# Three neat resources on QM. Deduction-based vs. historically oriented approaches in teaching physics. A new “ambition”, + miscellaneous.

Okaay… So… How are you?

… On my side, I’ve been having quite some fun studying QM. I’ve reached a certain point in my studies, and it seems like this is a right time to take a little break, and write down an update, and thereby keep some momentum going at this blog.

Today I am going to write a little bit about three neat resources on QM, and also share some random thoughts, which occurred to me as a result of my wondering as to why I find these resources useful. In the process, I am going to touch a bit on the various approaches for systematically presenting a difficult topic like QM. The approaches I have in mind are: deduction-oriented, historically sequenced, and some combination of the two. Finally, I will also write a bit about a new ambition that has arisen in my mind. … OK, so let’s get going…

0. Preliminaries:

Lecture notes (and even full text-books) on QM tend to be organized in a highly deductive manner, especially when it comes to the topic of the QM spin. OTOH, many of you probably are very well aware that I tend to dislike any deduction-heavy treatments, even if these come from Nobel laureates and are highly praised, e.g., The Feynman Lectures vol. 2 and 3. … If fully or almost fully deductive, then sometimes, I even hate them—I mean such notes and  books, the prime example here being: Dirac’s book on QM! … So… Where do we begin?

Ummm… First, a word about how the element of deduction can be put to some good use too!…

1. Some textbooks / lecture-notes / course-videos lean towards a deductive approach but still are good:

One good point with a somewhat deductively organized presentation is that it can achieve a greater efficiency in teaching—assuming that a class-room teacher (not to mention TAs for conducting the recitation sessions) are available to the student for clarifying his doubts and difficulties right then and there in an informal, personal, settings. A somewhat deduction-oriented pedagogy has its uses, given the time constraints of the typical university schedules—provided that it’s done right—and provided that resources like systematic recitation (or tutorial) sessions are available.

However, in general, if the organization of topics is more heavily slanted towards deduction, then, even if TAs are available, the main teacher himself has to be very careful. He has to exercise special care, especially on these two counts:

• he has to keep on giving at least some “intuitive” feel for the logical starting point of each topic separately, again and again, and
• the scope of the logical “fanning out” to implications, starting from the major premise(s) selected for deduction, also has to be kept judiciously delimited.

Deduction is powerful—too powerful, in fact! Just like a sharp, double-edged sword. If handled right, it can work wonders. In teaching-learning, it means: Deduction is useful to the student provided he already knows the general outline and meaning of a topic and its scope. But if you are completely new to a topic, then a deduction-heavy treatment is more likely to induce in you a large number of small but enduring misconceptions. Reason: Objectively speaking, induction has primacy. Deny it its rightful role by suppressing it, and it’s going to try and figure out some way out of the suppression. Splintering of knowledge, therefore, is a very easy possibility—if not splintering of the mind as well!

However, if due care is exercised on the aforementioned two counts, then a semi-deductive treatment can come in quite handy. Let me give you an example from engineering, just to illustrate what I have in mind here.

Consider the very first course in engineering mechanics, viz., the vector mechanics of Newtonian particles and rigid bodies.

For more than two centuries, physicists actually theorized certain natural phenomena, and solved problems related to these, without ever using the idea of vectors. They explicitly worked with systems of equations: three coupled scalar equations, one each formulated along a principal coordinate axis.

So, a pedagogical approach that sticks to a purely historical order would have to teach all the topics in applied mechanics—not just acceleration in $3D$, and angular momentum, and the Coriolis forces, and everything else, but eventually, also the entirety of Maxwell-Lorentz EM—using only systems of coupled scalar equations! The task of teaching would become unwieldy in practice (and require black-boards on all sides of a class-room, with student-chairs that can rotate through $360$ degrees). The task of learning would become even harder, and therefore, knowledge would become accessible to relatively very few students.

However, experience shows that a combination of the historical and the deductive approaches does work great for engineering mechanics. It definitely takes less amount of time to generate a good grasp of the subject for most students. (To get to the best possible grasp, you have no choice but to look up the history and fill in the details from such sources, purely on your own.)

So, what you do, for courses in engineering mechanics, using the “combination” approach, is this:

You begin with the separate scalar component equations, and list them once. But you actually use them only in the simplest cases like motion of particles in $1D$ and $2D$ (e.g. the parabolic path of a projectile, the uniform circular motion, etc.). Then, soon enough, you take a jump of approximately one–two centuries, and immediately introduce the idea of vectors right at this stage. You don’t get into all the complications of the concept, like the distinction between a true vector and a pseudo-vector, right at this stage. And, you certainly don’t give a formal definition of vector spaces at all. But, you do begin deploying the idea of vectors in calculations. And, you do so using only a simplified (or “curtailed”) sense of the term “vector” (which is: as a directed line segment). Then, as the physical systems to be analyzed become more and more complex, you also go on expanding and clarifying the idea of vectors and operations with them further. You introduce the scalar and cross products of vectors, and explain the necessity of their differences in reference to different aspects of physical phenomena, then you go on to the calculus of vector-valued functions, etc…

So, all in all, you are going back and forth in history a bit, but without necessarily creating too much confusion about the proper hierarchical relations among the physical concepts. Going back and forth in this way is OK, up to an extent, if the jumping around is kept limited to the concepts and techniques of mathematics. However, I would definitely say, such jumping around does not work for the concepts and contents of physics as such.

Now, an interesting fact here is that a lot of ideas from physics also have a heavy methodological, even mathematical, context to them. For example, ideas like: the variational principles (taught initially as the “energy” principles), operators, and of course, the spin in QM.

Teaching such topics too can become more efficient using the “combination” approach, but then, doing so requires a teacher who is comparatively more skillful, and also, much more careful.

The combination approach might be characterized, using a slightly more fundmental terminology, as the following:

Start with the phenomenological knowledge, and use induction to introduce certain important facts that are also generally applicable. Then, translate these preliminary ideas into more formal concepts. These new, formal concepts might themselves be of a far greater methodological scope, but start using them anyway, without pondering over all aspects of the expansion in scope which such a formalization implicitly brings in. Then start working out simple mathematical manipulations, while using the greatly generalized formalism, but only in the simpler contexts, and thereby make students comfortable with the rules of manipulations as well as the hieroglyphics (i.e. symbols). Then progressively go on fleshing up the meaning behind the symbolism as the student understanding deepens (and his facility in using the rules and symbols improves). And, all through this activity, always keep on dropping small bits of physical insights (or at least some hints) which show where the “floor” of the “ocean” lies. Do that frequently.

It works. Provided that, what the teacher is aiming at is only a more systematic treatment, i.e., if, emphatically, he has not sold out his soul to deduction as such. Not all teachers or textbook writers are of the latter kind.

Now, the trouble with today’s teachers (and text-book writers) is that they just don’t know how to stop short of being an outright slave to deduction—and in the process, they also pull their students in the same vortex. Their short-coming is especially evident as the physics gets more and more complicated and therefore more and more abstract, starting right from the fields idea in EM, and up to the modern physics of special relativity and QM.

[Aside: IMO, not even that great teacher—Feynman—could manage this challenge right, always. In evidence, see his deductive treatment in the Volume 2 of his Lectures, and compare it not just with Resnick and Halliday, or Sears and Zemansky, but also with Purcell (recently updated by Morin). I don’t know about you, but I would always go in for the latter three as my primary sources for learning. Once you have already learnt the topics, then Feynman does become good—especially for the occasional insights that are hard to find elsewhere (and more occasionally, for your cocktail party points). Personally, I have avoided The Feynman Lectures’ second volume (after a rapid but careful browsing some three decades ago). Aside over.]

So, yes, using some degree of a deductively oriented organization can lead to efficiency in terms of classroom time. Actually, the gain is in terms of generating an averagely good sort of competency, in the least possible time, for more number of students. But such methodology also is a bit more demanding on the main teacher.

And, this problem becomes an order of magnitude worse when it comes to teaching quantum mechanics.

That’s why, when one runs into a mildly-deductive treatment of QM that’s also done competently, one not only appreciates it, one also wants to applaud it. In this post, I will mention two such examples.

2. “Notes on Quantum Mechanics” by Prof. Daniel Schroeder:

This is a set of notes for an introductory UG course on QM at the Weber State University, available here [^].

The organization and presentation style followed in these notes is such that a definite slant towards deduction is very easy to make out. Yet, the writing is such that the notes remain very easy to follow—even in the absence of a class-room teacher, i.e., even for a “pure” self-study mode (without any video recordings and all).

I had noticed these notes some couple of years ago or so, but in the sea of all the material available on the ‘net for QM, I had come to postpone reading through it, back then. Then, some time later, I somehow came to forget about these notes.

Recently, I checked out Prof. Schroeder’s Web pages after a while; found a new version of these notes; and immediately downloaded the freely available PDF [ (PDF) ^]. A quick browsing later, I now decided to keep the MIT course-work on a further hold, and instead to go through these notes first.

Turns out that it was a very good decision to make.

By now, I have gone through the angular momentum- and spin-related parts of this book. (I skipped most of the initial parts of the book, simply because I knew those topics pretty well.) I have found the treatment of the QM spin here to be outstanding.

It in fact is the best introduction to spin among all notes and books I have seen so far.

Yes, the treatment of the spin here is, IMO, better than that in Townsend’s text-book. Reason: Schroeder’s notes are short, more readable, and the problems too are “doable”. In contrast, Townsend’s book is big in size, too big in fact. (I think it’s 800 pages long. I have merely browsed through it once or twice, but have not properly read through even one section completely. (TBD later!))

The treatment of spin in Schroeder’s notes also is better than that in Eisberg and Resnick (and many other text-books). An important reason: Schroeder’s notes have a distinctly modern “flavour”, and so, you can so easily transition from the introductory QM to reading the special-purpose books and literature (say on the QC) without much effort.

Another plus point:

These notes are the only source I know of which shows how to “implement” two-state systems using the spatial wavefunctions (i.e. without at all using the spin).

[Aside: To tell you the truth, I had independently figured out something like this—two-state systems using only $\Psi(x,t)$ some time ago, as also the fact that entanglement can be explained in reference to the spin-less particles too.

… The earliest memory I have about thinking of entanglement with only spatial wavefunctions, i.e., without involving any spin at all, goes back at least to November 2014, when I was teaching engineering courses in Mumbai; and then, a highlight also occurred around Diwali-time in 2017. … But my ideas were rather “conceptual” in nature. Actually, my ideas were relatively vague, though they were not quite “floating” abstractions. And remember, all my studies and research in QM has been purely on a part-time basis, except for the last one year (since the Covid-19 began).

Anyway, when I saw Schroeder’s paper, “Entanglement isn’t just for spin” [^] (which was may be in 2018 or so), I remember, how I had marveled at it. Now, coming back to the present, to these notes, the marvel repeated. I mean, it was pleasant to read a description which was also physics-wise fully correct! Aside over.]

The only “downside” (if it can be called that) which I found with Schroeder’s notes is this:

There is no coverage of topics like the “total” wavefunction (spatial + spinor) for the many-particle systems. … May be it was not practical for them to cover this topic during their regular university coursework. However, an additional chapter dealing with the details of this topic would have been very helpful.

Added attraction: The simulation applets written by the author himself. (His HTML5 code is clean!)

All in all:

Strongly recommended, especially for the topic of the spin.

If you come from a BS in CS sort of a background, but have never studied QM beyond the Modern Physics courses, and still, if you have somehow grown very enthusiastic about the QC, and are championing it around, then, for God’s sake, let me dispense away this completely gratuitous and unsought advice to you:

Don’t even consider opening your mouth to champion the QC until you have have already mastered this book, cover-to-cover, complete with solving the section-end problems too. (As to for my opinion about the pre-requisites required for this book, see the section 4. below.)

And yes, I mean it!

3. “Quirky Quantum Concepts: The Anti-Textbook” by Prof. Eric Michelsen:

I’ve forgotten the track of where I gathered about Prof. Michelsen’s background (even if I did it within the last fortnight!). Anyway, here it is, in brief. Michelsen started out as an engineer. He spent quite some time (“decades”) in engineering industry (IIRC, in electronics / semiconductors). Then, I gather, he also founded start-ups in software. Then, he turned to research, and did a PhD in lasers and astrophysics, from UCSD. He is now a professor at UCSD [^].

… In short, I might say that I am sort of like him (or he is sort of like me), minus his practical success. [See the endnote at the end of this section.]

…Anyway, to come back to the “Anti-Textbook” by Prof. Michelsen….

Anyway, so… A couple of weeks ago or so, I once again downloaded the latest copy of Michelsen’s draft. (The book has been published by Springer, but the draft version is still available for free, here [ (PDF) ^] ).

I am still going through it. However, by now, I have read significant parts from: chapter 2 (“Riding the Wave: More on Wave Mechanics”), chapter 4 (“Matrix Mechanics”), and some initial parts from chapter 5 (“Angular Momentum”). In particular, I’ve not yet completed the portion on the spin, and haven’t even begun with the next chapter (which is on the many-particle systems).

My impression?

I am plain astounded at the richness of the insights offered here, full stop. This is one of the best resources for understanding the subtle aspects of QM.

Again, sometimes, I was even stunned to find the same insights as I myself had come to develop, independently. … Not so mysterious! The very approach of engineers is like that. Engineers (I mean: people who have worked as engineers for long enough of time so as to internalize the peculiar approach of engineering) do tend to think in a subtly (but definitely) different way than “pure” physicists do. (And we won’t even mention mathematicians here!) That’s why, it shouldn’t come as a surprise that when two engineers think deeply about the same new subject, there is a considerably similarity in terms of all: how they approach it, what they find interesting in it, and what they choose to highlight or take up for detailed considerations.

Of course, with his further formal training in physics (at the level of a PhD in physics), Michelsen has a much better knowledge of the mainstream QM than I do. He certainly has far more insights to offer on the more advanced aspects of the mainstream QM. These are difficult topics, and my studies of QM itself are relatively much more limited. I am not even aware of some of the topics whose quirkiness he notes. Yet, since his thinking retains the characteristic fold of an engineer’s thought processes, I have not found major difficulty in getting his points—even if these are quite quirky!

So, all in all, I think I can say this about my impression of this book (at this point of time):

I can always understand what Michelsen is saying, and often times, I also find myself having already worked through to precisely the same (or very similar) conclusions. However, I don’t always anticipate all his insights pertaining to the peculiarities of the mainstream QM.

But, yes, one way or the other, I find that his book is packed with insights. Even if you are not an engineer, you should benefit tremendously from this book. … Don’t take my word for it. Just go through the book and see for yourself. … OK. Let me copy-paste just one insight (just to help concretize this point); the following excerpt is from the draft copy page 49 (i.e. p. 51 of the PDF file):

Observable Operators Do Not Produce States Resulting From Measurements:

The mathematical result of an observable operator acting on a state is very different from the state resulting from actually measuring that observable.

Many people confuse the result of an observable operator on a state with the act of measuring that observable. These are very different things!

Note that the act of measurement is a nonlinear operation on the wave function; it can not be represented by a linear operator acting on the wave function. Recall that the whole point of a linear operator is to produce a superposition of results based on the superposition that composes the given function (or ket). In contrast, the consequence of a measurement is to choose one specific state out of a superposition of eigenstates.

A measurement eliminates a superposition, in favor of a more definite state. Therefore, a measurement is not a linear operation on the state; it is inherently nonlinear.

[Emphasis in bold original.]

See, see, why I am so impressed with this book? (And if you can’t figure out the reason, then check out my Outline document, here [(PDF) ^].)

… The entire book is filled with such nuggets.

No, this book is not at all induction-primary; it’s not even a historically sequenced presentation. In fact, this book isn’t even your usual text-book. Read the Preface to see why the author himself calls it an “Anti-Textbook”. …So, yes, this book is quirky! But yes, it’s quite rich in insights too.

So far, all that I’ve done is to rapidly read through the aforementioned chapters. But I don’t think that I’ve had enough opportunity to ponder over every subtle point, every nuance. I will have to read through the remaining parts, and then, I will have to return and re-read some parts again (may be 2–3 times).

The whole book is a kind of a teaser, as it were, to me. (Yes, my hard-copy is full of underlines, margin notes, scribblings, and all.)

Yes, this book is going to keep me engaged for quite some time to come.

However, no, do not bring up some points from this book for discussion with me. Not right away. I still have to learn a lot, and I am definitely quite a distance away from mastering the pre-requisite contents. I am also not likely to attempt mastering it any time soon. Reason: Many of these topics are not relevant to the research on Foundations of QM as such, even though the book deals with many advanced or subtle aspects of the mainstream QM, in a very admirable way.

As to me: First, I have to complete the first version of my document on my new approach to QM. Before that, I’ve to complete the MIT course 08.05 (i.e., watching videos). Before that, I’ve to complete the second half of MIT 08.04 too.

So, the bottomline is this:

Bottomline: One of the best books for really learning the subtleties of QM that I’ve ever come across, at such a level that it should be accessible even to the undergraduate students.

But make sure that you have completed the pre-requisites.

[Endnote to section 4: Why do I say that I am like Dr. Michelsen, minus his practical success? … Well, some of you should know the background behind that statement already, but in case you’ve just begun visiting this blog in the more recent times, there is a story about me and UCSD (which doesn’t come up on my CV)…:

After my failure in the PhD qualifiers at UAB in 1993, I was admitted to the mechanics program at UCSD, for the academic year beginning Fall ’93. … Not bad for a guy with a “mere” 8.25/10.00 CGPA at IITM and a 3.16/04.00 GPA at UAB. … OTOH, in fact, people at UCSD were (very) impressed by me. Reason? A literature review document which I had (on my own) attached to my application. It was on the micro-mechanics of fracture in ceramic composites. … Ummm, yes, the document was pretty good. So, they had decided to have me on board as soon as the funding arrives (which was around the May–June 1993 times).

However, roughly around the same time, even their on-going funding got cut down. So, formally, they said that they will keep the offer open for me for at least a year (which, eventually, they did), and informally, they called me to discuss the situation in all its detail with me. (By they, I mean the professors, not the staff at the graduate school.) Some of their on-going PhD students, supported previously via the funded projects which were now cancelled, began working in Pizza Hut, I gathered. I was willing to follow the suit. But starting completely afresh just on Pizz-Hut, i.e., without any prior savings, was not at all practical, someone (an IITian) doing his PhD there told me. The professors confirmed this assessment too, during the several telephonic discussions I had with them. … Anyway, all in all, I had to let go their offer, and return to India (which was in the last week of August ’93).

… BTW, my admission letter at UCSD was signed by Prof. Nemat-Nasser, the then Chair of the program. He and his colleagues had already brought this mechanics PhD program to a very good reputation; it was already ranked within the top 10 US-based programs or so. For comparison: My earlier program at UAB was ranked 60+ at that time (within about 70 PhD programs, in all, in the USA). … BTW, eventually, Prof. Nemat-Nasser also went on to receive the Timoshenko Medal. As to this medal …Well, yes, you may think of it as the “Nobel” of applied mechanics [^] [^]. … So, that’s the part related to UCSD

… But coming back to the other aspects of practical success: As to my later experience in software, in particular, in the SF Bay Area… Well, ask me some other time, preferably in private (so I can be a bit free-er in my… err… expressions). Footnote over. ]

4. Revised recommended sequence for learning QM through self-studies alone:

If you want to study QM through the self-study mode (i.e. completely on your own, without any personal guidance from any one), then condensing down everything (including whatever I have said about this topic in the past, and now revising it), here is my advice in a nut-shell:

Resnick and Halliday / Sears and Zemansky + if necessary, Purcell (updated by Morin) $\Rightarrow$ Beiser (Modern Physics) $\Rightarrow$ First half of McQuarry (Quantum Chemistry), up to and including the Helium atom + augmented readings for the same topics from Atkins (Molecular Chem.) $\Rightarrow$ Schroeder’s notes $\Rightarrow$ Eisberg and Resnick, Alastair I. M. Rae, and may be an occasional look into others like Griffiths / Gasiorowicz $\Rightarrow$ Michelsen.

For the last two stages, you can start with Michelsen and “dip back” into Griffiths etc. as the need arises. Also, consider watching the video series by ViaScience (mentioned in my earlier post here [^]) any time after you are past the books on quantum chemistry (including the He atom).

[Aside: Once my viewing of the MIT course-work (08.04 and 08.05) is over, it’s possible that I will revise the above sequence. So far, I’m half-way through 08.04 (2013 version), and I’m impressed with it. However, I don’t think that I am going to include the MIT course work in the shortest sequence presented above, and I don’t think I am eventually going to drop anything from the above sequence either. So, there. Aside over.]

Addendum: If your interest is in the Foundations of QM, rather than in QM itself or the QC, then my advice would depend on your background.

If you come with a background in physical / engineering sciences, then go through the above mentioned sequence at least up to and including Schroeder’s notes. Then, follow this sequence:

Travis Norsen (Foundations of QM) $\Rightarrow$ Tim Maudlin (Philosophy of Physics: Quantum Theory), both augmented with David Harriman (Induction in Physics).

(BTW, I have only browsed through some initial parts of Maudlin’s book, but I can definitely recommend it without any reservations.)

If you come with a background in other sciences or philosophy, then follow this sequence:

David Harriman (Induction in Physics) + Tim Maudlin (Philosophy of Physics: Quantum Theory), in any order $\Rightarrow$  Travis Norsen (Foundations of QM).

If you cannot understand the physics part of Harriman’s or Maudlin’s book even after a second or a third reading, then I would suggest: Quit pursuing Foundations of QM; this field is not for you. The field of Foundations of QM has a far greater basis in physics rather than philosophy—regardless of what other people might have led you to believe.

But if you still must persist with this field (Foundations of QM) at any cost, then quit pursuing all philosophical and popular science books on this topic (including those by Bell, and Bohmians), and instead, begin with the first sequence (given above), right from Resnick & Halliday etc., and going up to (and including) Schroeder’s notes. Once you are through with it (which should take at least a couple of years, may be 3–4 years), then once again check out Maudlin. If you can follow it right on the first read, then you may follow either of the two sequences given for the Foundations of QM.

Good luck!

5. A new, personal, long-term ambition:

Now that I had accidentally “re-discovered” the two gems mentioned above (the notes by Schroeder and Michelsen), I chastened myself a bit. Then, straining my memory, I remembered about

Malcom Longair’s book: “Quantum Concepts in Physics: An Alternative Approach to the Understanding of Quantum Mechanics”.

I had bought this book way back [^]. However, I have never been able to take it up for a systematic and comprehensive reading. … All that I’ve been able to do is sometimes to “take a dip” into it, may be for some 2–3 pages at a time, only then to toss it aside once again. “No right time for this book!” That’s what I’ve been saying to myself, invariably…

The last time I checked Longair’s book was, may be, 3+ years ago. It certainly was at least months before February 2019, which is when I wrote the Outline document [^].

A few days ago, I picked up this book once again. It was the first time I was touching it after the Outline document was posted. Presently, I came to a conclusion.

But before telling you the conclusion, let me ask: Remember Dr. Jennifer Coopersmith’s book “The Lazy Universe”? I had mentioned it pretty recently, just a few posts ago, here [^]. In the preface to her book, Coopersmith says that her book is like a simplified version of Lanczos’ book on the variational calculus. Now, Lanczos’ book is the Ultimate One, when it comes to the calculus of variations (i.e. the “energy” principles). And Coopersmith seems to have handled the simplification very well…

Now, coming back to Longair’s book and my long-term ambition.

I would like to write a simplified account of QM, based on Longair’s book.

The write up would be in a text-book like manner—complete with some solved problems and some review questions.

I would like to work at it slowly, one section at a time, and also irregularly, as my free time permits. I would like to post each section (or sub-section) in a GitHub repo or so. It will be, by intention, a long-term, and also irregular, hobby project.

But why rewrite a book if the original itself is so great?

Well, there are multiple reasons for that:

Yes, Longair has done a very admirable, scholarly, job in his book. However, he also gives greater detail of the initial analyses of experimental results, events and personalities than what a modern UG student of QM could possibly handle. If the goal is to simplify the presentation, one could omit many such details—precisely because Longair’s book is there!

Another point. Just the way the teacher has to exercise great care when presenting anything with a deduction-oriented approach (e.g. vector mechanics), similarly, the teacher also has to exercise a great care when presenting anything with a historically-oriented approach. Reason: Following the historical sequence helps in achieving a focus on the inductive roots of concepts and ideas. However, the former does not automatically ensure the latter. Isolation of the inductive roots is a separate task by itself.

With my enhanced understanding of induction (as brought about by David Harriman’s book “Induction in Physics”), I think that I can have a good shot at simplification. …

Let me be clear: I wouldn’t be explaining, let alone proving, how this or that development does have inductive roots; that’s not my goal. But I would like to present the physics points in such a way that their inductive roots become easier to grasp (even if they don’t become inescapable). I might not always do a great job for this aim; it’s too lofty for me. But I do think that I stand a good chance in converting the description from a mainly historically oriented account to one that is (i) simplified and (ii) highlights the nature of the respective inductive generalizations performed by the physicists at various stages of the main development. I think I can do that, to an acceptable degree.

One more point: I also think that, in the process of developing my new approach to QM, I’ve achieved a good clarity regarding what the mainstream QM theory is trying to say. This greater clarity, brought about by my new approach, should help in my goal, even if the explicit concern remains only with the development of the mainstream theory.

OK, so… I will begin working on it some time after the upcoming document on my new approach is done. I will work at it purely at leisure, purely as a hobby, and by intention, without any explicit plan… I think one should have some long-term hobby project like that going at any time…

6. But when will I start writing my planned document on the new approach?

Oh well. You tell me. … I’ve already told you the status as of now, and also the plan. Here it is—the plan—once again:

A few days more for going through Schroeder and Michelsen  $\Rightarrow$ II half of MIT 08.04 (videos) $\Rightarrow$ MIT 08.05 $\Rightarrow$ Some rest $\Rightarrow$ Thinking as to how my new approach holds up—if it does! And, if it does hold—and I see no reason why it shouldn’t hold, including for the spin–then (and only then) $\Rightarrow$ Some planning for writing $\Rightarrow$ Start actually writing.

So, whaddaya think? When will I begin writing (that goddman document on that goddamn new approach of mine)? …

Well… Want to consult astrologers / tarot card readers / psychics?  … Fine by me! Just let me know what they think (if they do), and then, also, what you think, if you do, after you have heard from them. … Or may be, want to consult some AI program? may be after you implementing a rough-and-ready one? Fine by me, again! … Or, perhaps, want to put to a practical use some certifiably random RNG (random number generator)? simply on the grounds that QM is supposed to be fundamentally random, and physics is universal? Or simpler still: want to toss a coin a few times? … Once again, fine by me. Whatever floats your boat! … From my sides, I’m all ears…

As to me, from my side, I will come back with a status update some time after watching the videos for the MIT 08.04 course is over, and watching for 08.05 is already in progress… That is, may be after two weeks or so (unless I have some brief update to post or so)…

In the meanwhile, take care, and bye for now…

Two songs I like:

Actually, I’ve two songs this time around. … Let me first give the credits and the links for both:

(Hindi) हम प्यार में जलने वालों को… (“ham pyaar mein jalane waalon ko…”)
Singer: Lata Mangeshkar
Lyrics: Rajinder Krishan

A good audio for the original version is here [^]. There are other versions, including a so-called “revival” series version, but I won’t bother to give you the links to them.

The second song I have in mind is this:

(Hindi) फिर तेरी कहानी याद आयी… (“phir teri kahaani yaad aayee…”)
Singer: Lata Mangeshkar

A good quality audio is here [^].

Both are Hindi film songs based on Indian classical music. Both are sung by Lata. Both have been composed by highly acclaimed music composers. The two songs also have a certain unmistakable kind of a similarity in terms of certain turns of the tune, certain phrases of melody, so to speak. Both are serious kind of songs, evoking a sombre kind of a mood, one that borders on sadness but in a somewhat abstract sense. Further, people usually describe both these songs in quite superlative terms, and generally speaking, I quite agree with such assessments too.

However, personally, I happen to like one of them a bit more than the other. The question is: Which one? And, why, i.e., for what aspects / reasons? (And we consider only the audio aspects of the two songs here.) …

… I will let you think for a bit about it, and only then tell you my answer, in briefest possible terms, say in one sentence or two at the most, say via an update to this post (which may occur after 2–3 days or so). … In the meanwhile, happy guessing and / or consulting (once again) astrologers / tarot card readers / psychics / whoever, or even using (or implementing) an AI, or using RNGs / tossing coins. … Optionally, thinking too!…

BTW, if the songs are new to you, see if you enjoy any of them, or both…

Bye for now and take care…

History:
— 2021.05.31 20:01 IST: First published
— 2021.06.01 19:39 IST: Added a sub-section in section 4, covering a recommended sequence for Foundations of QM. Also, generally streamlined content, with some minor additions throughout.
— 2021.06.02 13:22 IST: Some more streamlining and fixing of typo’s. Now, I am done with this post.

I. A general update regarding my on-going research work (on my new approach to QM):

1.1 How the development is actually proceeding:

I am working through my new approach to QM. These days, I write down something and/or implement some small and simple Python code snippets (< 100 LOC Python code) every day. So, it’s almost on a daily basis that I am grasping something new.

The items of understanding are sometimes related to my own new approach to QM, and at other times, just about the mainstream QM itself. Yes, in the process of establishing a correspondence of my ideas with those of the mainstream QM, I am getting to learn the ideas and procedures from the mainstream QM too, to a better depth. … At other times, I learn something about the correspondence of both the mainstream QM and my approach, with the classical mechanics.

Yes, at times, I also spot some inconsistencies within my own framework! It too happens! I’ve spotted several “misconceptions” that I myself have had—regarding my own approach!

You see, when you are ab initio developing a new theory, it’s impossible to pursue the development of the theory very systematically. It’s impossible to be right about every thing, right from the beginning. That’s because the very theory itself is not fully known to you while you are still developing it! The neatly worked out structure, its best possible presentations, the proper hierarchical relations… all of these emerge only some time later.

Yes, you do have some overall, “vaguish” idea(s) about the major themes that are expected to hold the new theory together. You do know many elements that must be definitely there.

In my case, such essential themes or theoretical elements go, for example, like: the energy conservation principle, the reality of some complex-valued field, the specific (natural) form of the non-linearity which I have proposed, my description of the measurement process and of Born’s postulate, the role that the Eulerian (fixed control volume-based) formulations play in my theorization, etc.

But all these are just elements. Even when tied together, they still amount to only an initial framework. Many of these elements may eventually turn out to play an over-arching role in the finished theory. But during the initial stages (including the stage I am in), you can’t even tell which element is going to play a greater role. All the elements are just loosely (or flexibly) held together in your mind. Such a loosely held set does not qualify to be called a theory. There are lots and lots (and lots) of details that you still don’t even know exist. You come to grasp these only on the fly, only as you are pursuing the “fleshing out” of the “details”.

Once the initial stage gets over, and you are going through the fleshing out stage, the development has a way of progressing on multiple threads of thought, simultaneously.

There are insights or minor developments, or simply new validations of some earlier threads, which occur almost on a daily basis. Each is a separate piece of a small little development; it makes sense to you; and all such small little pieces keep adding up—in your mind and in your notebooks.

Still, there is not much to share with others, simply because in the absence of a knowledge of all that’s going through your mind, any pieces you share are simply going to look as if they were very haphazard, even “random”.

1.3. At this stage, others can easily misunderstand what you mean:

Another thing. There is also a danger that someone may misread you.

For example, because he himself is not clear on many other points which you have not noted explicitly.

Or, may be, you have noted your points somewhere, but he hasn’t yet gone through them. In my case, it is the entirety of my Ontologies series [^]. … Going by the patterns of hits at this blog, I doubt whether any single soul has ever read through them all—apart from me, that is. But this entire series is very much alive in my mind when I note something here or there, including on the Twitter too.

Or, sometimes, there is a worse possibility too: The other person may read what you write quite alright, but what you wrote down itself was somewhat misleading, perhaps even wrong!

Indeed, recently, something of this sort happened when I had a tiny correspondence with someone. I had given a link to my Outline document [^]. He went through it, and then quoted from it in his reply to me. I had said, in the Outline document, that the electrons and protons are classical point-particles. His own position was that they can’t possibly be. … How possibly could I reply him? I actually could not. So, I did not!

I distinctly remember that right when I was writing this point in the Outline document, I had very much hesitated precisely at it. I knew that the word “classical” was going to create a lot of confusions. People use it almost indiscriminately: (i) for the ontology of Newtonian particles, (ii) for the ontology of Newtonian gravity, (iii) for ontology of the Fourier theory (though very few people think of this theory in the context of ontologies), (iv) for ontology of EM as implied by Maxwell, (v) for ontology of EM as Lorentz was striving to get at and succeeded brilliantly in so many essential respects (but not all, IMO), etc.

However, if I were to spend time on getting this portion fully clarified (first to myself, and then for the Outline document), then I also ran the risk of missing out on noting many other important points which also were fairly nascent to me (in the sense, I had not noted them down in a LaTeX document). These points had to be noted on priority, right in the Outline document.

Some of these points were really crucial—the $V(x,t)$ field as being completely specified in reference to the elementary charges alone (i.e. no arbitrary PE fields), the non-linearity in $\Psi(x,t)$, the idea that it is the Instrument’s (or Detector’s) wavefunction which undergoes a catastrophic change—and not the wavefunction of the particle being measured, etc. A lot of such points. These had to be noted, without wasting my time on what precisely I meant when I used the word “classical” for the point-particle of the electron etc.

Yes, I did identify that I the elementary particles were to be taken as conditions in the aether. I did choose the word “background object” merely in order to avoid any confusion with Maxwell’s idea of a mechanical aether. But I myself wasn’t fully clear on all aspects of all the ideas. For instance, I still was not familiar with the differences of Lorentz’ aether from Maxwell’s.

All in all, a document like the Outline document had to be an incomplete document; it had to come out in the nature of a hurried job. In fact, it was so. And I identified it as such.

I myself gained a fuller clarity on many of these issues only while writing the Ontologies series, which happened some 7 months later, after putting out the Outline document online. And, it was even as recently as in the last month (i.e., about 1.5 years after the Outline document) that I was still further revising my ideas regarding the correspondence between QM and CM. … Indeed, this still remains a work in progress… I am maintaining handwritten notes and LaTeX files too (sort of like “journal”s or “diaries”).

All in all, sharing a random snapshot of a work-in-progress always carries such a danger. If you share your ideas too early, while they still are being worked out, you might even end up spreading some wrong notions! And when it comes to theoretical work, there is no product-recall mechanism here—at all! Detrimental to your goals, after all!

1.3 How my blogging is going to go, in the next few weeks:

So, though I am passing through a very exciting phase of development these days, and though I do feel like sharing something or the other on an almost daily basis, when I sit down and think of writing a blog post, unfortunately, I find that there is very little that I can actually share.

For this very reason, my blogging is going to be sparse over the coming weeks.

However, in the meanwhile, I might post some brief entries, especially regarding papers/notes/etc. by others. As in this post.

OTOH, if you want something bigger to think about, see the Q&A answers from my last post here. That material is enough to keep you occupied for a couple of decades or more… I am not joking. That’s what’s happened to others; it has happened to me; and I can guarantee you that it would happen to you too, so long as you keep forgetting whatever you’ve read about my new approach. You could then very easily spend decades and decades (and decades)…

Anyway, coming back to some recent interesting pieces by others…

2.1. Luboš Motl on TerraPower, Inc.:

Dr. Luboš Motl wrote a blog-post of the title “Green scientific illiteracy enters small nuclear reactors, too” [^]. This piece is a comment on TerraPower’s proposal. In case you didn’t know, TerraPower is a pet project of Bill Gates’.

My little note (on the local HDD), upon reading this post, had said something like, “The critics of this idea are right, from an engineering/technological viewpoint.”

In particular, I have too many apprehensions about using liquid sodium. Further, given the risk involved in distributing the sensitive nuclear material over all those geographically dispersed plants, this idea does become, err…, stupid.

In the above post, Motl makes reference to another post of his, one from 2019, regarding the renewable energies like the solar and the wind. The title of this earlier post read: “Bill Gates: advocates of dominant wind & solar energy are imbeciles” [^]. Make sure to go through this one too. The calculation given in it is of a back-of-the-envelop kind, but it also is very impeccable. You can’t find flaw with the calculation itself.

Of course, this does not mean that research on renewable energies should not be pursued. IMO, it should be!

It’s just that I want to point out a few things: (i) Motl chooses the city of Tokyo for his calculation, which IMO would be an extreme case. Tokyo is a very highly dense city—both population-wise and on the count of geographical density of industries (and hence, of industrial power consumption). There can easily be other places where the density of power consumption, and the availability of the natural renewable resources, are better placed together. (ii) Even then, calculations such as that performed by Motl must be included in all analyses—and, the cost of renewable energy must be calculated without factoring in the benefit of government subsidies. … Yes, research on renewable energy would still remain justified. (iii) Personally, I find the idea of converting the wind/solar electricity into hydrogen more attractive. See my 2018 post [^] which had mentioned the idea of using the hydrogen gas as a “flywheel” of sorts, in a distributed system of generation (i.e. without transporting the wind-generated hydrogen itself, over long distances).

2.2. Demonstrations on coupled oscillations and resonance at Harvard:

As to the relevance of this topic to my new approach to QM: The usual description of resonance proceeds by first stating a homogeneous differential equation, and then replacing the zero on the right hand-side with a term that stands for an oscillating driving force [^]. Thus, we specify a force-term for the driver, but the System under study is still being described with the separation vector (i.e. a displacement) as the primary unknown.

Now, just take the driver part of the equation, and think of it as a multi-scaled effect of a very big assemblage of particles whose motions themselves are fundamentally described using exactly the same kind of terms as those for the particles in the System, i.e., using displacements as the primary unknown. It is the multi-scaling procedure which transforms a fundamentally displacement-based description to a basically force-primary description. Got it? Hint below.

[Hint: In the resonance equation, it is assumed that form of the driving force remains exactly the same at all times: with exactly the same $F_0$, $m$, and $\omega$. If you replace the driving part with particles and springs, none of the three parameters characterizing the driving force will remain constant, especially $\omega$. They all will become functions of time. But we want all the three parameters to stay constant in time. …Now, the real hint: Think of the exact sinusoidal driving force as an abstraction, and multi-scaling as a means of reaching that abstraction.]

2.3 Visualization of physics at the University of St. Andrews:

Again, very neat [^]. The simulations here have very simple GUI, but the design of the applets has been done thoughtfully. The scenarios are at a level more advanced than the QM simulations at PhET, University of Colorado [^].

2.4. The three-body problem:

The nonlinearity in $\Psi(x,t)$ which I have proposed is, in many essential ways, similar to the classical $N$-body problem.

The simplest classical $N$-body problem is the $3$-body problem. Rhett Allain says that the only way to solve the $3$-body problem is numerically [^]. But make sure to at least cursorily note the special solutions mentioned in the Wiki [^]. This Resonance article (.PDF) [^] seems quite comprehensive, though I haven’t gone through it completely. Related, with pictures: A recent report with simulations, for search on “choreographies” (which is a technical term; it refers to trajectories that repeat) [^].

Sure there could be trajectories that repeat for some miniscule number of initial conditions. But the general rule is that the $3$-body problem already shows sensitive dependence on initial conditions. Search the ‘net for $4$-body, $5$-body problems. … In QM, we have $10^{23}$ particles. Cool, no?

2.5.1: Max Born in IISc Bangalore:

Check out a blog post/article by Karthik Ramaswamy, of the title “When Raman brought Born to Bangalore” [^]. (H/t Luboš Motl [^].)

2.5.2: Academic culture in India in recent times—a personal experience involving the University of Pune, IIT Bombay, IIT Madras, and IISc Bangalore:

After going through the above story, may I suggest that you also go through my posts on the Mechanical vs. Metallurgy “Branch Jumping” issue. This issue decidedly came up in 2002 and 2003, when I went to IIT Bombay for trying admission to PhD program in Mechanical department. I tried multiple times. They remained adamant throughout the 2002–2003 times. An associate professor from the Mechanical department was willing to become my guide. (We didn’t know each other beforehand.) He fought for me in the department meeting, but unsucessfully. (Drop me a line to know who.) One professor from their CS department, too, sympathetically listened to me. He didn’t understand the Mechanical department’s logic. (Drop me a line to know who.)

Eventually, in 2003, three departments at IISc Bangalore showed definite willingness to admit me.

One was a verbal offer that the Chairman of the SERC made to me, but in the formal interview (after I had on-the-spot cleared their written tests—I didn’t know they were going to hold these). He even offered me a higher-than-normal stipend (in view of my past experience), but he said that the topic of research would have to be one from some 4–5 ongoing research projects. I declined on the spot. (He did show a willingness to wait for a little while, but I respectfully declined it too, because I knew I wanted to develop my own ideas.)

At IISc, there also was a definite willingness to admit me by both their Mechanical and Metallurgy departments. That is, during my official interviews with them (which once again happened after I competitively cleared their separate written tests, being short-listed to within 15 or 20 out of some 180 fresh young MTech’s in Mechanical branch from IISc and IITs—being in software, I had forgotten much of my core engineering). Again, it emerged during my personal interviews with the departmental committees, that I could be in (yes, even in their Mechanical department), provided that I was willing to work on a topic of their choice. I protested a bit, and indicated the loss of my interest right then and there, during both these interviews.

Finally, at around the same time (2003), at IIT Madras, the Metallurgical Engg. department also made an offer to me (after yet another written test—which I knew was going to be held—and an interview with a big committee). They gave me the nod. That is, they would let me pursue my own ideas for my PhD. … I was known to many of them because I had done my MTech right from the same department, some 15–17 years earlier. They recalled, on their own, the hard work which I had put in during my MTech project work. They were quite confident that I could deliver on my topic even if they at that time they (and I!) had only a minimal idea about it.

However, soon enough, Prof. Kajale at COEP agreed to become my official guide at University of Pune. Since it would be convenient for me to remain in Pune (my mother was not keeping well, among other things), I decided to do my PhD from Pune, rather than broach the topic once again at SERC, or straight-away join the IIT Madras program.

Just thought of jotting down the more recent culture at these institutes (at IIT Bombay, IIT Madras, and IISc Bangalore), in COEP, and of course, in the University of Pune. I am sure it’s just a small slice in the culture, just one sample, but it still should be relevant…

Also relevant is this part: Right until I completely left academia for good a couple of years ago, COEP professors and the University of Pune (not to mention UGC and AICTE) continued barring me from becoming an approved professor of mechanical engineering. (It’s the same small set of professors who keep chairing interview processes in all the colleges, even universities. So, yes, the responsibility ultimately lies with a very small group of people from IIT Bombay’s Mechanical department—the Undisputed and Undisputable Leader, and with COEP and University of Pune—the  Faithful Followers of the former).

2.5.3. Dirac in India:

BTW, in India, there used to a monthly magazine called “Science Today.” I vaguely recall that my father used to have a subscription for it right since early 1970s or so. We would eagerly wait for each new monthly issue, especially once I knew enough English (and physics) to be able to more comfortably go through the contents. (My schooling was in Marathi medium, in rural areas.) Of course, my general browsing of this magazine had begun much earlier. [“Science Today” would be published by the Times of India group. Permanently gone are those days!]

I now vaguely remember that one of the issues of “Science Today” had Paul Dirac prominently featured in it. … I can’t any longer remember much anything about it. But by any chance, was it the case that Prof. Dirac was visiting India, may be TIFR Bombay, around that time—say in mid or late 1970s, or early 1980’s? … I tried searching for it on the ‘net, but could not find anything, not within the first couple of pages after a Google search. So, may be, likely, I have confused things. But would sure appreciate pointers to it…

PS: Yes, I found this much:

“During 1973 and 1975 Dirac lectured on the problems of cosmology in the Physical Engineering Institute in Leningrad. Dirac also visited India.’‘ [^].

… Hmm… Somehow, for some odd reason, I get this feeling that the writer of this piece, someone at Vigyan Prasar, New Delhi, must have for long been associated with IIT Bombay (or equivalent thereof). Whaddaya think?

2.6. Jim Baggott’s new book: “Quantum Reality”:

I don’t have the money to buy any books, but if I were to, I would certainly buy three books by Jim Baggott: The present book of the title “Quantum Reality,” as well as a couple of his earlier books: the “40 moments” book and the “Quantum Cookbook.” I have read a lot of pages available at the Google books for all of these three books (may be almost all of the available pages), and from what I read, I am fully confident that buying these books would be money very well spent indeed.

Dr. Sabine Hossenfelder has reviewed this latest book by Baggott, “Quantum Reality,” at the Nautil.us; see “Your guide to the many meanings of quantum mechanics,” here [^]. … I am impressed by it—I mean this review. To paraphrase Hossenfelder herself: “There is nothing funny going on here, in this review. It just, well, feels funny.”

Dr. Peter Woit, too, has reviewed “Quantum Reality” at his blog [^] though in a comparatively brief manner. Make sure to go through the comments after his post, especially the very first comment, the one which concerns classical mechanics, by Matt Grayson [^]. PS: Looks like Baggott himself is answering some of the comments too.

Sometime ago, I read a few blog posts by Baggott. It seemed to me that he is not very well trained in philosophy. It seems that he has read philosophy deeply, but not comprehensively. [I don’t know whether he has read the Objectivist metaphysics and epistemology or not; whether he has gone through the writings/lectures by Ayn Rand, Dr. Leonard Peikoff, Dr. Harry Binswanger and David Harriman or not. I think not. If so, I think that he would surely benefit by this material. As always, you don’t have to agree with the ideas. But yes, the material that I am pointing out is by all means neat enough that I can surely recommend it.]

Coming back to Baggott: I mean to say, he delivers handsomely when (i) he writes books, and (ii) sticks to the physics side of the topics. Or, when he is merely reporting on others’ philosophic positions. (He can condense down their positions in a very neat way.) But in his more leisurely blog posts/articles, and sometimes even in his comments, he does show a tendency to take some philosophic point in a something of a wrong direction, and to belabour on it unnecessarily. That is to say, he does show a certain tendency towards pedantry, as it were.  But let me hasten to add: He seems to show this tendency only in some of his blog-pieces. Somehow, when it comes to writing books, he does not at all show this tendency—well, at least not in the three books I’ve mentioned above.

So, the bottomline is this:

If you have an interest in QM, and if you want a comprehensive coverage of all its interpretations, then this book (“Quantum Reality”) is for you. It is meant for the layman, and also for philosophers.

However, if what you want is a very essentialized account of most all of the crucial moments in the development of QM (with a stress on physics, but with some philosophy also touched on, and with almost no maths), then go buy his “40 Moments” book.

Finally, if you have taken a university course in QM (or are currently taking it), then do make sure to buy his “Cookbook” (published in January this year). From what I have read, I can easily tell: You would be doing yourself a big favour by buying this book. I wish the Cookbook was available to me at least in 2015 if not earlier. But the point is, even after developing my new approach, I am still going to buy it. It achieves a seemingly impossible combination: Something that makes for an easy reading (if you already know the QM) but it will also serve as a permanent reference, something which you can look up any time later on. So, I am going to buy it, once I have the money. Also, “Quantum Reality”, the present book for the layman. Indeed all the three books I mentioned.

(But I am not interested in relativity theory, or QFT, standard model, etc. etc. etc., and so, I will not even look into any books on these topics, written by any one.)

OK then, let me turn back to my work… May be I will come back with some further links in the next post too, may be after 10–15 days. Until then, take care, and bye for now…

A song I like:

(Marathi) घन घन माला नभी दाटल्या (“ghan ghan maalaa nabhee daaTalyaa”)
Singer: Manna Dey
Music: Vasant Pawar

[A classic Marathi song. Based on the (Sanskrit, Marathi) राग मल्हार (“raaga” called “Malhaara”). The best quality audio is here [^]. Sung by Manna Dey, a Bengali guy who was famous for his Hindi film songs. … BTW, it’s been a marvellous day today. Clear skies in the morning when I thought of doing a blog post today and was wondering if I should add this song or not. And, by the time I finish it, here are strong showers in all their glory! While my song selection still remains more or less fully random (on the spur of the moment), since I have run so many songs already, there has started coming in a bit of deliberation too—many songs that strike me have already been run!

Since I am going to be away from blogging for a while, and since many of the readers of this blog don’t have the background to appreciate Marathi songs, I may come back and add an additional song, a non-Marathi song, right in this post. If so, the addition would be done within the next two days or so. …Else, just wait until the next post, please! Done, see the song below]

(Hindi) बोल रे पपीहरा (“bol re papiharaa”)
Singer: Vani Jairam
Music: Vasant Desai
Lyrics: Gulzar

[I looked up on the ‘net to see if I can get some Hindi song that is based on the same “raaga”, i.e., “Malhaar” (in general). I found this one, among others. Comparing these two songs should give you some idea about what it means when two songs are said to share the same “raaga”. … As to this song, I should also add that the reason for selecting it had more to do with nostalgia, really speaking. … You can find a good quality audio here [^].

Another thing (that just struck me, on the fly): Somehow, I also thought of all those ladies and gentlemen from the AICTE New Delhi, UGC New Delhi, IIT Bombay’s Mechanical Engg. department, all the professors (like those on R&R committees) from the University of Pune (now called SPPU), and of course, the Mechanical engg. professors from COEP… Also, the Mechanical engineering professors from many other “universities” from the Pune/Mumbai region. … पपीहरा… (“papiharaa”) Aha!… How apt are words!… Excellence! Quality! Research! Innovation! …बोल रे, पपीहरा ऽऽऽ (“bol re papiharaa…”). … No jokes, I had gone jobless for 8+ years the last time I counted…

Anyway, see if you like the song… I do like this song, though, probably, it doesn’t make it to my topmost list. … It has more of a nostalgia value for me…

Anyway, let’s wrap up. Take care and bye for now… ]

History:
— First published: 2020.09.05 18:28 IST.
— Several significant additions revisions till 2020.09.06 01:27 IST.
— Much editing. Added the second song. 2020.09.06 21:40 IST. (Now will leave this post in whatever shape it is in.)

# String theory of engineers, for physicists and mathematicians

A Special note for the Potential Employers from the Data Science field:

Recently, in April 2020, I achieved a World Rank # 5 on the MNIST problem. The initial announcement can be found here [^], and a further status update, here [^].

All my data science-related posts can always be found here [^]

1. You know the classical wave equation:

You know the classical wave equation, right?

What’s there in it? Just:

$u(t) = \sin( \omega t )$!

or, OK, to make it more general…

$u(t) = A \cos( \omega t ) + B \sin( \omega t )$

Something like that might have passed in your mind, first. When someone says “wave”, people think the water waves, say the ocean waves. Or, they think of the light or sound waves, the interference experiments, even the wave-particle duality. Yet, curiously, when you say “wave equation”, people tend to think of the SHM (simple harmonic motion)—the oscillations of a point-mass, but not the waves in continua.

So, to make it clear, suppose I ask you:

Ah, I see what you mean. Pretty simple, too. But now it makes sense to get into a little bit of the complex algebra:

$u(x,t) = A e^{i( \vec{k}\;\cdot\;\vec{x} - \omega\,t)}$

You are likely to continue…

…Remember the Euler identity? The minus sign, because we want to have a wave that travels to the right? Oops, in the positive $x$-direction…

Ummm…

You know,

I would have to say at this juncture,

the wave equation? I mean, the differential equation. The linear one!

To which, you are likely to retort back

What a ridiculous question! Of course I know it!

OK, it goes like this…

You might then proceed to jot down the following equation in a hurried manner, more or less to get done and be over with my questioning:

$\dfrac{\partial^2 u}{\partial x^2} = \dfrac{1}{c^2} \dfrac{\partial^2 u}{\partial t^2}$

Yeah, of course, so you do seem to know it. That’s what I was saying!

You studied the topic as early as in XI or XII standard (if not in your high-school). You had mastered it—right back then. You aced your exams, always. You then went to a great engineering school, and studied waves that were a lot more complicated. Like, may be, the EM waves radiated by a radio antenna, or may be, the vibrations in the machinery and cars, whatever …. You have even mastered the simulation techniques for them. Not just FDM but also FEM, BEM, pseudo-spectral methods, and all that.

Or, may be, you weren’t driven by the lowly commercial considerations. You were really interested in the fundamentals. So, you were interested in physics.

“Fundamentals”, you remember you had said some time ago in a distant past, as if to just once again re-affirm your conviction, all in the silence of your mind. And so, obviously, it would have to be physics! It couldn’t possibly have been chemistry for you! And that’s how, you went ahead and attended a great university. Just to pursue physics.

You calculated a lot of quantum wavefunctions but only while you were in UG years—and only in order to clear those stupid exams. But you already knew that fundamental physics is where your focus really was. Real physics. Mathematical physics. Maths!

That’s why, you zipped past that ridiculously simple stage of those mere wavefunctions. You still remember that way before your formal coursework covered it, you had mastered the Dirac notation, the Heisenberg formulation (where operators are time-dependent, not the stupid wavefunction, you had announced to your stupid class-mates), the Uncertainty Principle (uh!), the Poisson brackets, and all that… You had studied it all completely on your own. Then, you had gone into the relativistic QM as well—the Klein-Gordon equation, Dirac’s equation, Feynman’s path integral formulation… All of that. Also GR. Even QFT… May be you even landed into the string theory right while you still were a high-school or UG student.

… It was long ago that you had left those idiotic wavefunctions and all way behind you. They were best left for others to look after, you just knew. That’s what you had thought, and that’s how you’d come to that conclusion.

2. Will you be able to explain its derivation, now?:

So, whether you are an engineer or a physicist, now, it indeed seems that it’s been a long time since you studied the wave equation. That’s why, if someone now asks you to explain the derivation of the wave equation, you might perhaps narrow your eyes a bit. The reason is, unless you’ve been teaching courses to UG students in the recent times, you may not be able to do it immediately. You may have to take a look at the text-book, perhaps just the Wiki? … The Wiki may not be reliable, but since your grasp has been so solid, it wouldn’t take much to mentally go on correctingt the Wiki even as you are reading through it. …Yes, it might take a little bit of time now, but not much. May be a few minutes? Half an hour at the most? May be. But that’s only because you are going to explain it to someone else…

All the same, you are super-duper-damn confident that given the derivation in the text-books (those XII standard or UG level text-books), you are going to zip through it.

Given a brilliant school-kid, you would obviously be able to explain him the derivation all the way through: each and every step of it, and all the assumptions behind them, and even the mathematical reasonability of all those assumptions, too, in turn. You could easily get it all back right in a moment—or half an hour. … “It’s high-school classical physics, damnit”—that’s what you are likely to exclaim! And, following Feynman, you think you are going to enjoy it too…

You are right, of course. After all, it’s been more than 200 years that the $1D$ wave equation was first formulated and solved. It has become an inseparable part of the very intuition of the physicist. The great physicists of the day like d’Alembert and Euler were involved in it—in analyzing the wave phenomena, formulating the equation and inventing the solution techniques. Their thought processes were, say, a cut above the rest. They couldn’t overlook something non-trivial, could they? especially Euler? Wasn’t he the one who had first written down that neat identity which goes by his name? one of the most beautiful equations ever?

That’s what you think.

Euler, Lagrange, Hamilton, … , Morse and Feschback, Feynman…

They all said the same thing, and they all couldn’t possibly be careless. And you had fully understood their derivations once upon a time.

So, the derivation is going to be a cake-walk for you now. Each and every part of it.

Well, someone did decide to take a second look at it—the derivation of the classical wave equation. Then, the following is what unfolded.

3. A second look at the derivation. Then the third. Then the fourth. …:

3.1. Lior Burko (University of Alabama at Huntsville, AL, USA) found some problems with the derivation of the transverse wave equation. So, he wrote a paper:

Burko, Lior M. (2010), “Energy in one-dimensional linear waves in a string,” European Journal of Physics, Volume 31, Number 5. doi: [^]. PDF pre-print here [^].

Abstract: “We consider the energy density and energy transfer in small amplitude, one-dimensional waves on a string and find that the common expressions used in textbooks for the introductory physics with calculus course give wrong results for some cases, including standing waves. We discuss the origin of the problem, and how it can be corrected in a way appropriate for the introductory calculus-based physics course.”

In this abstract and all the ones which follow, the emphasis in italicized bold is mine.

3.2. Eugene Butikov (St. Petersburg State University, St. Petersburg, Russia) found issues with Burko’s arguments. So, he wrote a paper (a communication) by way of a reply in the same journal:

Butikov, Eugene I. (2011) “Comment on Energy in one-dimensional linear waves in a string’,” European Journal of Physics, Volume 32, Number 6. doi: [^] . PDF e-print available here [^].

Abstract: “In this communication we comment on numerous erroneous statements in a recent letter to this journal by Burko (Eur. J. Phys. 2010 31 L71–7) concerning the energy transferred by transverse waves in a stretched string.”

3.3. C. E. Repetto, A. Roatta, and R. J. Welti (Vibration and Wave Laboratory, Physics Department, Faculty of Exact Sciences, Engineering and Surveying [per Google Translate] (UNR), Rosario Santa Fe, Argentina, and Institute of Physics, Rosario, Argentina) also found issues with Burko’s paper, and so, they too wrote another paper, which appeared in the same issue as Butikov’s:

Repetto, Roatta and Welti (2011), “Energy in one-dimensional linear waves,” European Journal of Physics, Volume 32, Number 6. doi: [^] . PDF available here [^].

Abstract: “This work is based on propagation phenomena that conform to the classical wave equation. General expressions of power, the energy conservation equation in continuous media and densities of the kinetic and potential energies are presented. As an example, we study the waves in a string and focused attention on the case of standing waves. The treatment is applicable to introductory science textbooks.”

Though they didn’t mention Burko’s paper in the abstract, the opening line made it clear that this was a comment on the latter.

3.4. Burko, the original author, replied back to both these comments. All the three were published in the same issue of the same journal:

Burko, Lior M. (2011) “Reply to comments on Energy in one-dimensional linear waves in a string’,” European Journal of Physics, Volume 31, Number 6. doi: [^]. PDF eprint available here [^].

Abstract: “In this reply we respond to comments made by Repetto et al and by Butokov on our letter (Burko 2010 Eur. J. Phys. 31 L71–7), in which we discussed two different results for the elastic potential energy of a string element. One derived from the restoring force on a stretched string element and the other from the work done to bring the string to a certain distorted configuration. We argue that one cannot prefer from fundamental principles the former over the latter (or vice versa), and therefore one may apply either expression to situations in which their use contributes to insight. The two expressions are different by a boundary term which has a clear physical interpretation. For the case of standing waves, we argue that the latter approach has conceptual clarity that may contribute to physical understanding.”

3.5. David Rowland (University of Queensland, Brisbane, Australia) also wrote a reply, which too was published in the same issue of the same journal.

Rowland, David R. (2011) “The potential energy density in transverse string waves depends critically on longitudinal motion,” European Journal of Physics, Volume 31, Number 6. doi: [^]. The author’s pre-print (pre-publication version) is available here, [^].

Abstract: “The question of the correct formula for the potential energy density in transverse waves on a taut string continues to attract attention (e.g. Burko 2010 Eur. J. Phys. 31 L71), and at least three different formulae can be found in the literature, with the classic text by Morse and Feshbach (Methods of Theoretical Physics pp 126–127) stating that the formula is inherently ambiguous. The purpose of this paper is to demonstrate that neither the standard expression nor the alternative proposed by Burko can be considered to be physically consistent, and that to obtain a formula free of physical inconsistencies and which also removes the ambiguity of Morse and Feshbach, the longitudinal motion of elements of the string needs to be taken into account,even though such motion can be neglected when deriving the linear transverse wave equation. Two derivations of the correct formula are sketched, one proceeding from a consideration of the amount of energy required to stretch a small segment of string when longitudinal displacements are considered, and the other from the full wave equation. The limits of the validity of the derived formulae are also discussed in detail.”

3.6. Butikov wrote another paper, a year later, now in Physica Scripta.

Butikov, Eugene I. (2012) ”Misconceptions about the energy of waves in a strained string,” Physica Scripta, Vol. 86, Number 3, p. 035403. doi: [^]. PDF ePrint available here [^]:

Abstract: “The localization of the elastic potential energy associated with transverse and longitudinal waves in a stretched string is discussed. Some misunderstandings about different expressions for the density of potential energy encountered in the literature are clarified. The widespread opinion regarding the inherent ambiguity of the density of elastic potential energy is criticized.

3.7. Rowland, too, seems to have continued with the topic even after the initial bout of papers. He published another paper in 2013, continuing in the same journal where earlier papers had appeared:

Rowland, David R. (2013) “Small amplitude transverse waves on taut strings: exploring the significant effects of longitudinal motion on wave energy location and propagation,” European Journal of Physics, Volume 34, Number 2. doi: [^] . PDF ePrint is available here [^].

Abstract: “Introductory discussions of energy transport due to transverse waves on taut strings universally assume that the effects of longitudinal motion can be neglected, but this assumption is not even approximately valid unless the string is idealized to have a zero relaxed length, a requirement approximately met by the slinky spring. While making this additional idealization is probably the best approach to take when discussing waves on strings at the introductory level, for intermediate to advanced undergraduate classes in continuum mechanics and general wave phenomena where somewhat more realistic models of strings can be investigated, this paper makes the following contributions. First, various approaches to deriving the general energy continuity equation are critiqued and it is argued that the standard continuum mechanics approach to deriving such equations is the best because it leads to a conceptually clear, relatively simple derivation which provides a unique answer of greatest generality. In addition, a straightforward algorithm for calculating the transverse and longitudinal waves generated when a string is driven at one end is presented and used to investigate a $cos^2$ transverse pulse. This example illustrates much important physics regarding energy transport in strings and allows the `attack waves’ observed when strings in musical instruments are struck or plucked to be approximately modelled and analysed algebraically. Regarding the ongoing debate as to whether the potential energy density in a string can be uniquely defined, it is shown by coupling an external energy source to a string that a suggested alternative formula for potential energy density requires an unphysical potential energy to be ascribed to the source for overall energy to be conserved and so cannot be considered to be physically valid.

3.8. Caamaño-Withall and Krysl (University of California, San Diego, CA, USA) aimed for settling everything. They brought in a computational engineer’s perspective too:

Caamaño-Withall, Zach and Krysl, Petr (2016) “Taut string model: getting the right energy versus getting the energy the right way,” World Journal of Mechanics, Volume 6, Number 2. doi: [^]. This being an open-access article, the PDF is available right from the doi.

Abstract: “The initial boundary value problem of the transverse vibration of a taut stringisa classic that can be found in many vibration and acoustics textbooks. It is often used as the basis for derivations of elementary numerical models, for instance finite element or finite difference schemes. The model of axial vibration of a prismatic elastic baralso serves in this capacity, often times side-by-side with the first model. The stored (potential) energy for these two models is derived in the literature in two distinct ways. We find the potential energy in the taut string model to be derived from a second-order expression of the change of the length of the string. This is very different in nature from the corresponding expression for the elastic bar, which is predictably based on the work of the internal forces. The two models are mathematically equivalentin that the equations of one can be obtained from the equations of the other by substitution of symbols such as the primary variable, the resisting force and the coefficient of the stiffness. The solutions also have equivalent meanings, such as propagation of waves and standing waves of free vibration. Consequently, the analogy between the two models can and should be exploited, which the present paper successfully undertakes. The potential energy of deformation of the string was attributed to the seminal work of Morse and Feshbachof 1953. This book was also the source of a misunderstanding as to the correct expression for the density of the energy of deformation. The present paper strives to settle this question.”

4. A standard reference:

Oh, BTW, for a mainstream view prevalent before Burko’s paper, check out a c. 1985 paper by Mathews, Jr. (Georgetown University):

Mathews Jr., W. N. (1985) “Energy in a one‐dimensional small amplitude mechanical wave,” American Journal of Physics, Volume 53, 974. doi: [^].

Abstract: We present a discussion of the energy associated with a one‐dimensional mechanical wave which has a small amplitude but is otherwise general. We consider the kinetic energy only briefly because the standard treatments are adequate. However, our treatment of the potential energy is substantially more general and complete than the treatments which appear in introductory and intermediate undergraduate level physics textbooks. Specifically, we present three different derivations of the potential energy density associated with a one‐dimensional, small amplitude mechanical wave. The first is based on the ‘‘virtual displacement’’ concept. The second is based on the ideas of stress and strain as they are generally used in dealing with the macroscopic elastic properties of matter. The third is based on the principle of conservation of energy, and also leads to an expression for the energy flux of the wave. We also present an intuitive and physical discussion based on the analogy between our system and a spring.

I could not access it, but it was quoted by most (all?) of the papers cited above (which I could).

5. Is it a settled matter, now?:

Have these last few papers settled all the issues that were raised?

Ummm… Why don’t you read the papers and decide by yourself?

6. Why bother?

“But why did you get into all this exasperating thing / stupidity / mess, when all engineers have anyway been using the wave equation to design everything from radios, TVs, Internet router hardware to cars, washing machines, and what not?”

Many of you are likely to phrase your question that way.

My answer is: Well, simply because I ran into these papers while thinking something else about the wave equation and waves. I got puzzled a bit about one very simple and stupid physical idea that had struck me. Far, far simpler than what’s discussed in the above papers. Even just a conceptual analysis of my stupid-simple idea seemed pretty funny to me. So, I’d googled on the related topics just in order to know if any one had thought of along the same lines. Which then led me to the above selection of papers.

What was that idea?

Not very important. Let me mention it some other time. I think there is much more than enough material already in this post!

In the meanwhile, browse through these papers and see if you get all the subtle arguments—all of them being accessible to engineers too, not just to physicists or mathematicians.

Come to think of it, it might be a good idea to post a shortened version of this entry at iMechanica too. … May be a few days later…

In the meanwhile, take care and bye for now…

A song I like:

(Western, Pop): “karma chamelion”
Band: Culture Club

[A Western song that is also hummable! … As always, I couldn’t (and still can’t!) make out words, though today I did browse the lyrics [^] and the Wiki on the song [^]. Back in the 1980s, it used to be quite popular in Pune. Also in IIT Madras. … I like this song for its “hummability” / “musicality” / ”lyricality” / melody or so. Also, the “texture” of the sound—the bass and the rhythm blends really well with the voices and other instrumentals. A pretty neat listen…]