# Why Is Quantum Mechanics So Difficult?

Common Topics: qm, understanding, mathematical, classical, existing

## 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 upon what we have already understood. This is something crucial to keep in mind because, in the study of QM, this part is missing! I am certain that for most non-physicists, this is the most common reason why QM is puzzling, and why quacks and other people who are trying to use QM 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 only upon two complete revolutions? We simply have to accept that we use the same word, but to ONLY mean that it produces a magnetic moment. It has nothing to do with anything that’s spinning classically. We can’t build the understanding of the QM spin using 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 as 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 on.

What this implies clearly is that, if one lacks the understanding of the mathematical formalism of QM, one really hasn’t understood QM at all! One ends up with all these weird, unexplained, unfamiliar, and frankly, rather strange ideas on how the world works. These conceptual 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 weely neely.

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

## 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 be a smooth transition to build upon. Without this, QM will not make “sense”.

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254 replies
1. microsansfil says:

[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.[/QUOTE]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

2. bhobba says:

[quote="kith, post: 4826221"]The question is do you want to learn about the physics or the metaphysics?[/QUOTE]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

3. bhobba says:

[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.[/QUOTE]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

4. kith says:

[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?[/QUOTE]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.

5. kith says:

[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.[/QUOTE]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.

6. bhobba says:

[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.[/QUOTE]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?[/QUOTE]He doesn't do that.[quote="bolbteppa, post: 4825624"]To be clear, I haven't read Ballentine.[/QUOTE]You should.[quote="bolbteppa, post: 4825624"]Are you guys aware this is how deep into the rabbit hole you have to go?[/QUOTE]Your misunderstandings are not flaws – just misunderstandings.ThanksBill

7. bhobba says:

8. bolbteppa says:

[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:[QUOTE]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)[/QUOTE](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[/QUOTE]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?

9. atyy says:

[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.[/QUOTE]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.

10. Nugatory says:

[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.[/QUOTE]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.

11. atyy says:

[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.[/QUOTE]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.

12. atyy says:

[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.[/QUOTE]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.

13. Nugatory says:

[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. [/QUOTE]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.

14. atyy says:

[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[/QUOTE]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.

15. vanhees71 says:

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.).

16. bhobba says:

[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…[/QUOTE]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

17. bhobba says:

[quote="atyy, post: 4825495"]Ballentine is the most misleading book on quantum mechanics I have ever read.[/QUOTE]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

18. atyy says:

[quote="bolbteppa, post: 4825395"]Regarding Ballentine, are these claims really true:[/QUOTE]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…[/QUOTE]I too would pick Landau and Lifshitz, or Weinberg for correct presentations of quantum mechanics.

19. bolbteppa says:

Regarding Ballentine, are these claims really true:[QUOTE]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[/QUOTE]Edit: I see on the wikihttp://en.wikipedia.org/wiki/Ensemble_interpretation#Single_particlesa rebuttal&[QUOTE]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[/QUOTE]They seem like pretty fatal flaws to me, or at least good reasons to choose Landau instead of this potentially shaky stuff…

20. bhobba says:

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?[/QUOTE]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[/QUOTE]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.[/QUOTE]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

21. Barry911 says:

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

22. bhobba says:

[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.[/QUOTE]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

23. bhobba says:

[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[/QUOTE]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

24. microsansfil says:

[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.[/QUOTE]"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

25. thegreenlaser says:

[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. [/QUOTE]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.

26. atyy says:

[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[/QUOTE]There are problems with quantum mechanics, but you have not diagnosed them accurately.  http://www.tau.ac.il/~quantum/Vaidman/IQM/BellAM.pdf

27. Barry911 says:

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

28. bhobba says:

M[quote="Barry911, post: 4823428"]no observables, no predictions….sounds like an elegant theory of pure mathematics.[/QUOTE]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

29. Barry911 says:

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

30. RUTA says:

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.

31. stevendaryl says:

[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…[/QUOTE]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.

32. atyy says:

[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 ;-)).[/QUOTE]In fact Landau and Lifshitz introduce philosophy early and correctly in their QM book, which is what makes it so wonderful.

33. microsansfil says:

[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.[/QUOTE]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

34. bhobba says:

[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 – …[/QUOTE]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

35. microsansfil says:

[quote="bhobba, post: 4821770"]I think philosophers worry more about that sort of thing more than physicists or mathematicians.[/QUOTE]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

36. stevendaryl says:

[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[/QUOTE]I assume you mean the idea expressed by the sentence:[QUOTE]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.[/QUOTE]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.

37. stevendaryl says:

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.

38. bhobba says:

[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[/QUOTE]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

39. microsansfil says:

[quote="WannabeNewton, post: 4821588"]It is just a cookbook on calculations.[/QUOTE]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

40. bhobba says:

[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.[/QUOTE]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

41. vanhees71 says:

[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.[/QUOTE]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 ;-)).

42. atyy says:

[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.[/QUOTE]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!

43. bhobba says:

[quote="WannabeNewton, post: 4821588"]Indeed one of the professors I know basically called Griffiths' book a cookbook in differential equations. [/QUOTE]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

44. WannabeNewton says:

[quote="atyy, post: 4821599"]Is it easy?[/QUOTE]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.

45. atyy says:

[quote="WannabeNewton, post: 4821596"]What o.O[/QUOTE]Is it easy?

46. WannabeNewton says:

[quote="atyy, post: 4821595"]Hmmm, I still can't derive the Stefan-Boltzmann whatever – chills down my spine.  How is that easy?[/QUOTE]What o.O

47. atyy says:

Hmmm, I still can't derive the Stefan-Boltzmann whatever – chills down my spine.  How is that easy?

48. WannabeNewton says:

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.

49. bohm2 says:

I don't think there's one best way. Some learn better using the approach advocated here. Others learn better the other way around.

50. bhobba says:

[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. [/QUOTE]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