Why Is Quantum Mechanics So Difficult? - Comments

[microsansfil]

That's the whole point - they are semantic neutral.

Why can we derive the formalism of quantum theory from information-theoretic axioms, design with other very different concept ? The foundation of mathematic can also be buid with Category theory rather then Set theory.

Built physics with the Wheeler's «it from bit» point of view is also an other modern view.


Patrick

 

[bhobba]

Why can we derive the formalism of quantum theory from information-theoretic axioms, design with other very different concept ?

Its just the way things are - many roads lead to Rome.

BTW that's not an endorsement of the validity of any approach I haven't studied in detail.

But many physical theories such as classical mechanics have different but equivalent starting points.

Take a look at the Cox and Kolmogorov axioms - they are equivalent. Its simply the nature of the beast.

Thanks
Bill

 

[microsansfil]

Its just the way things are - many roads lead to Rome.

Logician would say : "The sense fails in nonsense like rivers into the sea". This means that semantics are determined by the syntax.

Example :

A circle is a set of points with a fixed distance, called the radius, from a point called the center.

I understand your point of view.

Patrick

 

[bhobba]

This is not a problem, unless we had the completely unjustified belief that the theory was exactly right.

As usual Frederk hit the nail on the head.

To apply it you need some rules to make sense of the math.

Its fairly obvious semantics won't resolve the type of issues Frederic pointed out.

That's where you need to add something like we ignore probabilities below a certain very small level as being irrelevant.

There are probably other ways, and discussing that may be interesting.

Thanks
Bill

 

[bhobba]

Logician would say : "The sense fails in nonsense like rivers into the sea"

That's probably a philosophical logician like Wittgenstein.

He had some well known debates about it with the mathematical logician, and very great mathematician (and Wittgenstien was equally as great - and - while not well known was actually well trained in the applied math of aeronautics - he started a Phd in it before being influenced by Russell and switched to philosophy) - Turing.

By 'it' I mean the foundations of applied math.

It was judged as a debate Wiggenstein may have won it - but later appraisal (by mathematicians of course ) gave it to Turing.

But this is getting into philosophy - which is off topic here.

If you want to pursue it the philosophy forums would be a better choice.

Thanks
Bill

 

[microsansfil]

If you want to pursue it the philosophy forums would be a better choice.

This is not a good argument.

No, behind there is the question about : can we reduce the physics to the mathematical axiomatic ( Proof theory ) ?

Patrick

 

[bhobba]

The mathematical theory of probability is now included in mathematical theory of measure.

Yea - Lebesgue integration and all that.

Fortunately in discussing the foundations of QM you don't need to worry about that because its enough to deal with finite discreet variables.

One then uses the Rigged Hilbert Space formalism to handle the continuous case.

Thanks
Bill

 

[bhobba]

This is not a good argument.

Its not an argument - its a statement of fact.

Philosophy is off-topic here.

If you go down that path, I will not respond, and the moderators will take action.

Discussing the modern axiomatic view of math would be on topic, the philosophy behind it, such as for example Wittgenstein's conventionalism, wouldn't.

Thanks
Bill

 

[microsansfil]

Philosophy is off-topic here.

This is why I'm not talking about philosphy. Why you see philosphy in my speech ? Is it a Straw man argument to impose your philosophy ?

The question is about axiomatize the physics

Discussing the modern axiomatic view of math would be on topicl

This is the point of the discussion.

Now this may be beyond the scope of this thread ?

Patrick

 

[bhobba]

Logician would say : "The sense fails in nonsense like rivers into the sea". This means that semantics are determined by the syntax.

I am not going to get into an argument about it - but stuff like the above IMHO is philosophy pure and simple.

I will not be drawn into it.

Thanks
Bill

 

[microsansfil]

I am not going to get into an argument about it - but stuff like the above IMHO is philosophy pure and simple.

It was a metaphor in response to your (you failed to write). I give a mathematical example in this context : http://en.wikipedia.org/wiki/Taxicab_geometry

Again Can we reduce physics to mathematical aximomatics ? Physical reduce itself to an applied science of mathematics?

Patrick

 

[bhobba]

Can we reduce physics to mathmematical aximomatics

You really need to start a new thread about that - its getting off topic.

But, as the only comment I will make here on it, attempts to do it, for example in QFT, leads to some extremely mind numbing math.

I used to ask questions like that in my degree.

The answer I got was I can give you some books that do just that - but you wouldn't read them.

He was right and it cured me.

BTW its nothing to do with semantics - its to do with rigour and reasonableness.

As an example it isn't hard to derive a Weiner process, but showing such actually exists is mathematically quite difficult. That's the difference between pure and applied math. Physically, because of the process it models, you believe it exists. But rigorously proving it is another matter.

Thanks
Bill

 

[stevendaryl]

This is not a problem, unless we had the completely unjustified belief that the theory was exactly right.

I waffle back and forth about the importance of understanding what QM is all about. If you take the sensible point of view that QM is not the ultimate theory, but a "good enough" theory, then a lot of the debate about foundations seems beside the point. Whether you believe in collapse of the wave function or not, whether you believe in Many Worlds or not, whether you believe in Bohmian nonlocal interactions or not, it just doesn't matter. When it comes to applying QM, we pretty much all agree on how to do it. We have a recipe for applying QM, and that recipe tells us enough about the meaning of QM to get on with doing science. There are lots of puzzling aspects of the various interpretations: What's special about measurement? What's happening between observations? How do these nonlocal correlations come about? Etc. But if you take the point of view that QM is just an incomplete theory, with operational semantics, and not anything ultimate, then it's really not that important that it answer all those questions. If you don't expect it to answer those questions, then it hardly matters what interpretation of QM you use.

On the other hand, the thing that is puzzling about QM as an incomplete theory is that there are no hints as to the limits of its applicability. There are no hints as to what more complete theory might replace it.

 

[stevendaryl]

As an example it isn't hard to derive a Weiner process, but showing such actually exists is mathematically quite difficult. That's the difference between pure and applied math. Physically, because of the process it models, you believe it exists. But rigorously proving it is another matter.

I'm a little puzzled about the role of rigor in physics. It seems that there are times when there are rigorous proofs that a certain thing is impossible, and physicists go ahead and do it, anyway. The example that comes to mind is Haag's theorem. I don't complete understand it, but based on a very superficial understanding, it seems to be saying that the techniques that physicists use in QFT, namely, starting with the free particle Hilbert space and viewing particle interactions via perturbation theory, can't work. But physicists do it and seem to get reasonable results. So what exactly is Haag's theorem telling us?

 

[microsansfil]

You really need to start a new thread about that - its getting off topic.

OK

Your speech on the proselytism of Ballentine is on this topic ?Patrick

 

[bhobba]

It seems that there are times when there are rigorous proofs that a certain thing is impossible, and physicists go ahead and do it, anyway.

Mate that is a deep question I have no answer for.

Zee says, correctly, there are many good physicists with the technological ability to do things like long mind numbing computations. But that doesn't make a great physicist - it's the ability to see into the heart of a problem. They are magicians - you can't go where they go. There have only been a few - Feynman, Landau, Einstein, Von Neumann come to mind.

Many people marvel at the technical virtuosity of Von-Neumann, but what really set him apart and made great mathematicians like Poyla scared of him was this magical ability to see to the heart of things - "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper.'

Feynman, no slouch in the Magician area himself, freely admitted Von-Neumann was his better.

Now we come to Einstein. Technically all those others I mentioned were way ahead of Einstein - they were all mathematical virtuosos. Not so Einstein - his math ability was quite ordinary - competent - but not spectacular. But his ability to see to the heart of an issue was above all those other greats - and that's what made him a greater physicist.

As they said about Feynman 'Feynman seemed to possesses a frightening ease with the substance behind the equations, like Albert Einstein at the same age, like the Soviet physicist Lev Landau—but few others.' That's the real key - the substance behind the math. Few have it - and its those that somehow, magically, know what to ignore, and what's important, that are great.

Thanks
Bill

 

[bhobba]

Your speech on the proselytism of Ballentine is on this topic ?

Have you actually been reading what I have been saying?

I have issues with Ballentine.

Its the best book on QM I have read - but perfect it aren't.

Look the exact divide between on and off topic is obviously a matter of opinion.

But I think most would say a discussion on the axiomatisation of physics is far wider than Why is QM So Difficult.

A discussion of exactly how Ballentine tackles the topic of QM would seem quite relevant

Its dead simple to start another thread - why get worried about it?

Thanks
Bill

 

[microsansfil]

Have you actually been reading what I have been saying?

I have issues with Ballentine.

The topic is about : Why Is Quantum Mechanics So Difficult ? isn't it ?

Perhaps that the possible divergence of view is an answer ?

Patrick

 

[bhobba]

Perhaps that the possible divergence of view is an answer ?

Indeed it is an answer - the semantic waffling of no actual mathematical content clouds the issue - as I have been discussing.

But the general axiomatisation of physics is beyond that.

Simply start a new thread.

It wouldn't be in the QM section - it would be in the general physics section.

Thanks
Bill

 

[bolbteppa]

Since I'm in the extremely small minority that dislikes Ballentine's book, let me say that I don't think the criticisms from Neumaier and Motl are that relevant to my point of view (although Neumaier and Motl may be correct, but I won't comment on that, since Ballentine's Ensemble interpretation itself appears to have changed between his famous erroneous review and the book, and Neumaier and Motl might be commeting on the review). Neither is the issue about the interpretation of probability important to me. Clearly, Copenhagen works despite its acknowledged problem of having to postulate an observer as fundamental. One cannot just declare that individual systems don't have states, or that collapse is wrong, since that would mean Copenhagen is wrong (Ballentine erroneously claims that Copenhagen is wrong, but my point if that even if we forgive him that, that does not fix his problems). The major approaches to interpretation never claim that Copenhagen is wrong. Rather, they seek to derive Copenhagen, but remove the observer as a fundamental component of the postulates. Ballentine doesn't even try to do that, and his theory has a Heisenberg cut, so it is not really an interpretation. Rather it is at best a derivation of Copenhagen or "Operational Quantum Theory" from axioms other than those found in Landau and Lifshitz, Shankar, Sakurai and Napolitano, Weinberg, or Nielsen and Chuang. Excellent examples in this spirit are those of Hardy http://arxiv.org/abs/quant-ph/0101012 or Chribella, D'Ariano and Perinotti http://arxiv.org/abs/1011.6451. So the question is does Ballentine's derivation work? I believe it doesn't, and that it is technically flawed.

The key question is whether Ballentine is able to derive his Eq 9.30. For comparison, one may see Laloe's treatment of the same equation in http://arxiv.org/abs/quant-ph/0209123, where it is Eq 37. If Ballentine did derive that equation, I think the other mistakes could be overlooked. If he did not, his interpretation has a hole and is not quantum mechanics.

Now should all approaches to interpretation be without flaw? No, but they should be clear where their flaws and issues are. For example, Wallace makes clear that the issue of how probability arises at all in Many-Worlds is still an issue, even if his derivation of the Born rule were to be correct. Similarly, there is the well known limitation that Bohmian Mechanics at present sits uncomfortably with exact Lorentz invariance. For the same reason, Landau and Lifshitz and Weinberg are excellent Copenhagen books because they explicitly point out the Heisenberg cut, rather than sweeping it under the rug.

Finally a bit of substance regarding this book. So Ballentine a) not only doesn't make the flaws explicit, b) he actually goes and claims Copenhagen is wrong? Mix that with c) You have to use a different system of probability (apparently equivalent after you do a ton of work and change your entire perspective of probability), d) you have to treat single particle systems in some weird way, & a potential e) your only benefit is fewer axioms at the expense of a less general form of QM, where as you say it's even questionable that he can achieve QM at all. I haven't read any of the guys bragging about Ballentine on here mention any of this stuff, these are such serious issues that I'm amazed tbh...

Why put yourself through such nonsense when you've got Landau, Dirac and Von Neumann sitting right there... I guess QM is so hard because people ignore the good books.

Thanks man

 

[microsansfil]

But the general axiomatisation of physics is beyond that.

But In the spécific of QM axiomatic is only your speech ? From the same axiomatic we can build different semantics. In mathematics is Model theory. The link between semantic and syntax is build by Gödel's completeness theorem.

Patrick

 

[atyy]

But In the spécific of QM axiomatic is only your speech ? From the same axiomatic we can build different semantics. In mathematics is Model theory. The link between semantic and syntax is build by Gödel's completeness theorem.

Yes, the derivations must put in some "semantics", or rather "physics". Semantics is the assignment of sets (and to use sets we have to have natural language) to meaningless symbols and grammar. Physics is the assignment of things we see and things we do to meaningless symbols and grammar. Even Euclidean geometry has different physical interpretations because of the duality between lines and points in the theory, so a physical line can correspond to a point in the theory. The derivations of Hardy or Chiribella et al start from the same physics background as standard Copenhagen - we assume a commonsense macroscopic world, and we know what a measurement (a little black box that takes an input and gives an output). They are alternative axioms for Copenhagen, in the same sense that the Hilbert action, the Palatini action and the Einstein field equations are different axioms for the same classical theory of gravity.

 

[bhobba]

he actually goes and claims Copenhagen is wrong?

Yes that's an error - one of its, fortunately, minor ones.

Why put yourself through such nonsense when you've got Landau, Dirac and Von Neumann sitting right there... I guess QM is so hard because people ignore the good books.

You mean Von-Neumann's thrashing of the Dirac Delta function that Ballentine rectifies? Things have moved on a lot since that classic was penned.

I am not going into the issues with the others, but will point out Ballentine is the only one of those that explains the true foundation of Schroedinger's equation etc - the symmetries of the POR.

Otherwise it looks basically like it's pulled out of a hat.

Dirac comes closest with his algebraic approach to Poisson Brackets but it doesn't explain why it holds. The POR is a general law applicable to all physics.

Thanks
Bill

 

[atyy]

Finally a bit of substance regarding this book. So Ballentine a) not only doesn't make the flaws explicit, b) he actually goes and claims Copenhagen is wrong? Mix that with c) You have to use a different system of probability (apparently equivalent after you do a ton of work and change your entire perspective of probability), d) you have to treat single particle systems in some weird way, & a potential e) your only benefit is fewer axioms at the expense of a less general form of QM, where as you say it's even questionable that he can achieve QM at all. I haven't read any of the guys bragging about Ballentine on here mention any of this stuff, these are such serious issues that I'm amazed tbh...

Why put yourself through such nonsense when you've got Landau, Dirac and Von Neumann sitting right there... I guess QM is so hard because people ignore the good books.

Thanks man

OK, maybe I was a bit hard on Ballentine claiming that Copenhagen is wrong. Strictly, speaking he only claims that his caricature of Copenhagen is wrong. But as you can see, even bhobba who likes the book makes far stronger criticisms of Ballentine's earlier interpretation - the claim that the earlier Ensemble Interpretation is secretly Bohmian is very strong criticism. Nothing wrong with being Bohmian of course, but the assumption should be stated clearly. Ballentine is vague enough, and doesn't even mention the Heisenberg cut, unlike Landau and Lifshitz or Weinberg, that I don't know if I agree with bhobba. But yes, if Ballentine is secretly Bohmian that would make a lot of sense, since one would then not need to add an assumption that proper and improper mixtures are equivalent, an assumption Ballentine makes in his book but fails to state. It also seems that Ballentine is secretly Many-Worlds, since he seems to want to have unitary evolution of the wave function and nothing else. Maybe he is secretly Bohmian Many-Worlds, which is possible, since Bohmian mechanics has unitary evolution of the wave function.

 

[bhobba]

Strictly, speaking he only claims that his caricature of Copenhagen is wrong. But as you can see, even bhobba who likes the book makes far stronger criticisms of Ballentine's earlier interpretation - the claim that the earlier Ensemble Interpretation is secretly Bohmian is very strong criticism.

Very true. BTW the BM thing is fixed in the book - but at a cost.

Don't get me wrong.

It has issues eg I think that propensity stuff is a crock of the proverbial - I wouldn't touch it with a barge pole.

But you have to look at it overall.

His explanation of the math, for example, is simply a cut above, even giving an overview of the important Rigged Hilbert Space formalism.

Thanks
Bill

 

[TrickyDicky]

On the other hand, the thing that is puzzling about QM as an incomplete theory is that there are no hints as to the limits of its applicability. There are no hints as to what more complete theory might replace it.

And if there were you couldn't mention them here anyway, so there are reasons to waffle on about beside the point interpretational debates.

 

[atyy]

Very true. BTW the BM thing is fixed in the book - but at a cost.

Don't get me wrong.

It has issues eg I think that propensity stuff is a crock of the proverbial - I wouldn't touch it with a barge pole.

But you have to look at it overall.

His explanation of the math, for example, is simply a cut above, even giving an overview of the important Rigged Hilbert Space formalism.

Thanks
Bill

Yes, I agree that Ballentine's presentation of the symmetries in the first few chapters is valuable, and hard to find elsewhere. So I would say use Ballentine for the "maths" (I put it in quotes because he presents it in a nice physicky way, which I don't know if strict mathematicians will like), but not so much for the interpretation, which is (at best) Copenhagen renamed.

 

[stevendaryl]

And if there were you couldn't mention them here anyway, so there are reasons to waffle on about beside the point interpretational debates.

Well, what I mean is this: People are pretty sure that General Relativity has to break down when it comes to conditions where both gravity and quantum mechanics are important. People knew that Schrodinger's equation wouldn't work relativistically. Fermi knew that his original model for weak interactions had to break down at high energy (because it wasn't renormalizable). Balmer knew that his formula for the energy spectrum of hydrogen can't possibly be the final theory, because it was clearly ad hoc. Einstein knew from early on that Special Relativity wouldn't work in cases where gravity was important. So a lot of theories of physics are provisional, and the people who create them already know that they aren't the final answer, and they often know the conditions under which their theories will turn out to be wrong. But QM is very different in this regard, in that nobody has a clue as to what conditions would cause it to break down.

 

[atyy]

Well, what I mean is this: People are pretty sure that General Relativity has to break down when it comes to conditions where both gravity and quantum mechanics are important. People knew that Schrodinger's equation wouldn't work relativistically. Fermi knew that his original model for weak interactions had to break down at high energy (because it wasn't renormalizable). Balmer knew that his formula for the energy spectrum of hydrogen can't possibly be the final theory, because it was clearly ad hoc. Einstein knew from early on that Special Relativity wouldn't work in cases where gravity was important. So a lot of theories of physics are provisional, and the people who create them already know that they aren't the final answer, and they often know the conditions under which their theories will turn out to be wrong. But QM is very different in this regard, in that nobody has a clue as to what conditions would cause it to break down.

Yes, there are two sorts of theories: those which can be a theory of some universe, and so experiment, and experiment alone tell us it must break down (eg. Newtonian gravity), while there are others where the theory itself tells us it must breakdown (eg. QED, if there is no asymptotic safety). Copenhagen itself suggests QM must breakdown, since Copenhagen typically does not acknowledge a wave function of the universe. Interpretations such as Bohmian Mechanics would place QM together with QED, and so far these are the only interpretations that are known to be without technical flaw (except maybe for chiral interactions). Bohmian Mechanics says that QM must break down, because it requires the quantum equilibrium condition, which is analogous to equilibrium in statistical mechanics. For the ensembles to emerge from a single reality, there has to be non-equilibrium in reality, but not detectable over the resolutions that we are able to access at the moment. If pure Many-Worlds works, then QM could conceivably be a theory of some universe, just like Newtonian gravity.

 

[TrickyDicky]

Well, what I mean is this: People are pretty sure that General Relativity has to break down when it comes to conditions where both gravity and quantum mechanics are important. People knew that Schrodinger's equation wouldn't work relativistically. Fermi knew that his original model for weak interactions had to break down at high energy (because it wasn't renormalizable). Balmer knew that his formula for the energy spectrum of hydrogen can't possibly be the final theory, because it was clearly ad hoc. Einstein knew from early on that Special Relativity wouldn't work in cases where gravity was important. So a lot of theories of physics are provisional, and the people who create them already know that they aren't the final answer, and they often know the conditions under which their theories will turn out to be wrong. But QM is very different in this regard, in that nobody has a clue as to what conditions would cause it to break down.

I'm not sure what you mean by saying that QM is very different in this regard, what are you calling QM exactly? Because the endless interpretational debates are mostly about "Schroedinger's QM", that you cite as an example of theory for which we we know what it means to break down.

 

[stevendaryl]

I'm not sure what you mean by saying that QM is very different in this regard, what are you calling QM exactly? Because the endless interpretational debates are mostly about "Schroedinger's QM", that you cite as an example of theory for which we we know what it means to break down.

I'm using QM in a more general sense than Schrodinger's equation. QFT is the quantum mechanics of fields.

 

[atto]

Strangely enough, QM's formalism isn't any more difficult than other areas of physics. The mathematics of the "standard" QM isn't any worse than, let's say, electromagnetism. Yet, to many people, especially non-physicists, QM presents a very daunting effort to understand.

Probably there is nothing to understand for a man which expects some new, original knowledge.
People want a new knowledge, but there is nothing new knowledge in the QM, but just a concept, convention, ie. a model with unrealistic ideas, entities like the half-spin, which is just a fundamental thing in this model.

You can't explain these using existing classical concepts. The line between these two is not continuous, at least, not as of now. How does one use classical idea of a "spin" to explain a spin 1/2 particle in which one only regains the identical symmetry only upon two complete revolutions? We simply have to accept that we use the same word, but to ONLY mean that it produces a magnetic moment. It has nothing to do with anything that's spinning classically. We can't build the understanding of the QM spin using existing classical spin that we have already understood.

That's the problem: the half-spin is just a mathematical sketch.

It is no a coincidence the Sommerfeld solution is identical with the solutions of the Dirac equation for the hydrogen like atom, despite Sommerfeld doesn't used any intrinsic spin concept.

The QM is just too much primitive, because completely artificial - numerical concept, thus this is unsatisfactory for people which are looking for a theory, ie. understanding, not a computational machine only.

QM is good enough maybe for engineers, but not for the real scientists.

 

[Sugdub]

… I strongly believe that it all comes down to how we understand things and how we expand our knowledge... One lacks any connection with the existing reality that one has understood...

In case you wish to consider the view of a non-physicist, I would suggest to introduce QM to newcomers as a pure phenomenology, since it is nothing else in a first instance:

1- the mathematical formalism of QM deals with a range of experiments which have a “potential” for producing a flow of random discrete events amongst a well defined set, each experiment being characterised by: i) a reproducible statistical distribution, and ii) a continuous variation of this distribution in response to a continuous variation of a single operational parameter, for example the relative orientation or the distance between two devices in the experimental set-up.
The “potential” (understood as a property of the experiment, not as a property of an hypothetical “system” located in the world) can be formally represented by the orientation of a unit vector in a Hilbert space (the list of cosines which define this orientation therefore corresponds to the list of coordinates of the usual “state vector”).

2- one can infer the general form of the equation which predicts the evolution of the potential in response to a continuous change of the experimental context, namely when the variable parameter changes value in a continuous way, under the assumption that this evolution is independent of the initial state.

3- an extension of this formalism can be derived dealing with nested experimental setups such as the addition of new “analysers” in a series. These are typically non-continuous changes of the experimental set-up, and they naturally translate into discontinuous evolutions of the potential insofar it gets projected onto a different base of the same Hilbert space. It is essential to note that as long as it is not interpreted as a property of “something” located inside the experimental device, the potential is a-local. Therefore the famous “measurement problem” cannot arise (the “collapse” of the state vector is assumed to occur inside the experimental device).

4- a further extension of the formalism deals with the combination of two contexts of the same family, leading to the combination of the contributing potentials into a new one. The distribution observed derives from the new combined potential, not from a direct combination of distributions.

My recommendation would be to proceed through this purely phenomenological presentation of the QM formalism which never suggests that the “potential” might represent “something of the world”, and clearly refrains from promoting the belief that QM ought to be a “physics theory”, I mean a theory describing what there is in the world, how it works or what happens there inside the experimental device. This approach would be extremely concrete, directly connected to a series of well-known experiments. Emphasis would be made on clarifying which subset of the QM formalism can be derived on the basis of pure phenomenological considerations, taking due account of the symmetries within each context of the experimental setup and within the family of all contexts explored through varying the operational parameters: students should be taught the exemplary rationality of QM as a phenomenology before being prompted with the intricacies and paradoxes resulting from its interpretations as a physics theory. I think this approach could resolve the issue raised by the OP whereby: “One lacks any connection with the existing reality that one has understood.”

 

[Sandalwood]

From my experience, undergraduate QM wasn't too difficult. Yes, there were things that weren't fully explained, but if you were willing to take them for granted and follow a few simple rules, it wasn't bad at all. The real hard stuff comes at the graduate level, and I think you need a really good grasp of classical mechanics to truly understand what's going on. Bohm's book Quantum Theory is quite possibly the best QM text I've come across. He highlights the parallels between CM and QM and also draws from what was known from experiments at the time. So it doesn't feel like you're learning QM by pulling random stuff out of thin air. Everything is explained very clearly. I highly recommend the book.

 

[Rabin D Natha]

Author: ZapperZ
Originally posted on Jun16-14

Strangely enough, QM's formalism isn't any more difficult than other areas of physics. The mathematics of the "standard" QM isn't any worse than, let's say, electromagnetism. Yet, to many people, especially non-physicists, QM presents a very daunting effort to understand.

I strongly believe that it all comes down to how we understand things and how we expand our knowledge. Typically, when we teach students new things, what we do is build upon their existing understanding. We hope that a student already has a foundation of knowledge in certain areas, such as basic mathematics, etc., so that we can use that to teach them about forces, motion, energy, and other fun stuff in intro physics. Then, after they understand the basic ideas, we show them the same thing, but with more complications added to it.

The same thing occurs when we try to help a student doing a homework problem. We always try to ask what the student know already, such as the basic principle being tested in that question. Does the student know where to start? What about the most general form of the equation that is relevant to the problem? Once we know a starting point, we then build on that to tackle that problem.

The common thread in both cases is that there exists a STARTING point as a reference foundation on which, other "new" stuff are built upon. We learn new and unknown subject based upon what we have already understood. This is something crucial to keep in mind because in the study of QM, this part is missing! I am certain that for most non-physicists, this is the most common reason why QM is puzzling, and why quacks and other people who are trying to use QM into other areas such as "metaphysics" or mysticism, are using it in a completely hilarious fashion.

There is a complete disconnect between our "existing" understanding of the universe based on classical understanding, and QM. There is nothing about our understanding of classical mechanics that we can build on to understand QM. We use the identical words such as particle, wave, spin, energy, position, momentum, etc... but in QM, they attain a very different nature. You can't explain these using existing classical concepts. The line between these two is not continuous, at least, not as of now. How does one use classical idea of a "spin" to explain a spin 1/2 particle in which one only regains the identical symmetry only upon two complete revolutions? We simply have to accept that we use the same word, but to ONLY mean that it produces a magnetic moment. It has nothing to do with anything that's spinning classically. We can't build the understanding of the QM spin using existing classical spin that we have already understood.

Now interestingly enough, the MATHEMATICAL FORMULATION of QM is quite familiar! The time-dependent Schrodinger equation has the same structure as a standard wave equation. We call the energy operator as the Hamiltonian not for nothing since it looks very familiar with the hamiltonian approach to classical mechanics. The matrix formulation also isn't anything new. What this means is that while the conceptual foundation of QM is completely disconnected with our traditional conceptual understanding, the mathematical formulation of QM completely follows from our existing understanding! Mathematically, there is no discontinuity. We build the formalism of QM based on our existing understanding!

This is why, in previous threads in PF, I disagree that we should teach students the concepts of QM FIRST, rather than the mathematical formulation straightaway. There is nothing to "build on" in terms of conceptual understanding. We end up telling the students what they are out of thin air. The postulates of QM did not come out of our classical understanding of our world. Instead, the mathematical formalism is the only thing that saves us from dangling in mid air. It is the only thing in which our existing understanding can be built on.

What this implies clearly is that, if one lacks the understanding of the mathematical formalism of QM, one really hasn't understood QM at all! One ends up with all these weird, unexplained, unfamiliar, and frankly, rather strange ideas on how the world works. These conceptual description QM may even appear "mystical". It is not surprising that such connections are being made between QM and various forms of mysticism. One lacks any connection with the existing reality that one has understood. So somehow, since QM can do this, it seems as if it's a license to simply invent stuff weely neely.

The mathematical formalism of QM is what defines the QM description. The "conceptual description" is secondary, and is only present because we desire some physical description based on what we already have classically. It is why people can disagree on the interpretation of QM, yet they all agree on the source, the mathematical formalism of QM.

This, however, does not mean that QM is nothing more than "just mathematics". This is no more true than saying the musical notes on a sheet of paper are just scribbles. The notes are not the important object. Rather, it is the sound that it represents that's the main point. The musical notes are simply a means to convey that point clearly and unambiguously. Similarly, the mathematics that is inherent in QM and in all of physics, is a means to convey an idea or principle. It is a form of communication, and so far it is the ONLY form of communication accurate and unambiguous enough to describe our universe. It reflects completely our understanding of a phenomena. So a mathematical formulation isn't "just math".

You cannot use your existing understanding of the universe to try to understand the various concepts of QM. There is a discontinuity between the two. It is only via the mathematical continuity of the description can there be a smooth transition to build upon. Without this, QM will not make "sense".

Author: ZapperZ
Originally posted on Jun16-14

QM seeks to explain the real, rational physical universe. A good teacher can explain it in real, rational, physical language. Too often, specialists create their own unique worldview and lose touch with the ordinary uniververse. If you can't explain it, it has no value outside its unique community. In the ordinary world, Schrödinger's Cat is stuck in a poorly conceived experiment with a nonsensical hypothesis.