Why Is Quantum Mechanics So Difficult?
Table of Contents
Quantum Mechanics Key points:
- Quantum mechanics (QM) is often perceived as difficult, especially by non-physicists.
- The difficulty lies in the conceptual foundation of QM, as it doesn’t connect well with classical understanding.
- Unlike other areas of physics, there’s no direct continuity between classical concepts and QM.
- While the conceptual understanding is challenging, the mathematical formulation of QM is familiar and follows from existing knowledge.
- Mathematical formalism is crucial in QM, as it provides a bridge between classical and quantum worlds.
- QM’s mathematical formalism is the foundation on which our understanding is built.
- Disagreements often arise in the interpretation of QM, but the source, mathematical formalism, remains consistent.
- The mathematical aspect of QM is a means of conveying ideas and principles accurately, akin to musical notes conveying music.
- In QM, mathematics is a form of communication that accurately describes our universe.
- Quantum mechanics doesn’t fully make “sense” without embracing its mathematical continuity.
QM’s formalism
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 knows 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 is built upon. We learn new and unknown subjects based on 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 in other areas such as “metaphysics” or mysticism, are using it in a completely hilarious fashion.
The Disconnect
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 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 the classical idea of a “spin” to explain a spin 1/2 particle in which one only regains the identical symmetry 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 the 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 the Hamiltonian not for nothing since it looks very familiar with the Hamiltonian approach to classical mechanics. The matrix formulation isn’t anything new. What this means is that while the conceptual foundation of QM is completely disconnected from 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!
Mathematical formulation
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, mathematical formalism is the only thing that saves us from dangling in mid-air. It is the only thing on which our existing understanding can be built.
What this implies is that, if one lacks the understanding of the mathematical formalism of QM, one 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 descriptions of 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..
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.
QM as musical notes
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 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 is a smooth transition to build upon. Without this, QM will not make “sense”.
PhD Physics
Accelerator physics, photocathodes, field-enhancement. tunneling spectroscopy, superconductivity
[quote="microsansfil, post: 4826939"]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 :tongue::tongue::tongue::tongue:) 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.ThanksBill
[quote="Fredrik, post: 4826676"]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 wont 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.ThanksBill
[quote="bhobba, post: 4826931"]Its just the way things are – many roads lead to Rome.[/Quote]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
[quote="microsansfil, post: 4826920"]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.ThanksBill
[quote="bhobba, post: 4826913"]That's the whole point – they are semantic neutral.[/Quote]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
[quote="microsansfil, post: 4826909"]I don't know Ballentine "Point of view". Is it an oher interpretation of MQ or is it a new axiomatic of MQ ?”His view is similar to Popper – and would not be my choice of how to attack it.The key point I am trying to get across is his arguments depend on the axioms – not how you interpret them.I have already posted my derivation of the two axioms that starts with a single axiom:'An observation/measurement with possible outcomes i = 1, 2, 3 ….. is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.'That way you don't have to show its compatible with probability – its there right from the start – without any semantic baggage.It's clearer IMHO what's going on that way.Of course Ballentine isn't wrong – but as this thread shows it gets caught up in semantic baggage.ThanksBill
[quote="microsansfil, post: 4826884"]Formal systems seem to be rigid because purely syntactic, but their semantics embedded in the axioms is unspoken.”That's the whole point – they are semantic neutral.Again – read what Feller said:'We shall no more attempt to explain the true meaning of probability than the modern physicist dwells on the real meaning of the mass and energy or the geometer discusses the nature of a point. Instead we shall prove theorem's and show how they are applied'This is the modern view.BTW when I say modern it developed during the 19th century where a more cavalier attitude caused problems (eg 1 – 1 + 1 – 1 ….. converged in naive Fourier series) and permeated all of modern pure and applied math – including physics. Many say the pure guys went a bit too far, which led to a bit of good natured ribbing between applied and pure camps, but both have taken on the central lesson.ThanksBill
[quote="bhobba, post: 4826904"]Mate all I am asking is for you to detail the point you are trying to make because I am confused about it.What Ballentine does is show the probability axioms are consistent with his two axioms. “I don't know Ballentine "Point of view". Is it an oher interpretation of MQ or is it a new axiomatic of MQ ?Patrick
[quote="bhobba, post: 4826864"]What probability is, is defined by the Kolmogorov axioms.”The area of relevance of a formal system is confined – by design – to the field of relevance of a hidden semantic, whose presence is unspoken.indeed,there is a comparability between other formalism like Cox-Jaynes’s approach to probability and de Finetti. Yet as written E.T Jaynes [Quote]In summary, we see no substantive conflict between our system of probability and Kolmogorov’s as far as it goes; rather, we have sought a deeper conceptual foundation which allows it to be extended to a wider class of applications, required by current problems of science. [/Quote]Patrick
[quote="microsansfil, post: 4826895"]What is the meaning of "work" in the context of interpretation ?”Mate all I am asking is for you to detail the point you are trying to make because I am confused about it.What Ballentine does is show the probability axioms are consistent with his two axioms. He calls probability propensity, but that's not really relevant; philosophers get caught up in that sort of thing but mathematically it the axioms whatever it is obeys that's important. He uses the Cox axioms, but they are equivalent to the Kolmogorov axioms.That implies the existence of ensembles which is all that is required – its got nothing to do with the semantics of the situation.Is that what you mean by information theoretic?If so information theoretic is not what I would use – axiomatic based would be my description.Added Later:While I was penning the above you did another post that hopefully clarified what you had in mind. Will address that. ThanksBill
[quote="bhobba, post: 4826880"] about Ballentine's interpretation?”About : " So the question is does Ballentine's derivation work?" included im my quote is simply a mistake of cut and paste.What is the meaning of "work" in the context of interpretation ?Patrick
[quote="bhobba, post: 4826880"]Can you detail the relevance to Atty's statement about Ballentine's interpretation?”Formal systems seem to be rigid because purely syntactic, but their semantics embedded in the axioms is unspoken. In MQ i agree with the point of view that axiomatization has to be based on postulates that can be precisely translated in mathematical terms but not vice versa. The Alexei Grinbaum's work is an example among others. Patrick
[quote="microsansfil, post: 4826870"]Alexei Grinbaum "THE SIGNIFICANCE OF INFORMATION IN QUANTUM THEORY"”Can you detail the relevance to Atty's statement about Ballentine's interpretation?ThanksBill
[quote="atyy, post: 4826810"]Chribella, D'Ariano and Perinotti http://arxiv.org/abs/1011.6451. So the question is does Ballentine's derivation work? “Alexei Grinbaum "THE SIGNIFICANCE OF INFORMATION IN QUANTUM THEORY"[Quote]Interest toward information-theoretic derivations of the formalism of quantum theory has been growing since early 1990s thanks to the emergence of the field of quantum computation.In Part II we derive the formalism of quantum theory from information-theoretic axioms. After postulating such axioms, we analyze the twofold role of the observer as physical system and as informational agent. Quantum logical techniques are then introduced, and with their help we prove a series of results reconstructing the elements of the formalism. One of these results, a reconstruction theorem giving rise to the Hilbert space of the theory, marks a highlight of the dissertation. Completing the reconstruction, the Born rule and unitary time dynamics are obtained with the help of supplementary assumptions. We show how the twofold role of the observer leads to a description of measurement by POVM, an element essential in quantum computation.”Patrick
[quote="microsansfil, post: 4826545"]Here a critique of Popper's interpretation of quantum mechanics and the claim that the propensity interpretation of probability resolves the foundational problems of the theory”Without even reading it, its fairly obvious calling probability propensity, plausibility or any other words you can think of, will not change anything.What probability is, is defined by the Kolmogorov axioms.The rest is simply philosophical waffle IMHO.Those axioms all by themselves are enough, via the law of large numbers, to show Ballintines ensembles conceptually exist, which is all that required to justify his interpretation.If you think of probability as some kind of plausibility then you get something like Copenhagen – although the law of large numbers still applies and you can also conceptually define ensembles if you wish.I sometimes say guys with a background in applied math like me and philosophers sometimes talk past one another.Here's an example from Rub's paper:'The propensity interpretation may be understood as a generalization of the classical interpretation. Popper drops the restriction to "equally possible cases," assigning "weights" to the possibilities as "measures of the propensity, or tendency, of a possibility to realize itself upon repetition." He distinguishes probability statements from statistical statements. Probability statements refer to frequencies in virtual (infinite) sequences of well-defined experiments, and statistical statements refer to frequencies in actual (finite) sequences of experiments. Thus, the weights assigned to the possibilities are measures of conjectural virtual frequencies to be tested by actual statistical frequencies: "In proposing he propensity interpretation I propose to look upon probability statements as statements about some measure of a property (a physical property, comparable to symmetry or asymmetry) of the whole experimental arrangement; a measure, more precisely, of a virtual frequency'My view is just like Fellers:'We shall no more attempt to explain the true meaning of probability than the modern physicist dwells on the real meaning of the mass and energy or the geometer discusses the nature of a point. instead we shall prove theorem's and show how they are applied'Conceptual infinite ensembles are easily handled by simply assuming there is a very small probability below which it is indistinguishable in practical terms from zero. If you do that the law of large numbers leads to large, but finite ensembles. For example we know there is a very small probability all the atoms in a room will go in the same direction at once and levitate a chair into the air – but in practice it never happens – we can safely assumes probabilities that small can be neglected – just like in calculus at an applied level we often think of dx as a small increment in x such that dx^2 can be ignored.That's why guys with my background and those with a philosophical bent sometimes talk past each other.ThanksBill
[quote="atyy, post: 4826810"]since Ballentine's Ensemble interpretation itself appears to have changed between his famous erroneous review”It did.He had to take on board Kochen-Specker.He assumed, initially (in his original review article), it had the property when measured. Kochen-Specker says you cant do that. Fredrick put his finger on it – originally it was basically BM in disguise.However with decoherence you can do that – but of course it still doesn't fully resolve the measurement problem – which looked like was his hope.ThanksBill
[quote="bhobba, post: 4826703"]We are advocating the modern version where it is based on the Kolmogorov axioms (or equivalent) “This leave with the impression that Kolmogorov’s axiomatization was born full grown. Kolmogorov only translates probability concept, well known many years later, into an axiomatic/formal mathematical language. The mathematical theory of probability is now included in mathematical theory of measure.The measurement theory is the branch of mathematics that deals with measured spaces and is the axiomatic foundation of probability theory.The basic intuition in probability theory remain the notion of randomness based on the notion of random variable.There are certain ‘non commutative’ versions that have their origins in quantum mechanics, for instance K. R.Parthasarathy (an introduction to quantum stochastic calculus), that are generalizations of the Kolmogorov Model. Patrick
[quote="bolbteppa, post: 4826363"]From all the comments on Ballentine I've read on here that stick in my head, the only benefit compared to Landau is that a) it's easier than Landau, b) you can prove one or two things Landau assumes (though apparently at the price of a less general form of QM) as long as you take a different interpretation of QM to that of Landau, an interpretation that, at best, is ultimately no more justifiable than Landau's perspective, and at worse is less general. In that light, it seems like the book is a waste of time, but I'm happy to be wrong.”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.
[quote="bolbteppa, post: 4826363"]That's an extremely important distinction in the sense that, logically, it's very different from taking the crass frequentist interpretation that you implied”That's the precise problem. Ballentine and I are not advocating a 'crass' frequency interpretation. We are advocating the modern version where it is based on the Kolmogorov axioms (or equivalent) and applying the law of large numbers.It matters not if you call it propensity, plausibility, or leave it it semantically open, it implies exactly the same thing. ThanksBill
[quote="bolbteppa, post: 4826363"]what about the issue of uniqueness of the limit that me and Ballentine brought up? “It's not an issue. The assignment of probabilities in the purely mathematical part of the theory, is just an assignment of relative sizes to subsets. These assignments tell us nothing about the real world on their own. That's why the theory consists of the mathematics and a set of correspondence rules that tell us how to interpret the mathematics as predictions about results of experiments. Those rules tell us that the relative frequency of a particular result in a long sequence of identical measurements, will be equal to the probability that has been assigned to (a subset that represents) that particular result.The correspondence rules can't just say that probabilities are propensities, because we need to know how to test the accuracy of the theory's predictions. If we can't, it's not a theory.The non-existence of a limit wouldn't be relevant even if we had a theory that has a chance of being exactly right, because1. You can't perform an infinite sequence of measurements.2. The measurements won't be perfectly accurate.3. The measurements won't be identical.4. If a very long sequence of identical measurements would (for example) sometimes go into the interval 1.000000001-1000000002 and then hop around inside it, and in another experiment go into the interval 1.0000000005-10000000006 and then hop around inside it, the conclusion would be that somewhere around the tenth decimal, we're hitting the limits of the theory's domain of validity. This is not a problem, unless we had the completely unjustified belief that the theory was exactly right.
[quote="stevendaryl, post: 4826375"]So in the context of QM in the density-matrix approach, pure states represent propensities, while mixed states combine propensities and subjective probabilities?”Here a critique of Popper's interpretation of quantum mechanics and the claim that the propensity interpretation of probability resolves the foundational problems of the theory, by Jeffrey Bub.Patrick
I mentioned–either in this thread, or another–the "propensity" interpretation of probabilities, but in my opinion, it's not an interpretation, at all. It's just another word for "probability". Maybe it's supposed to be that part of probability that is left over after all probabilities due to ignorance are stripped away. So in the context of QM in the density-matrix approach, pure states represent propensities, while mixed states combine propensities and subjective probabilities?
[quote="bhobba, post: 4825898"]Sure they need not remain homogeneous – but the conceptualisation is they do – its a straw man argument.Many, many books explain the validity of the frequentest interpretation when backed by the Kolmogorov axioms eg http://www.amazon.com/Introduction-Probability-Theory-Applications-Edition/dp/0471257087[/quote]Having looked through Feller, he actually doesn't claim that the frequency interpretation of probability is justified by Kolmogorov's axioms, and just to be clear – if such a passage actually existed then it would imply both me and Ballentine are wrong when we say frequentist probability is flawed. Ballentine mentions this issue uniqueness of the limit on page 32:”One of the oldest interpretations is the limit frequency interpretation. If the conditioning event C can lead to either A or ∼A, and if in n repetitions of such a situation the event A occurs m times, then it is asserted that P(A|C) = limn→∞(m/n). This provides not only an interpretation of probability, but also a definition of probability in terms of a numerical frequency ratio. Hence the axioms of abstract probability theory can be derived as theorems of the frequency theory. In spite of its superficial appeal, the limit frequency interpretation has been widely discarded, primarily because there is no assurance that the above limit really exists for the actual sequences of events to which one wishes to apply probability theory.The defects of the limit frequency interpretation are avoided without losing its attractive features in the propensity interpretation. The probability P(A|C) is interpreted as a measure of the tendency, or propensity, of the physical conditions describe by C to produce the result A. It differs logically from the older limit-frequency theory in that probability is interpreted, but not redefined or derived from anything more fundamental. It remains, mathematically, a fundamental undefined term, with its relationship to frequency emerging, suitably qualified, in a theorem. It also differs from the frequency theory in viewing probability (propensity) as a characteristic of the physical situation C that may potentially give rise to a sequence of events, rather than as a property (frequency) of an actual sequence of events.”Calling my argument a strawman argument is calling Ballentine's argument a strawman argument. I notice you only focused on homogeneity, but what about the issue of uniqueness of the limit that me and Ballentine brought up? [quote="bhobba, post: 4825898"]He does use propensity – but I think he uses it simply as synonymous with probability – most certainly in the equations he writes that's its meaning.[/quote]As the quote from Ballentine given above shows, it seems he uses this word as a way to give the closest thing to a frequentist interpretation possible, but qualifies this by saying it's merely a word given to a theorem proven from Cox's axioms. That's an extremely important distinction in the sense that, logically, it's very different from taking the crass frequentist interpretation that you implied, and doubly important since you are claiming both that frequentist probability can be justified by Kolmogorov's axioms and that Ballentine is taking a frequentist interpretation when he clearly says he isn't…So he's not using frequentist probability, he's using Cox's probability axioms and just interpreting some theorems in a way that lies closest to a frequentist interpretation possible. That's fine, but had I not checked that out I'd be left with a completely wrong impression of Ballentine based on this thread.[quote="bhobba, post: 4825898"]To be frank I don't even understand Lubos's criticism – mind carefully explaining it to me?[/quote]All I'm going off is the conclusion which is that all we really get from the ensemble interpretation is a restricted and modest view of the power of QM. Hopefully someone who understands it fully will be able to challenge it.[quote="bhobba, post: 4825898"]You should. But what leaves me scratching my head is you seem to have all these issues with it – but haven't gone to the trouble to actually study it. I could understand that if it was generally considered crank rubbish – but it isn't. Its a very well respected standard textbook. It is possible for sources of that nature to have issues – and it does have a couple – but they are very minor.[/quote]Well I want to find out about the book, which is why I'm posting. Thus far I have been given the impression that it's based on frequentist probability, and been told such a position can be justified by Kolmogorov's axioms, when in fact the book explicitly says it's not based on frequentist probability and actually uses Cox's axioms. Then we have the two main issues, one about the theory applying to a single particle, which may be more complicated than Neumaier implied http://physics.stackexchange.com/a/15553/25851 and also Lubos' claim that all we really get from the ensemble interpretation anyway is just a restricted and modest view of the power of QM. Sounds awfully unappealing at this stage.From all the comments on Ballentine I've read on here that stick in my head, the only benefit compared to Landau is that a) it's easier than Landau, b) you can prove one or two things Landau assumes (though apparently at the price of a less general form of QM) as long as you take a different interpretation of QM to that of Landau, an interpretation that, at best, is ultimately no more justifiable than Landau's perspective, and at worse is less general. In that light, it seems like the book is a waste of time, but I'm happy to be wrong.
[quote="bhobba, post: 4826312"]just a bit different.”In his book E.T Jaynes write.[Quote]Foundations: From many years of experience with its applications in hundreds of real problems, our views on the foundations of probability theory have evolved into something quite complex, which cannot be described in any such simplistic terms as pro-this" or anti-that." For example, our system of probability could hardly be more different from that of Kolmogorov, in style,philosophy, and purpose. What we consider to be fully half of probability theory as it is needed in current applications the principles for assigning probabilities by logical analysis of incomplete information|is not present at all in the Kolmogorov system.As noted in Appendix A, each of his axioms turns out to be, for all practical purposes, derivable from the Polya-Cox desiderata of rationality and consistency. In short, we regard our system of probability as not contradicting Kolmogorov's; but rather seeking a deeper logical foundation that permits its extension in the directions that are needed for modern applications. In this endeavor, many problems have been solved, and those still unsolved appear where we should naturally expect them: in breaking into new ground.However, our system of probability differs conceptually from that of Kolmogorov in that we do not interpret propositions in terms of sets, but we do interpret probability distributions as carriers of incomplete information. Partly as a result, our system has analytical resources not present at all in the Kolmogorov system. This enables us to formulate and solve many problems- particularly the so-called "ill posed" problems and "generalized inverse" problems – that would be considered outside the scope of probability theory according to the Kolmogorov system. These problems are just the ones of greatest interest in current applications.[/Quote]E.T Jaynes purposefully do not use the term “random variable”, as it is a much too restrictive a notion, and carries with it all the baggage of the Kolmogorov approach to probability theory, but a random variable seem to be an example of an unknown/incomplete information.Possible point of view : Quantum mechanics is basically a mathematical recipe on how to construct physical models. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics.For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion.Patrick
[quote="microsansfil, post: 4826284"]it seems to me that you have also an other possibility develop by E.T.Jaynes "probability theory as an extension of logic". In this context probability is not reduce to random variables. “That's the Bayesian view where its how plausible something is.What you do is come up with reasonable axioms on what plausibility should be like – these are the so called Cox axioms. They are logically equivalent to the Kolmogorov axioms where exactly what probability is is left undefined.Ballentine bases it on those axioms but called it propensity – which isn't really how Coxes axioms are usually viewed. It's logically sound since its equivalent to Kolomogorovs axioms – just a bit different.In applied math what's usually done is simply to associate this abstract thing called probability defined by the Kolmogerov axioms with independent events. Then you have this thing called the law of large numbers (and its a theorem derivable from those axioms) which basically says if you do a large number of trials the proportion of outcomes tends toward the probability. That's how you make concrete this abstract thing and its certainly how I suspect most people tend to view it.Basically what Ballentine does it look at probability as a kind of propensity obeying the Cox axioms. Then he uses the law of large numbers to justify his ensemble idea.There is no logical issues with this, but personally I wouldn't have used propensity – simply an undefined thing as per Kolomogorov's axioms.But really its no big deal.ThanksBill
[quote="bhobba, post: 4826233"]If you think in terms of plausibility, states of knowledge etc, you get something like Copenhagen. If, regardless of how you view probability, plausibility, something abstract as in the Kolmogorov axioms, it doesn't really matter, but apply the law of large numbers you get something like the ensemble, which is very frequentest like.”it seems to me that you have also an other possibility develop by E.T.Jaynes "probability theory as an extension of logic". In this context probability is not reduce to random variables. A proof of Cox’s Theorem.Patrick
[quote="kith, post: 4826221"]The question is do you want to learn about the physics or the metaphysics?”I dug up my copy of Feller and reacquainted myself with what he says.From page 3'We shall no more attempt to explain the true meaning of probability than the modern physicist dwells on the real meaning of the mass and energy or the geometer discusses the nature of a point. instead we shall prove theorem's and show how they are applied'And that's exactly what going on here. I mentioned the fundamental axiom I applied Gleason to:'An observation/measurement with possible outcomes i = 1, 2, 3 ….. is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.'Probability is an assumed primitive of the theory. Its described by the Kolmogorov axioms. You can apply the law of large numbers and get a frequentest view – that would be Balentines Ensemble. You can call it 'propensity' – what difference it makes is beyond me. My view is Fellers – its simply an assumed primitive. You can view it as plausibility, state of knowledge and get something like Copenhagen. That's fine. I like Ballentine because it's pictorially nice – you simply view an observation as selecting an element from an ensemble. But that's all there is to it – it simply appeals to me.ThanksBill
[quote="kith, post: 4826213"]As often, I am skeptical about whether this criticism is specific to the quantum case. It seems to me that Lubo's thought experiment is not much different to the throwing of real (non-identical) coins. If a certain probability interpretation can be applied to this situation I think it can also be applied to the quantum case.”Mate – I can't follow it at all – I have zero idea what he is driving at.I am also scratching my head at Bolbteppa's exact concern.As far as I can see it's that Ballentine uses the term 'propensity' to describe probability, rather than say plausibility like Bayesian's do.I think philosophers get caught up in terms like that, but my background is applied math, and I really can't see the point. If you think in terms of plausibility, states of knowledge etc, you get something like Copenhagen. If, regardless of how you view probability, plausibility, something abstract as in the Kolmogorov axioms, it doesn't really matter, but apply the law of large numbers you get something like the ensemble, which is very frequentest like.I think most applied math types with a background in stochastic modelling (which is what I have) view it in a frequentest way backed by the Kolmogorov axioms via the law of large numbers. Most certainly books like Feller, and Ross (Introduction to Probability Models) that I have view it that way. For example its the simplest way to view the important limit theorems of Markov chains.There is an issue with the law of large numbers in that it converges in probability or almost assuredly so a bit of care is required. But it's not a particularly difficult thing – you simply assume that some very small probability is for all practical purposes zero – its the type of thing you do in applied math all the time. I have discussed this sort of thing before, but I still dont understand why people worry about it – I guess it's a philosophy thing.ThanksBill
[quote="bolbteppa, post: 4825624"]To be clear, I haven't read Ballentine. Lubos's issues already turned me off a while ago, so I posted here to find a reason to give it a chance. After just finding out I also have to throw away Kolmogorov probability, I'm now even less inclined, but I'd still consider it if there's good enough a reason. Are you guys aware this is how deep into the rabbit hole you have to go?”The question is do you want to learn about the physics or the metaphysics?For the first part, Ballentine is an excellent book. I know a good deal of standard textbooks and the only other book which gave me a similar feeling of understanding important things about the physics is Sakurai. Ballentine talks about quite a few things which I haven't read anywhere else and he goes more into detail than Sakurai (for example when he examines the implications of Galilean symmetry). On the other hand I really like Sakurai's writing style. I recommend to just try which book suits you better. As for Landau / Liflshitz and Weinberg, I have only skimmed them. They seem to be good books but I can't comment on them in detail.For the second part, working through Ballentine completely is probably overkill. There are however many thought-provoking bits in different parts of the book and he is very outspoken about his opinion on interpretational issues. I think his view makes sense but even if you have issues with it, it will probably be enriching to read what he thinks. What I don't like is that he doesn't present it as an opinion.
[quote="bolbteppa, post: 4825624"]I can see how that applies to Neumaier's criticism, but it doesn't say anything about Lubos' criticism of the ensemble interpretation as being nothing but a restricted and modest view of the power of QM. I'm curious what people think of this criticism.”As often, I am skeptical about whether this criticism is specific to the quantum case. It seems to me that Lubo's thought experiment is not much different to the throwing of real (non-identical) coins. If a certain probability interpretation can be applied to this situation I think it can also be applied to the quantum case.
[quote="bolbteppa, post: 4825624"]but it doesn't say anything about Lubos' criticism of the ensemble interpretation as being nothing but a restricted and modest view of the power of QM. I'm curious what people think of this criticism.”To be blunt I dont even understand Lubos's crtitism – mind carefully explaining it to me?[quote="bolbteppa, post: 4825624"]In other words, why should we throw away both Kolmogorov probability and some axioms of standard quantum mechanics in favour of less axioms and another form of probability when all we get is a restricted and modest view of the power of QM?”He doesn't do that.[quote="bolbteppa, post: 4825624"]To be clear, I haven't read Ballentine.”You should.[quote="bolbteppa, post: 4825624"]Are you guys aware this is how deep into the rabbit hole you have to go?”Your misunderstandings are not flaws – just misunderstandings.ThanksBill
[quote="bolbteppa, post: 4825624"]Well first of all, this isn't correct – the limitations of the frequentest interpretation of probability”Sure they need not remain homogeneous – but the conceptualisation is they do – its a straw man argument.Many, many books explain the validity of the frequentest interpretation when backed by the Kolmogorov axioms eg http://www.amazon.com/Introduction-Probability-Theory-Applications-Edition/dp/0471257087[quote="bolbteppa, post: 4825624"]but the probability in Ballentine's book is not frequentist.”The conceptual ensemble the outcome is selected from is by definition frequentest.[quote="bolbteppa, post: 4825624"]Ballentine derives this propensity interpretation from Cox's probability axioms”Its true he doesn't use the usual Kolmogorov axioms – and uses the Cox axioms – but they are equivalent. Usually however when people talk about the Cox axioms they mean the interpretation based on plausibility – he is not doing that.He does use propensity – but I think he uses it simply as synonymous with probability – most certainly in the equations he writes that's its meaning.What he assumes is states apply to a very large number (an ensemble) of similarly prepared systems with a particular outcome of an observation. From the law of large numbers they occur in proportion to the probability of that outcome. That's the way its frequentest.He actually goes a bit further than that thinking of them as infinite. I personally have a bit of difficulty with that – and think of them as very large – but not infinite. In applying the law of large numbers you imagine some probability so close to zero for all practical purposes it is zero and we have a large, but finite, number of trials whose entries are in proportion to their probability. [quote="bolbteppa, post: 4825624"]Frequentist flaws aren't somehow fixed by Kolmogorov btw,”The law of large numbers says otherwise – again this is fully explained in books like Feller. I am pretty sure I know your issue – its concerned with the law of large numbers convergence in probability or almost assuredly – however simple assumptions made when applying it fix that issue. Again any good book on probability such as Feller will explain this – but its simple. There is obviously a probability below which its impossible in practice to tell from zero. That sort of assumption is made all the time in applying theories. That being the case in the law of large numbers you simple assume the conceptual outcome of a large number of trials is well below that level.[quote="bolbteppa, post: 4825624"]So Ballentine not only asks us to throw away axioms of quantum mechanics, he also asks us to throw away the most widely used and basic form of probability, Kolmogorov's probability”Errrr. He bases it on the two stated axioms in Chapter 2. Nothing is thrown out.In fact it can be based on one axiom as detailed in post 137 of the link I gave previously.Exactly what don't you get about Gleason and it showing (with the assumption of non-contextuality) that a state exists and it obeys the Born Rule?This, IMHO, is clearer than Ballentine's approach that assumes two axioms then shows they are compatible with the axioms of probability. [quote="bolbteppa, post: 4825624"]but it doesn't say anything about Lubos' criticism of the ensemble interpretation as being nothing but a restricted and modest view of the power of QM. I'm curious what people think of this criticism.”To be frank I don't even understand Lubos's criticism – mind carefully explaining it to me?[quote="bolbteppa, post: 4825624"]In other words, why should we throw away both Kolmogorov probability and some axioms of standard quantum mechanics in favour of less axioms and another form of probability when all we get is a restricted and modest view of the power of QM?”He doesn't do that.[quote="bolbteppa, post: 4825624"]To be clear, I haven't read Ballentine.”You should. But what leaves me scratching my head is you seem to have all these issues with it – but haven't gone to the trouble to actually study it. I could understand that if it was generally considered crank rubbish – but it isn't. Its a very well respected standard textbook. It is possible for sources of that nature to have issues – and it does have a couple – but they are very minor.It should be fairly obvious major issues with standard well respected textbooks are more than likely misunderstandings.[quote="bolbteppa, post: 4825624"]Are you guys aware this is how deep into the rabbit hole you have to go?”Your misunderstandings are not flaws – just misunderstandings.ThanksBill
[quote="bhobba, post: 4825505"]That's a complete misunderstanding of the ensemble interpretation.Its a conceptual ensemble, exactly the same as a conceptual ensemble in the frequentest interpretation of probability.If there is a flaw in it, there is a flaw in the frequentest interpretation of probability – which of course there isn't since circularity has been removed by basing it on the Kolmogorov axioms – it would mean a flaw in those axioms and many areas would be in deep doo doo.[/quote]Well first of all, this isn't correct – the limitations of the frequentist interpretation of probability:”Remark 1.8. (Limitations of Frequency Interpretation of Probability)1. If an experiment is repeated a very large number of times or indefinitely, then the conditions of the experiment need not remain homogeneous. As a consequence, the frequency ratio of A is subject to change.2. The frequency ratio of A, [itex]f_n(A) = tfrac{n(A)}{n}[/itex] need not converge [itex] lim f_n(A) = P(A)[/itex] to a unique value. Hence, P(A) is not well-defined.(In a random experiment E is repeated n times an event A occurs n(A) times thus [itex]f_n(A)[/itex] is it's frequency ratio)“(Ballentine mentions the second) mean that frequentist probability itself is flawed. Ballentine also mentions this, but the probability in Ballentine's book is not frequentist. The word propensity interpretation is used on page 32 of the 1st edition as a means to take the good and leave the bad in the frequentist interpretation. Ballentine derives this propensity interpretation from Cox's probability axioms which are similar to Kolmogorov's, but not the same…(Frequentist flaws aren't somehow fixed by Kolmogorov btw, they can't be fixed inside a frequentist perspective. If frequentist probability gives a correct result, you can derive it from Kolmogorov's axioms, but the issue is that a frequentist foundation leads to problems while Kolmogorov's foundation doesn't. This is all irrelevant though, as Ballentine is working from Cox's probability)So Ballentine not only asks us to throw away axioms of quantum mechanics, he also asks us to throw away the most widely used and basic form of probability, Kolmogorov's probability. My issue is the following: [quote="bhobba, post: 4825505"]The usual criticisms revolve around applying it to single systems – but as the article correctly says:'However, the "ensemble" of the ensemble interpretation is not directly related to a real, existing collection of actual particles, such as a few solar neutrinos, but it is concerned with the ensemble collection of a virtual set of experimental preparations repeated many times. This ensemble of experiments may include just one particle/one system or many particles/many systems. In this light, it is arguably, difficult to understand Neumaier's criticism, other than that Neumaier possibly misunderstands the basic premise of the ensemble interpretation itself'ThanksBill”I can see how that applies to Neumaier's criticism, but it doesn't say anything about Lubos' criticism of the ensemble interpretation as being nothing but a restricted and modest view of the power of QM. I'm curious what people think of this criticism.In other words, why should we throw away both Kolmogorov probability and some axioms of standard quantum mechanics in favour of less axioms and another form of probability when all we get is a restricted and modest view of the power of QM?To be clear, I haven't read Ballentine. Lubos's issues already turned me off a while ago, so I posted here to find a reason to give it a chance. After just finding out I also have to throw away Kolmogorov probability, I'm now even less inclined, but I'd still consider it if there's good enough a reason. Are you guys aware this is how deep into the rabbit hole you have to go?
[quote="Nugatory, post: 4825599"]The statement "If you perform the two measurements, the results will be correlated by cos[sup]2[/sup]Θ" is frame-independent and doesn't care about the temporal ordering of the two measurements.Whether it's satisfying or not is a different question.”Sure, but in any one frame there is a temporal ordering, and in any one frame there is wave function evolution. So if you use wave function evolution in any one frame, part of the correct evolution of the wave function in that frame involves collapse.Take a look at http://arxiv.org/abs/1007.3977.
[quote="atyy, post: 4825577"]However, in a Bell test, where there are simultaneously measurements at spacelike separation, those measurements will not be simultaneous in another reference frame. So if there are measurements at spacelike separation, and if any reference frame can be used in quantum mechanics, then there will be collapse in one frame.”The statement "If you perform the two measurements, the results will be correlated by cos[sup]2[/sup]Θ" is frame-independent and doesn't care about the temporal ordering of the two measurements.Whether it's satisfying or not is a different question.
[quote="Nugatory, post: 4825567"]Of course if you want something with deeper explanatory behavior, the ensemble interpretation is infuriating/exasperating/frustrating because it stubbornly refuses to say anything about why the probabilities are what they are.”It is the same with Copenhagen, except that since we already are agnostic about the reality of the wave function, and the wave function is just a tool to calculate the probabilities which we can observe, then there is nothing problematic about collapsing the wave function – it is just another tool like the wave function that we use to calculate the probabilities of outcomes.
[quote="Nugatory, post: 4825567"]I prefer "Copenhagen without collapse" to "Copenhagen renamed", because the ensemble interpretation doesn't carry along the additional and somewhat problematic notion of collapse. If there's no collapse I don't have to worry about how measurements cause collapse, and because I'm just using the theory to generate statements about the outcomes of interactions I can put the Von Neumann cut wherever I find it computationally convenient.”Yes, one can have Copenhagen without collapse, if one always pushes all measurements to the end of the experiment. If all measurements occur at the end, and in the same location, then there are no further measurements, no need for a quantum state after the measurement, and no collapse. In this viable view, one simply denies the existence of measurements at spacelike separation.However, in a Bell test, where there are simultaneously measurements at spacelike separation, those measurements will not be simultaneous in another reference frame. So if there are measurements at spacelike separation, and if any reference frame can be used in quantum mechanics, then there will be collapse in one frame.Here is one example of how collapse might be used to analyse measurements at spacelike separation: http://arxiv.org/abs/1007.3977.
[quote="atyy, post: 4825556"]If the ensemble is only notional, there is no difference between Ensemble and Copenhagen…. .So basically if Ensemble is correct, then it is just Copenhagen renamed. “I prefer "Copenhagen without collapse" to "Copenhagen renamed", because the ensemble interpretation doesn't carry along the additional and somewhat problematic notion of collapse. If there's no collapse I don't have to worry about how measurements cause collapse, and because I'm just using the theory to generate statements about the outcomes of interactions I can put the Von Neumann cut wherever I find it computationally convenient.Of course if you want something with deeper explanatory behavior, the ensemble interpretation is infuriating/exasperating/frustrating because it stubbornly refuses to say anything about why the probabilities are what they are.
[quote="bhobba, post: 4825505"]That's a complete misunderstanding of the ensemble interpretation.Its a conceptual ensemble, exactly the same as a conceptual ensemble in the frequentest interpretation of probability.If there is a flaw in it, there is a flaw in the frequentest interpretation of probability – which of course there isn't since circularity has been removed by basing it on the Kolmogorov axioms – it would mean a flaw in those axioms and many areas would be in deep doo doo.The Wikipedia article on it explains it quite well:http://en.wikipedia.org/wiki/Ensemble_interpretationThe usual criticisms revolve around applying it to single systems – but as the article correctly says:'However, the "ensemble" of the ensemble interpretation is not directly related to a real, existing collection of actual particles, such as a few solar neutrinos, but it is concerned with the ensemble collection of a virtual set of experimental preparations repeated many times. This ensemble of experiments may include just one particle/one system or many particles/many systems. In this light, it is arguably, difficult to understand Neumaier's criticism, other than that Neumaier possibly misunderstands the basic premise of the ensemble interpretation itself'ThanksBill”If the ensemble is only notional, there is no difference between Ensemble and Copenhagen, if we take the probabilities in Copenhagen to be frequentist. In Copenhagen, the state vector is not necessarily real, but the outcomes and their probabilities are, so a frequentist interpretation is allowed. So basically if Ensemble is correct, then it is just Copenhagen renamed. Unfortunately, Ballentine disparages Copengagen and wilfully deletes one axiom from it, rendering Ballentine's version of the Ensemble interpretation incorrect quantum mechanics. Basically, Ballentine appears to claim that Landau and Lifshitz and Weinberg are wrong! But I believe Landau and Lifshitz and Weinberg are correct, whereever there is a disagreement between Ballentine and them. There is one error in the tradition that Landau and Lifshitz and Weinberg come from, but that error (as far as I know) does not appear in their books. That error is the von Neumann proof against hidden variables, which came to light partly through Bohm and Bell, although it was known before. Since (as far as I know) this error does not appear in Landau and Lifshitz or Weinberg, I recommend their books as good presentations of quantum mechanics.
The "ensemble" is not only conceptual, it's created all the time when physicists measure things in the lab. They perform the experiment many times with as independent realizations as possible and measure always the same quantities again and again, evaluate the outcome via statistical methods and give the result of the measurement. Often the value of the measurement is simple compared to give a well estimated systematical error.For many-body systems you also have another type of "ensemble". The ensemble is realized by the many-body system itself. You do not ask about all positions of all gas molecules in a container (or the wave-function of a [itex]10^{24}[/itex]-particle systems) but look at pretty "coarse grained" quantities like the density, flow-velocity field, pressure, temperature, etc. Here the coarse-graining is over space-time volumes which can be taken as small on a scale over which such macroscopic quantities change considerable but large on a microscopic scale. It involves the average over some time interval and some volume containing still many particles. In this way you can derive the macroscopic behavior of everyday many-particle objects around us. The gas will be described by its thermodynamic equation of state (equilibrium or local equilibrium; hydrodynamical level of description) or by the Boltzmann(-Uehling-Uhlenbeck) equation (off-equilibrium; transport level of description), etc.Of course, there is a conceptual problem with physics (not only quantum theory!) concerning single events. You can only deduce physical laws from reproducible well-defined objective setups of an experiment. You cannot conclude much from a single event. E.g., the idea to ask for a "wave function/quantum state" of the entire universe is flawed, because whatever an answer you might give, how should you experimental verify or falsify this hypothesis? What we observe in cosmology are very small areas of a part of the universe like the measurement of the temperature fluctuations of the cosmic microwave background radiation or its polarization (COBE, WMAP, PLANCK satelites). Another example is the measurement of the redshift-distance relation of far-distant supernovae (Hubble space telescope etc.).
[quote="bolbteppa, post: 4825395"]They seem like pretty fatal flaws to me, or at least good reasons to choose Landau instead of this potentially shaky stuff…”That's a complete misunderstanding of the ensemble interpretation.Its a conceptual ensemble, exactly the same as a conceptual ensemble in the frequentest interpretation of probability.If there is a flaw in it, there is a flaw in the frequentest interpretation of probability – which of course there isn't since circularity has been removed by basing it on the Kolmogorov axioms – it would mean a flaw in those axioms and many areas would be in deep doo doo.The Wikipedia article on it explains it quite well:http://en.wikipedia.org/wiki/Ensemble_interpretationThe usual criticisms revolve around applying it to single systems – but as the article correctly says:'However, the "ensemble" of the ensemble interpretation is not directly related to a real, existing collection of actual particles, such as a few solar neutrinos, but it is concerned with the ensemble collection of a virtual set of experimental preparations repeated many times. This ensemble of experiments may include just one particle/one system or many particles/many systems. In this light, it is arguably, difficult to understand Neumaier's criticism, other than that Neumaier possibly misunderstands the basic premise of the ensemble interpretation itself'ThanksBill
[quote="atyy, post: 4825495"]Ballentine is the most misleading book on quantum mechanics I have ever read.”That is very much a minority view.Many regular posters around here, me, Strangerep, Vanhees and others rate it very highly.That said Landau is up there as well. But watch it – its terse and the problems challenging to say the least.ThanksBill
[quote="bolbteppa, post: 4825395"]Regarding Ballentine, are these claims really true:”Ballentine is the most misleading book on quantum mechanics I have ever read. In every place where he deviates structurally (I'm not talking about minor accidental errors) from the textbook presentation, it is Ballentine who is wrong and not the textbook.[quote="bolbteppa, post: 4825395"]They seem like pretty fatal flaws to me, or at least good reasons to choose Landau instead of this potentially shaky stuff…”I too would pick Landau and Lifshitz, or Weinberg for correct presentations of quantum mechanics.
Regarding Ballentine, are these claims really true:”Among the traditional interpretations, the statistical interpretation discussed by L.E. Ballentine,The Statistical Interpretation of Quantum Mechanics,Rev. Mod. Phys. 42, 358-381 (1970)is the least demanding (it assumes less than the Copenhagen interpretation and the Many Worlds interpretation) and the most consistent one. The statistical interpretation explains almost everything, and only has the disatvantage that it explicitly excludes the applicability of QM to single systems or very small ensembles (such as the few solar neutrinos or top quarks actually detected so far), and does not bridge the gulf between the classical domain (for the description of detectors) and the quantum domain (for the description of the microscopic system).In particular, the statistical interpretation does not apply to systems that are so large that they are unique. Today no one disputes that the sun is governed by quantum mechanics. But one cannot apply statistical reasoning to the sun as a whole. Thus the statistical interpretation cannot be the last word on the matter.http://www.mat.univie.ac.at/~neum/physfaq/topics/mostConsistent“Edit: I see on the wikihttp://en.wikipedia.org/wiki/Ensemble_interpretation#Single_particlesa rebuttal&”I chose not to label the "ensemble interpretation" as correct because the ensemble interpretation makes the claim that only the statistics of the huge repetition of the very same experiment may be predicted by quantum mechanics. This is a very "restricted" or "modest" claim about the powers of quantum mechanics and this modesty is actually wrong. Even if I make 1 million completely different experiments, quantum physics may predict things with a great accuracy.Imagine that you have 1 million different unstable nuclei (OK, I know that there are not this many isotopes: think about molecules if it's a problem for you) with the lifetime of 10 seconds (for each of them). You observe them for 1 second. Quantum mechanics predicts that 905,000 plus minus 1,000 or so nuclei will remain undecayed (it's not exactly 900,000 because the decrease is exponential, not linear). The relatively small error margin is possible despite the fact that no pair of the nuclei consisted of the same species!So it's just wrong to say that you need to repeat exactly the same experiment many times. If you want to construct a "nearly certain" proposition – e.g. the proposition that the number of undecayed nuclei in the experiment above is between 900,000 and 910,000 – you may combine the probabilistically known propositions in many creative ways. That's why one shouldn't reduce the probabilistic knowledge just to some particular non-probabilistic one. You could think it's a "safe thing to do". However, you implicitly make statements that quantum mechanics can't achieve certain things – even though it can.http://motls.blogspot.ie/2013/01/poll-about-foundations-of-qm-experts.html“They seem like pretty fatal flaws to me, or at least good reasons to choose Landau instead of this potentially shaky stuff…
Hi Barry[quote="Barry911, post: 4824753"]Do you find a fundamental problem with the idea that the wave function represents a pure probability description (Born) and yet represents all the information defining the "particle" of interest?”That's not what a wave-function is. Its simply a representation of the state. All a state is, is an aid to calculating the probabilities of outcomes. Those probabilities are all we can know.To fully appreciate it you need to comes to grips with Gleason – see post 137:https://www.physicsforums.com/showthread.php?t=763139&page=8A state is simply a requirement of the basic axiom:An observation/measurement with possible outcomes i = 1, 2, 3 ….. is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.Its something, that is required from that mapping, to aid us in calculating those probabilities.To go even deeper in the why of Quantum Mechanics you need to understand the modern view that its simply the most reasonable probability model that allows continuous transformations between pure states, which physically is a very necessary requirement:http://arxiv.org/pdf/quantph/0101012.pdfAt an even deeper level its what you get if you want entanglement:http://arxiv.org/abs/0911.0695In fact either the requirement of continuity or entanglement is enough to single out QM.[quote="Barry911, post: 4824753"]I have a problem with the word "determinism" it seems to imply the cause-effect relation of Newton”In this context it means are the outcomes of observations uniquely determined, or to be more specific defining a probability measure only of 0 and 1 is not possible under the Born Rule. Gleason shows this is impossible if you have non-contextuality. But in some interpretations, by means, for want of a better description, certain shenanigans, such as a pilot wave or many worlds, then the outcome is deterministic. However they all introduce extra stuff than the formalism either breaking non-contextuality, or a sneaky interpretation of decoherence like Many Worlds where you don't even really have an outcome. Consistent Histories is another sneaky way out as well by also not having an actual observation – but it is fundamentally stochastic, though in a conventional sense. [quote="Barry911, post: 4824753"]Also thanks for the recurring reference to Bellentine! I just bought the book. Ive just finished the math chapterand thought it excellent. It seems like a "superposition" of a textbook and an advanced popularizer.”Its simply the finest book on QM I have ever studied.Once you have gone through it you will have a thorough grasp of all the issues.Its not perfect though eg you will notice in his discussion of Copenhagen he assumes the wave-function in the interpretation exists in a real sence. Very few versions of Copenhagen are like that – in nearly all of them its simply subjective nowledge:http://motls.blogspot.com.au/2011/05/copenhagen-interpretation-of-quantum.htmlOnce you have gone through at least the first 3 chapters then you will have a good background to discuss what's going on in QM. For example you will understand Schroedinger's equation etc is simply a requirement of symmetry – the essence of QM lies in the two axioms Ballentine uses. Via Gleason that can be reduced to just one – all of quantum weirdness in just one axiom.But that revelation awaits you.ThanksBill
Hello bhobba:Couple of questions…1. Do you find a fundamental problem with the idea that the wave function represents a pure probability description (Born) and yet represents all the information defining the "particle" of interest?2. I have a problem with the word "determinism" it seems to imply the cause-effect relation of Newton and Laplace. I perhaps wrongly, assume you mean causality. Do you believe that the fundamental dynamics of our universe is "statistical causality"? It certainly satisfies the requirement of effect following cause but permits a limited variety of effects for an identical cause (identical in principle) and seems fundamental to QM.Also thanks for the recurring reference to Bellentine! I just bought the book. Ive just finished the math chapterand thought it excellent. It seems like a "superposition" of a textbook and an advanced popularizer.RespectfullyBarry911
[quote="thegreenlaser, post: 4824072"]My 3rd year quantum prof explained this concept in his first lecture, and that was a very big "aha" moment for me. Up until then, all my teachers had tried to explain quantum mechanics in terms of classical mechanics, and it never quite made sense. When someone finally explained that you can't really understand quantum mechanics in terms of classical mechanics, I felt like I was finally able to start learning.”That's very common in QM.I had read a lot of books on QM, including Ballentine's excellent text, and thought I had a pretty good grasp – but in my hubris I was mistaken.But every now and then you have these aha moments of insight that helps enormously.A big one for me was this semantic use of the word – observation, that you think from everyday use means some kind of human observer. Books often don't state it clearly, but in QM observation does not mean that at all. It means something that occurs in our everyday common-sense classical world.Another was the import of Gleason's theorem – the Born Rule is not pulled out of a hat – its actually required from what an observation in QM is. The key issue is non-contextuality.ThanksBill
[quote="Barry911, post: 4824050"]The outcome of single quantun outcomes does not yield a meaningful outcome. Iteration of "identicalexperiments" yields probability densities. P-densities do not predict where a quantum event will occur only statistical weightings”That QM is statistical is not one of its problems.As Atty says – it has problems, but that aren't one of them.Or rather, it would be more correct to say, whatever problem worries you, you can find an interpretation where is not an issue at all – the rub is you can't find an interpretation where all are fixed.For example, at first sight it may seem that QM's statistical nature is a problem if you have some preconceived view of how nature works that it must be deterministic. However we have Bohmian Mechanics that is totally deterministic – but at a cost – non-locality and a preferred frame. Interpretations are all like that – 6 of one, half a dozen of the other, no easy answer.ThanksBill
[quote="thegreenlaser, post: 4824072"]When someone finally explained that you can't really understand quantum mechanics in terms of classical mechanics, I felt like I was finally able to start learning.”"The paradox arises when using improper classical concepts to describe a quantum condition" dixit Serge HarocheYou don't perceive a elementary particle as you see an apple fall.Patrick
[quote="Greg Bernhardt, post: 4821250"] 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. “My 3rd year quantum prof explained this concept in his first lecture, and that was a very big "aha" moment for me. Up until then, all my teachers had tried to explain quantum mechanics in terms of classical mechanics, and it never quite made sense. When someone finally explained that you can't really understand quantum mechanics in terms of classical mechanics, I felt like I was finally able to start learning.
[quote="Barry911, post: 4824050"]Problem:The outcome of single quantun outcomes does not yield a meaningful outcome. Iteration of "identicalexperiments" yields probability densities. P-densities do not predict where a quantum event will occuronly statistical weightings”There are problems with quantum mechanics, but you have not diagnosed them accurately. http://www.tau.ac.il/~quantum/Vaidman/IQM/BellAM.pdf
Problem:The outcome of single quantun outcomes does not yield a meaningful outcome. Iteration of "identicalexperiments" yields probability densities. P-densities do not predict where a quantum event will occuronly statistical weightings
M[quote="Barry911, post: 4823428"]no observables, no predictions….sounds like an elegant theory of pure mathematics.”Pure math is the last thing QM is.At the axiomatic level the primitive of the theory is an observation eg see post 137:https://www.physicsforums.com/showthread.php?t=763139&page=8The fundamental axiom is:An observation/measurement with possible outcomes i = 1, 2, 3 ….. is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.When you get right down to it much of the difficulty of QM boils down to exactly what is an observation? Its generally taken to be something that occurs here in an assumed classical common-sense world. But QM is supposed to be the theory that explains that world – yet assumes its existence from the get-go.Much of the modern research into the foundations of QM has been how to resolve that tricky issue – with decoherence playing a prominent role.A lot of progress has been made – but issues still remain – although opinions vary as to how serious they are.ThanksBill
Good comments!However; There seems to be a Platonic trend among the "speculative" types including all the string theorists,i.e.-no observables, no predictions….sounds like an elegant theory of pure mathematics.Multiverses, "anthropic principle, demanding multiverses, Maldecena's conjecture ADS/cft also elegantbut lacking physical relevance. His holographic universe came about because he felt that information isconserved in two dimensions inside black holes! Have these people no humility?Q.M. requires more than analysis unless your limited to applied physics and just don't care.The power of QM of course lies in its mathematical formalism but it is a physical theory and requiresinterpretation. At this time, however (I'll say it again) all interpretation is premature. but even theextraordinarily inelegant interpretations are better than a strictly analytical i.e.-Platonic approach.Respectfully,Barry911
I start my undergrad QM course with Quantum Mechanics and Experience, David Z. Albert, Harvard Univ Press, 1992, ISBN 0-674-74113-7. It's not the dry math start that you find in, say, Principles of Quantum Mechanics, 2nd Ed., R. Shankar, Plenum Press, 1994, ISBN 0-306-44790-8. Don't get me wrong, I like Shankar and use it after the students do the calculations in Albert and some AJP papers cited below. I choose this intro because it involves some interesting phenomena that we can easily model mathematically. The phenomena is electron spin to include entanglement, so its "weirdness" tends to motivate the students to work on the matrix algebra needed to model it. And, the parameters in the matrix algebra correspond directly to Stern-Gerlach orientations and spatial locations of detector outcomes which are easy to visualize. Thus, while the outcomes are "mysterious," the modeling of the experiment is intuitive. I then have them reproduce the quantum calculations for each of Mermin's AJP papers on "no instruction sets":"Bringing home the quantum world: Quantum mysteries for anybody," N.D. Mermin, Am. J. Phys. 49, Oct 1981, 940-943. “Quantum mysteries revisited,” N.D. Mermin, Am. J. Phys. 58, Aug 1990, 731-734.“Quantum mysteries refined,” N.D. Mermin, Am. J. Phys. 62, Oct 1994, 880-887.Again, in each case, there is an easy-to-understand counterintuitive outcome that motivates the students to work with the simple, intuitive matrix modeling. We finish this intro by reproducing all the calculations in:“Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory,” D. Dehlinger and M.W. Mitchell, Am. J. Phys. 70, Sep 2002, 903-910to include the error analysis. That gives them a grounding in an actual experiment. Only after all that do we proceed to Shankar.
[quote="stevendaryl, post: 4821785"]When discussing the best approach to teaching something like quantum mechanics, I think you really have to consider the purpose in teaching it…”I just wanted to add that, whether or not the student is going to go on to become a physicist, there are certain ways to teach quantum mechanics that I think are just bad. There might be ways to teach a little bit of the feel of what quantum mechanics is about without getting into the mathematics that would be necessary to solve actual problems. But what is worse than useless is to skip the actual facts about quantum mechanics and instead teach people sound bites about how "Quantum mechanics teaches us that the mind creates its own reality" or whatever Deepak Chopra might say about it. However, the goal of giving the layman a flavor of quantum mechanics without being misleading is very difficult to pull off.
[quote="vanhees71, post: 4821738"]If you want to rise interpretational problems at all, you shouldn't do this in QM 1 or at least not too early. First you should understand the pure physics, and that's done with the minimal statistical interpretation. If you like Landau/Lifshits (all volumes are among the most excellent textbooks ever written, but they are for sure not for undergrads; this holds also true for the also very excellent Feynman lectures which are clearly not a freshmen course but benefit advanced students a lot), I don't understand why you like to introduce philosophy into a QM lecture. This book is totally void of it, and that's partially what it makes so good ;-)).”In fact Landau and Lifshitz introduce philosophy early and correctly in their QM book, which is what makes it so wonderful.
[quote="bhobba, post: 4821837"]But of relevance to this thread you will get a lot more out of that book if you know some of the real deal detail.”To understand the quantum theory in terms of mathematical language, we have in "France" some good free lecture like this one from "Ecole polytechnique" : http://www.phys.ens.fr/~dalibard/Notes_de_cours/X_MQ_2003.pdfon the other side there is not a unique look on its interpretation.Patrick
[quote="microsansfil, post: 4821829"]Erwin Schrodinger : Mind and matter – What Is Life? – My View of the World – …Werner Heisenberg : Physics and Philosophy: The Revolution in Modern Science – Mind and Matter – The physicist's conception of nature – …”Know both those books – but they are old mate.These days the following is much better at that sort of level:http://www.amazon.com/Understanding-Quantum-Mechanics-Roland-Omnès/dp/0691004358But of relevance to this thread you will get a lot more out of that book if you know some of the real deal detail.ThanksBill
[quote="bhobba, post: 4821770"]I think philosophers worry more about that sort of thing more than physicists or mathematicians.”probably not theory, but the people : Erwin Schrodinger : Mind and matter – What Is Life? – My View of the World – …Werner Heisenberg : Physics and Philosophy: The Revolution in Modern Science – Mind and Matter – The physicist's conception of nature – ……Patrick
[quote="bhobba, post: 4821770"]And I really do mean IDEA – not ideas – see post 137:https://www.physicsforums.com/showthread.php?t=763139&page=8“I assume you mean the idea expressed by the sentence:”An observation/measurement with possible outcomes i = 1, 2, 3 ….. is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.”I would say that that's a single sentence, but I'm not sure I would call it a single idea. There are many other ideas involved in understanding why we would want basis-independence, why we are looking for probabilities in the first place, why we want the outcome probabilities to be determined by [itex]E_i[/itex] (as opposed to depending on both the system being measured and the device doing the measurement), what is an "observation" or "measurement", why should it have a discrete set of possible results, etc.
When discussing the best approach to teaching something like quantum mechanics, I think you really have to consider the purpose in teaching it. Some of the people studying quantum mechanics are going to go on to become physics researchers, but my guess is that that is a tiny, tiny fraction. A small fraction of those who learn QM go on to get undergrad physics degrees, and a small fraction of them go on to get postgraduate physics degrees, and a small fraction of them go on to get jobs as physics researchers. So for the majority (I'm pretty sure it's a majority) who are not going to become physics researchers, what do we want them to know about quantum mechanics?I'm not asking these as rhetorical questions, I really don't know. But I think that if we want people to be able to solve problems in QM, there might be a best way to teach it to get them up to speed in solving problems. If we want them to understand the mathematical foundations, there might be a different way to teach it. If we want them to be able to apply QM to problems arising in other fields–say chemistry or biology or electronics–there might be another best way to teach it.So when people say things like "You shouldn't bring up X, because that will just confuse the student" or "The historical approach, with all of its false starts and blunders, is just not relevant to today's students", they need to get clear what, exactly, they want the student to get out of their course in QM. And I think that the answer to that question isn't always the same for all students.
[quote="microsansfil, post: 4821763"]May be QM is primarily predictive. Quantum mechanics construed as a predictive structure. After we try to interpret it with épistemic or ontological human sense. For example "The debate on the interpretation of quantum mechanics has been dominated by a lasting controversy between realists and empiricists" : http://michel.bitbol.pagesperso-orange.fr/transcendental.html“I think philosophers worry more about that sort of thing more than physicists or mathematicians.An axiomatic development similar to what Ballentine does is all that's really required, with perhaps a bit of interpretational stuff thrown in just to keep the key idea behind the principles clear.And I really do mean IDEA – not ideas – see post 137:https://www.physicsforums.com/showthread.php?t=763139&page=8It always amazes me exactly the minimal assumptions that goes into QM and what needs 'interpreting'.ThanksBill
[quote="WannabeNewton, post: 4821588"]It is just a cookbook on calculations.”May be QM is primarily predictive. Quantum mechanics construed as a predictive structure. After we try to interpret it with épistemic or ontological human sense. For example "The debate on the interpretation of quantum mechanics has been dominated by a lasting controversy between realists and empiricists" : http://michel.bitbol.pagesperso-orange.fr/transcendental.htmlPatrick
[quote="vanhees71, post: 4821738"]Well, the historic approach is bad. You are taught "old quantum mechanics" a la Einstein and Bohr only to be adviced to forget all this right away when doing "new quantum mechanics". I've never heard that it is a good didactical approach to teach something you want the students to forget. They always forget inevitably most important things you try to teach them anyway, but in a kind of Murphy's Law they remember all the wrong things being taught in the introductory QM lecture.”Abso-friggen-lutely.And to make matters worse they do not go back and show exactly how the correct theory accounts for the historical stuff and students are left with a sort of hodge podge, not knowing whats been replaced and what changed or the why of things like the double slit experiment.ThanksBill
[quote="WannabeNewton, post: 4821588"]Honestly I think at the undergraduate level QM is the easiest physics class one has to take. It is just a cookbook on calculations. Every book is uninspired and my class was certainly uninspired. It is an incredibly boring subject at this level. So I don't think difficulty is the issue. It is simply the lack of physical concepts and a healthy dose of philosophy that is avoided when teaching QM at the undergraduate level. Indeed one of the professors I know basically called Griffiths' book a cookbook in differential equations. A good book can go a long way. For me the saving grace was Landau and Lifshitz. It is the sole reason I started liking QM. Seriously the way undergrad QM is taught really isn't fun for the students. Boredom from a lack of intellectusl stimulation really isn't how a physics class should be.”Well, the historic approach is bad. You are taught "old quantum mechanics" a la Einstein and Bohr only to be adviced to forget all this right away when doing "new quantum mechanics". I've never heard that it is a good didactical approach to teach something you want the students to forget. They always forget inevitably most important things you try to teach them anyway, but in a kind of Murphy's Law they remember all the wrong things being taught in the introductory QM lecture.You see it in this forum: Most people remember the utmost wrong picture about photons, and it is very difficult to make them forget these ideas, because they are apparently simple. The only trouble is they are also very wrong. As Einstein said, you should explain things as simple as possible but not simpler.Concerning philosophy, I think the healthy dose is 0! Nobody tends to introduce some philosophy in the introductory mechanics or electrodynamics lecture. Why should one need to do so in introdutory QM?If you want to rise interpretational problems at all, you shouldn't do this in QM 1 or at least not too early. First you should understand the pure physics, and that's done with the minimal statistical interpretation. If you like Landau/Lifshits (all volumes are among the most excellent textbooks ever written, but they are for sure not for undergrads; this holds also true for the also very excellent Feynman lectures which are clearly not a freshmen course but benefit advanced students a lot), I don't understand why you like to introduce philosophy into a QM lecture. This book is totally void of it, and that's partially what it makes so good ;-)).
[quote="WannabeNewton, post: 4821605"]Im actually not sure what youre referring to. Are you talking about the Stefan Boltzmann law of radiation? Im not sure what that has to do with undergrad QM apart from historical impetus but there is a particularly lucid derivation in section 9.13 of Reif if youre interested. It's more of a statistical mechanics derivation. Which is good because statistical mechanics, both classical and quantum, is actually extremely interesting at the undergrad level.”Actually, I only dimly remember what it is, although it was very exciting. It sounds right that it should be in a stat mech book, because the whole point IIRC was that classical thermodynamics was able to derive all sorts of completely correct things about blackbody radiation, yet classical stat mech could not. Then miraculously when one switched to quantum stat mech everything fell in place with classical thermo. I remember the narrative, but none of the calculations except Planck's. The text we used was Gasiorowicz, and I think his chapter 1 is all about this.Apart from the Stefan-Boltzmann law, the other amazing derivation was Wien's displacement law. IIRC, these were all from classical thermodynamics, with no quantum mechanics, yet they are correct!
[quote="WannabeNewton, post: 4821588"]Indeed one of the professors I know basically called Griffiths' book a cookbook in differential equations. “That I agree with.I gave it away for health reasons no need to go into here. But I did enrol in a Masters in Applied Math at my old alma mater that included a good dose of QM. When mapping out the course structure with my adviser he said forget the intro QM course – since you have taken courses on advanced linear algebra, Hilbert spaces, partial differential equations etc it's completely redundant. Other students with a similar background to mine were totally bored. He suggested I start on the advanced course right away.Really I think it points to doing a math of QM course before the actual QM course where you study the Dirac notation etc – basically the first and a bit of the second chapter of Ballentine. You can then get stuck into the actual QM.And yes – I like Landau and Lifshitz too. Their Mechanics book was a revelation; QM, while good and better than most, wasn't quite as impressive to me as Ballintine. But like all books in that series it's, how to put it, terse, and the problems are, again how to put it, challenging, but to compensate actually relevant.ThanksBill
[quote="atyy, post: 4821599"]Is it easy?”Im actually not sure what youre referring to. Are you talking about the Stefan Boltzmann law of radiation? Im not sure what that has to do with undergrad QM apart from historical impetus but there is a particularly lucid derivation in section 9.13 of Reif if youre interested. It's more of a statistical mechanics derivation. Which is good because statistical mechanics, both classical and quantum, is actually extremely interesting at the undergrad level.
[quote="WannabeNewton, post: 4821596"]What o.O”Is it easy?
[quote="atyy, post: 4821595"]Hmmm, I still can't derive the Stefan-Boltzmann whatever – chills down my spine. How is that easy?”What o.O
Hmmm, I still can't derive the Stefan-Boltzmann whatever – chills down my spine. How is that easy?
Honestly I think at the undergraduate level QM is the easiest physics class one has to take. It is just a cookbook on calculations. Every book is uninspired and my class was certainly uninspired. It is an incredibly boring subject at this level. So I don't think difficulty is the issue. It is simply the lack of physical concepts and a healthy dose of philosophy that is avoided when teaching QM at the undergraduate level. Indeed one of the professors I know basically called Griffiths' book a cookbook in differential equations. A good book can go a long way. For me the saving grace was Landau and Lifshitz. It is the sole reason I started liking QM. Seriously the way undergrad QM is taught really isn't fun for the students. Boredom from a lack of intellectusl stimulation really isn't how a physics class should be.
I don't think there's one best way. Some learn better using the approach advocated here. Others learn better the other way around.
[quote="Greg Bernhardt, post: 4821250"]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. “Agree entirely.The mathematical formalism is required to understand the concepts.That is exactly the process taken in my favourite QM book, Ballentine, and is much more rational than the semi historical approach usually taken.The only problem with Ballentine is it is at graduate level. I have always thought a book like Ballentine, but accessible to undergraduate students, would be the ideal introduction.In particular it would have a 'watered' down version of the very important chapter 3 that explains the dynamics of QM from symmetry. Its a long hard slog even for math graduates like me – definitely not for undergraduates. But the key results and theorems can be stated, and their importance explained, without the proofs. I think its very important for beginning students to understand the correct foundation of Schroedinger's equation etc from the start – if the not the mathematical detail.ThanksBill