, the more the symmetry.

In condensed matter, the lower the energy, the more the symmetry.

Exactly!

Now vanhees71 will find important counterexamples, but that will not change the fact that the above statement greatly summarizes the general spirit in the two communities.

 

[vanhees71]

Speaking of symmetries, groups and representations, I was very disappointed with the new book by Zee
https://www.amazon.com/dp/0691162697/?tag=pfamazon01-20

In his previous books (Quantum Field Theory and Gravity) I have found a lot of new deep original insights, but that didn't happen with his last book on Group Theory. Perhaps it tells more about me than about the book, but I would certainly like to see what others think.

Hm, I was disappointed by all books by Zee, I've had a look at. I've not looked at the newest one yet. So I can't say, whether I like it or not.

 

[vanhees71]

I have heard one condensed matter physicist (Cenke Xu) explain it this way:

In HEP, the higher the energy, the more the symmetry.

In condensed matter, the lower the energy, the more the symmetry.

Can you explain the meaning of these enigmatic statements? Also in condensed matter physics the highest symmetry is reached at (asymptotic) high energies, where matter becomes an ideal gas of elementary particles (of quite probably yet unknown fundamental degrees of freedom), but that cannot be what the condensed-matter physicist wanted to express.

 

[Demystifier]

I was disappointed by all books by Zee

Interesting, you are the first person I know disappointed with the Zee's QFT in a Nutshell.
But that's OK. I guess I am one or rare persons who does not like Weinberg's QFT2. (Even though I do like his QFT1.)

 

[Demystifier]

Can you explain the meaning of these enigmatic statements? Also in condensed matter physics the highest symmetry is reached at (asymptotic) high energies, where matter becomes an ideal gas of elementary particles (of quite probably yet unknown fundamental degrees of freedom), but that cannot be what the condensed-matter physicist wanted to express.

I would put it this way. Both communities use symmetry to understand the most interesting part of their branch of physics. In high-energy physics the most interesting question is what happens at the highest energies, for this is were new particles are expected to appear. By contrast, in condensed matter the high-energy level (atoms) is well understood and quite boring, while the most interesting phenomena, including new quasi-particles, happen at low energies (large scales).

In general, of course, old symmetries may disappear and new ones appear in both directions in the energy-scale, in both particle physics and cond mat physics. But it has to do with psychology in the two communities, with what is considered "interesting" and "important". People look for symmetry in the "interesting" things, whatever they are.

But why then symmetry is considered less important in cond mat? The general rule is that larger scales usually involve more complexity, and that more complexity usually involves less symmetry. Therefore in cond mat the most interesting things involve more complexity and hence less symmetry.

 

[dextercioby]

Speaking of symmetries, groups and representations, I was very disappointed with the new book by Zee
https://www.amazon.com/dp/0691162697/?tag=pfamazon01-20

In his previous books (Quantum Field Theory and Gravity) I have found a lot of new deep original insights, but that didn't happen with his last book on Group Theory. Perhaps it tells more about me than about the book, but I would certainly like to see what others think.

I think you are not addressing the purpose of the book by comparing it to his former 2. This is a book about applied mathematics, not a book about physics, thus you need to judge it from a different angle, i.e. not how well and full of insightful facts it teaches physics, but how well it teaches mathematics to (present/future to be) physicists. Therefore, we have some questions to ask:
1. Is this book necessary in the context of the available literature on this subject?
2. Is this book too abstract ?
3. Does it contain new facts/discoveries from mathematics and physics compared to, let's say the much older books by Barut & Raczka and Cornwell ?
4. Is the exposition clear enough, or is it
simply a tough reading

 

[Demystifier]

I think you are not addressing the purpose of the book by comparing it to his former 2. This is a book about applied mathematics, not a book about physics, thus you need to judge it from a different angle, i.e. not how well and full of insightful facts it teaches physics, but how well it teaches mathematics to (present/future to be) physicists. Therefore, we have some questions to ask:
1. Is this book necessary in the context of the available literature on this subject?
2. Is this book too abstract ?
3. Does it contain new facts/discoveries from mathematics and physics compared to, let's say the much older books by Barut & Raczka and Cornwell ?
4. Is the exposition clear enough, or is it
simply a tough reading

My answers:
1. No.
2. No.
3. I haven't find any.
4. Perhaps it's no so tough, but it's boring.

What I was hoping for is to see group theory from a new angle. I hoped that his book might change the way I think about groups and representations. That didn't happen, and that's why I was disappointed.

 

[vanhees71]

Interesting, you are the first person I know disappointed with the Zee's QFT in a Nutshell.
But that's OK. I guess I am one or rare persons who does not like Weinberg's QFT2. (Even though I do like his QFT1.)

I was particularly disappointed about QFT in a Nutshell. It's just too superficial. Just not mentioning the subtleties properly doesn't mean they are not there! Weinberg QT of Fields is, in my opinion, the best book on relativistic QFT for experts. It's not so good to start with. My favorite intro textbook for QFT is M. Schwartz, Quantum Field Theory and the Standard Model.

 

[A. Neumaier]

How can unitary group representations help to understand the measurement problem?

I didn't claim it would. But as you probably know from the discussions in this forum, I don't think that there is a measurement problem in quantum mechanics. At least not one deeper than the corresponding classical measurement problem, which is usually taken to be absent. Everyone who measures something knows how to apply the theory to match experiments, and that's all needed.

 

[Dr. Courtney]

That's a very healthy approach. Too much philosophy hinders the understanding of science ;-)).

I certainly agree at the level of late high school and early undergraduate, which is what the original Insight article seemed to be discussing (the absence of a starting point). I proffered my experience that a great high school chemistry course that focused on the periodic table provided a pretty good starting point.

At some point, original research (PhD level and beyond) in QM likely requires wrestling with the philosophical bits, but an awful lot of the applications of QM (intro through a lot of PhD and beyond atomic and molecular physics) can be accomplished in a satisfactory manner without wrestling with the philosophy of it.

A lot of confusion arises because the philosophical bits get introduced too early, that is before there is a sound foundation of the parts that are more immediately experimentally testable and that serve as the necessary basis for the periodic table and atomic physics. For me, a good two semester undergrad course in QM is likely the necessary starting point to really understand the philosophical aspects.

 

[Demystifier]

At some point, original research (PhD level and beyond) in QM likely requires wrestling with the philosophical bits

Perhaps by philosophical bits you don't mean the same thing as most of us do? Or perhaps you are working in a field such as quantum optics where quantum bits are more important than in most other branches of quantum physics? To check this out, can you name a few philosophical bits which you have in mind?

 

[RUTA]

I took 11 courses (grad and undergrad) in QM and QFT (to include solid state, nuclear, and particle physics) using several different texts and different approaches (historical, Dirac notation, Schrodinger eqn, etc.). How you decide to teach quantum physics depends on what problems you want to solve. If you want atomic and molecular energy levels, you don't need to worry about the measurement problem or violations of the Bell inequality, for example. As I said I in post #22 of this thread, I teach QM based on foundations of physics, so I only teach the weird stuff. I have since added the following two problems to my course https://www.physicsforums.com/insights/weak-values-part-1-asking-photons/ (based on experiment published in Phys. Rev. Lett. in 2013) and https://www.physicsforums.com/insights/weak-values-part-2-quantum-cheshire-cat-experiment/ (based on experiment published in Nature Comm in 2014). These analyses would be worthless in a course on chemical physics, for example.

 

[vanhees71]

Perhaps by philosophical bits you don't mean the same thing as most of us do? Or perhaps you are working in a field such as quantum optics where quantum bits are more important than in most other branches of quantum physics? To check this out, can you name a few philosophical bits which you have in mind?

Hm, where do you need philosophy for quantum optics? For me the fascinating thing about quantum optics is that you just do the very fundamental experiments discussed as thought experiments only in older textbooks. There's no need for philosophy at all but quite basic manipulations of bras and kets to predict the outcome of measurements.

 

[Demystifier]

Hm, where do you need philosophy for quantum optics? For me the fascinating thing about quantum optics is that you just do the very fundamental experiments discussed as thought experiments only in older textbooks. There's no need for philosophy at all but quite basic manipulations of bras and kets to predict the outcome of measurements.

Well, you are certainly right that quantum optics can be done without philosophy. Nevertheless, if you think about typical quantum-optics experiments such as those that involve violation of Bell inequalities, weak measurements, or delayed choice quantum erasers, philosophic questions occur more naturally than in other branches of quantum physics. Yes, you can resist thinking about philosophical aspects of such experiments if you have a strong character, but temptation is quite strong. Some experimentalists in that field even call it - experimental metaphysics. If you read the introductions of experimental papers published in Nature or Science, you will see that they use a lot of philosophy to explain why their results are important.