A list of books for understanding the non-relativistic QM

TL;DR: NFY (Not for you).

In this post, I will list those books which have been actually helpful to me during my self-studies of QM.

But before coming to the list, let me first note down a few points which would be important for engineers who wish to study QM on their own. After all, my blog is regularly visited by engineers too. That’s what the data about the visit patterns to various posts says.

Others (e.g. physicists) may perhaps skip over the note in the next section, and instead jump directly over to the list itself. However, even if the note for engineers is too long, perhaps, physicists should go through it too. If they did, they sure would come to know a bit more about the kind of background from which the engineers come.

I. A note for engineers who wish to study QM on their own:

The point is this: QM is vast, even if its postulates are just a few. So, it takes a prolonged, sustained effort to learn it.

For the same reason (of vastness), learning QM also involves your having to side-by-side learn an entirely new approach to learning itself. (If you have been a good student of engineering, chances are pretty good that you already have some first-hand idea about this meta-learning thing. But the point is, if you wish to understand QM, you have to put it to use once again afresh!)

In terms of vastness, QM is, in some sense, comparable to this cluster of subjects spanning engineering and physics: engineering thermodynamics, statistical mechanics, kinetics, fluid mechanics, and heat- and mass-transfer.

I.1 Thermodynamics as a science that is hard to get right:

The four laws of thermodynamics (including the zeroth and the third) are easy enough to grasp—I mean, in the simpler settings. But when it comes to this subject (as also for the Newtonian mechanics, i.e., from the particle to the continuum mechanics), God lies not in the postulates but in their applications.

The statement of the first law of thermodynamics remains the same simple one. But complexity begins to creep in as soon as you begin to dig just a little bit deeper with it. Entire categories of new considerations enter the picture, and the meaning of the same postulates gets both enriched and deepened with them. For instance, consider the distinction of the open vs. the closed vs. the isolated systems, and the corresponding changes that have to be made even to the mathematical statements of the law. That’s just for the starters. The complexity keeps increasing: studies of different processes like adiabatic vs. isochoric vs. polytropic vs. isentropic etc., and understanding the nature of these idealizations and their relevance in diverse practical applications such as: steam power (important even today, specifically, in the nuclear power plants), IC engines, jet turbines, refrigeration and air-conditioning, furnaces, boilers, process equipment, etc.; phase transitions, material properties and their variations; empirical charts….

Then there is another point. To really understand thermodynamics well, you have to learn a lot of other subjects too. You have to go further and study some different but complementary sciences like heat and mass transfer, to begin with. And to do that well, you need to study fluid dynamics first. Kinetics is practically important too; think of process engineering and cost of energy. Ideas from statistical mechanics are important from the viewpoint of developing a fundamental understanding. And then, you have to augment all this study with all the empirical studies of the irreversible processes (think: the boiling heat transfer process). It’s only when you study such an entire gamut of topics and subjects that you can truly come to say that you now have some realistic understanding of the subject matter that is thermodynamics.

Developing understanding of the aforementioned vast cluster of subjects (of thermal sciences) is difficult; it requires a sustained effort spanning over years. Mistakes are not only very easily possible; in engineering schools, they are routine. Let me illustrate this point with just one example from thermodynamics.

Consider some point that is somewhat nutty to get right. For instance, consider the fact that no work is done during the free expansion of a gas. If you are such a genius that you could correctly get this point right on your very first reading, then hats off to you. Personally, I could not. Neither do I know of even a single engineer who could. We all had summarily stumbled on some fine points like this.

You see, what happens here is that thermodynamics and statistical mechanics involve entirely different ways of thinking, but they both are being introduced almost at the same time during your UG studies. Therefore, it is easy enough to mix up the some disparate metaphors coming from these two entirely different paradigms.

Coming to the specific example of the free expansion, initially, it is easy enough for you to think that since momentum is being carried by all those gas molecules escaping the chamber during the free expansion process, there must be a leakage of work associated with it. Further, since the molecules were already moving in a random manner, there must be an accompanying leakage of the heat too. Both turn out to be wrong ways of thinking about the process! Intuitions about thermodynamics develop only slowly. You think that you understood what the basic idea of a system and an environment is like, but the example of the free expansion serves to expose the holes in your understanding. And then, it’s not just thermo and stat mech. You have to learn how to separate both from kinetics (and they all, from the two other, closely related, thermal sciences: fluid mechanics, and heat and mass transfer).

But before you can learn to separate out the unique perspectives of these subject matters, you first have to learn their contents! But the way the university education happens, you also get exposed to them more or less simultaneously! (4 years is as nothing in a career that might span over 30 to 40 years.)

Since you are learning a lot many different paradigms at the same time, it is easy enough to naively transfer your fledgling understanding of one aspect of one paradigm (say, that of the particle or statistical mechanics) and naively insert it, in an invalid manner, into another paradigm which you are still just learning to use at roughly the same time (thermodynamics). This is what happens in the case of the free expansion of gases. Or, of throttling. Or, of the difference between the two… It is a rare student who can correctly answer all the questions on this topic, during his oral examination.

Now, here is the ultimate point: Postulates-wise, thermodynamics is independent of the rest of the subjects from the aforementioned cluster of subjects. So, in theory, you should be able to “get” thermodynamics—its postulates, in all their generality—even without ever having learnt these other subjects.

Yet, paradoxically enough, we find that complicated concepts and processes also become easier to understand when they are approached using many different conceptual pathways. A good example here would be the concept of entropy.

When you are a XII standard student (or even during your first couple of years in engineering), you are, more or less, just getting your feet wet with the idea of the differentials. As it so happens, before you run into the concept of entropy, virtually every physics concept was such that it was a ratio of two differentials. For instance, the instantaneous velocity is the ratio of d(displacement) over d(time). But the definition of entropy involves a more creative way of using the calculus: it has a differential (and that too an inexact differential), but only in the numerator. The denominator is a “plain-vanilla” variable. You have already learnt the maths used in dealing with the rates of changes—i.e. the calculus. But that doesn’t mean that you have an already learnt physical imagination with you which would let you handle this kind of a definition—one that involves a ratio of a differential quantity to an ordinary variable. … “Why should only one thing change even as the other thing remains steadfastly constant?” you may wonder. “And if it is anyway going to stay constant, then is it even significant? (Isn’t the derivative of a constant the zero?) So, why not just throw the constant variable out of the consideration?” You see, one major reason you can’t deal with the definition of entropy is simply because you can’t deal with the way its maths comes arranged. Understanding entropy in a purely thermodynamic—i.e. continuum—context can get confusing, to say the least. But then, just throw in a simple insight from Boltzmann’s theory, and suddenly, the bulb gets lit up!

So, paradoxically enough, even if multiple paradigms mean more work and even more possibilities of confusion, in some ways, having multiple approaches also does help.

When a subject is vast, and therefore involves multiple paradigms, people regularly fail to get certain complex ideas right. That happens even to very smart people. For instance, consider Maxwell’s daemon. Not many people could figure out how to deal with it correctly, for such a long time.

…All in all, it is only some time later, when you have already studied all these topics—thermodynamics, kinetics, statistical mechanics, fluid mechanics, heat and mass transfer—that finally things begin to fall in place (if they at all do, at any point of time!). But getting there involves hard effort that goes on for years: it involves learning all these topics individually, and then, also integrating them all together.

In other words, there is no short-cut to understanding thermodynamics. It seems easy enough to think that you’ve understood the 4 laws the first time you ran into them. But the huge gaps in your understanding begin to become apparent only when it comes to applying them to a wide variety of situations.

I.2 QM is vast, and requires multiple passes of studies:

Something similar happens also with QM. It too has relatively few postulates (3 to 6 in number, depending on which author you consult) but a vast scope of applicability. It is easy enough to develop a feeling that you have understood the postulates right. But, exactly as in the case of thermodynamics (or Newtonian mechanics), once again, the God lies not in the postulates but rather in their applications. And in case of QM, you have to hasten to add: the God also lies in the very meaning of these postulates—not just their applications. QM carries a one-two punch.

Similar to the case of thermodynamics and the related cluster of subjects, it is not possible to “get” QM in the first go. If you think you did, chances are that you have a superhuman intelligence. Or, far, far more likely, the plain fact of the matter is that you simply didn’t get the subject matter right—not in its full generality. (Which is what typically happens to the CS guys who think that they have mastered QM, even if the only “QM” they ever learnt was that of two-state systems in a finite-dimensional Hilbert space, and without ever acquiring even an inkling of ideas like radiation-matter interactions, transition rates, or the average decoherence times.)

The only way out, the only way that works in properly studying QM is this: Begin studying QM at a simpler level, finish developing as much understanding about its entire scope as possible (as happens in the typical Modern Physics courses), and then come to studying the same set of topics once again in a next iteration, but now to a greater depth. And, you have to keep repeating this process some 4–5 times. Often times, you have to come back from iteration n+2 to n.

As someone remarked at some forum (at Physics StackExchange or Quora or so), to learn QM, you have to give it “multiple passes.” Only then can you succeed understanding it. The idea of multiple passes has several implications. Let me mention only two of them. Both are specific to QM (and not to thermodynamics).

First, you have to develop the art of being able to hold some not-fully-satisfactory islands of understanding, with all the accompanying ambiguities, for extended periods of time (which usually runs into years!). You have to learn how to give a second or a third pass even when some of the things right from the first pass are still nowhere near getting clarified. You have to learn a lot of maths on the fly too. However, if you ask me, that’s a relatively easier task. The really difficult part is that you have to know (or learn!) how to keep forging ahead, even if at the same time, you carry a big set of nagging doubts that no one seems to know (or even care) about. (To make the matters worse, professional physicists, mathematicians and philosophers proudly keep telling you that these doubts will remain just as they are for the rest of your life.) You have to learn how to shove these ambiguous and un-clarified matters to some place near the back of your mind, you have to learn how to ignore them for a while, and still find the mental energy to once again begin right from the beginning, for your next pass: Planck and his cavity radiation, Einstein, blah blah blah blah blah!

Second, for the same reason (i.e. the necessity of multiple passes and the nature of QM), you also have to learn how to unlearn certain half-baked ideas and replace them later on with better ones. For a good example, go through Dan Styer’s paper on misconceptions about QM (listed near the end of this post).

Thus, two seemingly contradictory skills come into the play: You have to learn how to hold ambiguities without letting them affect your studies. At the same time, you also have to learn how not to hold on to them forever, or how to unlearn them, when the time to do becomes ripe.

Thus, learning QM does not involve just learning of new contents. You also have learn this art of building a sufficiently “temporary” but very complex conceptual structure in your mind—a structure that, despite all its complexity, still is resilient. You have to learn the art of holding such a framework together over a period of years, even as some parts of it are still getting replaced in your subsequent passes.

And, you have to compensate for all the failings of your teachers too (who themselves were told, effectively, to “shut up and calculate!”) Properly learning QM is a demanding enterprise.

II. The list:

Now, with that long a preface, let me come to listing all the main books that I found especially helpful during my various passes. Please remember, I am still learning QM. I still don’t understand the second half of most any UG book on QM. This is a factual statement. I am not ashamed of it. It’s just that the first half itself managed to keep me so busy for so long that I could not come to studying, in an in-depth manner, the second half. (By the second half, I mean things like: the QM of molecules and binding, of their spectra, QM of solids, QM of complicated light-matter interactions, computational techniques like DFT, etc.) … OK. So, without any further ado, let me jot down the actual list.  I will subdivide it in several sub-sections

II.0. Junior-college (American high-school) level:


  • Resnick and Halliday.
  • Thomas and Finney. Also, Allan Jeffrey

II.1. Initial, college physics level:

  • “Modern physics” by Beiser, or equivalent
  • Optional but truly helpful: “Physical chemistry” by Atkins, or equivalent, i.e., only the parts relevant to QM. (I know engineers often tend to ignore the chemistry books, but they should not. In my experience, often times, chemistry books do a superior job of explaining physics. Physics, to paraphrase a witticism, is far too important to be left to the physicists!)

II.2. Preparatory material for some select topics:

  • “Physics of waves” by Howard Georgi. Excellence written all over, but precisely for the same reason, take care to avoid the temptation to get stuck in it!
  • Maths: No particular book, but a representative one would be Kreyszig, i.e., with Thomas and Finney or Allan Jeffrey still within easy reach.
    • There are a few things you have to relearn, if necessary. These include: the idea of the limits of sequences and series. (Yes, go through this simple a topic too, once again. I mean it!). Then, the limits of functions.
      Also try to relearn curve-tracing.
    • Unlearn (or throw away) all the accounts of complex numbers which remain stuck at the level of how \sqrt{-1} was stupefying, and how, when you have complex numbers, any arbitrary equation magically comes to have roots, etc. Unlearn all that talk. Instead, focus on the similarities of complex numbers to both the real numbers and vectors, and also their differences from each. Unlike what mathematicians love to tell you, complex numbers are not just another kind of numbers. They don’t represent just the next step in the logic of how the idea of numbers gets generalized as go from integers to real numbers. The reason is this: Unlike the integers, rationals, irrationals and reals, complex numbers take birth as composite numbers (as a pair of numbers that is ordered too), and they remain that way until the end of their life. Get that part right, and ignore all the mathematicians’ loose talk about it.
      Study complex numbers in a way that, eventually, you should find yourself being comfortable with the two equivalent ways of modeling physical phenomena: as a set of two coupled real-valued differential equations, and as a single but complex-valued differential equation.
    • Also try to become proficient with the two main expansions: the Taylor, and the Fourier.
    • Also develop a habit of quickly substituting truncated expansions (i.e., either a polynomial, or a sum complex exponentials having just a few initial harmonics, not an entire infinity of them) into any “arbitrary” function as an ansatz, and see how the proposed theory pans out with these. The goal is to become comfortable, at the same time, with a habit of tracing conceptual pathways to the meaning of maths as well as with the computational techniques of FDM, FEM, and FFT.
    • The finite differences approximation: Also, learn the art of quickly substituting the finite differences (\Delta‘s) in place of the differential quantities (d or \partial) in a differential equation, and seeing how it pans out. The idea here is not just the computational modeling. The point is: Every differential equation has been derived in reference to an elemental volume which was then taken to a vanishingly small size. The variation of quantities of interest across such (infinitesimally small) volume are always represented using the Taylor series expansion.
      (That’s correct! It is true that the derivations using the variational approach don’t refer to the Taylor expansion. But they also don’t use infinitesimal volumes; they refer to finite or infinite domains. It is the variation in functions which is taken to the vanishingly small limit in their case. In any case, if your derivation has an infinitesimall small element, bingo, you are going to use the Taylor series.)
      Now, coming back to why you must learn develop the habit of having a finite differences approximation in place of a differential equation. The thing is this: By doing so, you are unpacking the derivation; you are traversing the analysis in the reverse direction, you are by the logic of the procedure forced to look for the physical (or at least lower-level, less abstract) referents of a mathematical relation/idea/concept.
      While thus going back and forth between the finite differences and the differentials, also learn the art of tracing how the limiting process proceeds in each such a case. This part is not at all as obvious as you might think. It took me years and years to figure out that there can be infinitesimals within infinitesimals. (In fact, I have blogged about it several years ago here. More recently, I wrote a PDF document about how many numbers are there in the real number system, which discusses the same idea, from a different angle. In any case, if you were not shocked by the fact that there can be an infinity of infinitesimals within any infinitesimal, either think sufficiently long about it—or quit studying foundations of QM.)

II.3. Quantum chemistry level (mostly concerned with only the TISE, not TDSE):

  • Optional: “QM: a conceptual approach” by Hameka. A fairly well-written book. You can pick it up for some serious reading, but also try to finish it as fast as you can, because you are going to relean the same stuff once again through the next book in the sequence. But yes, you can pick it up; it’s only about 200 pages.
  • “Quantum chemistry” by McQuarrie. Never commit the sin of bypassing this excellent book.
    Summarily ignore your friend (who might have advised you Feynman vol. 3 or Susskind’s theoretical minimum or something similar). Instead, follow my advice!
    A suggestion: Once you finish reading through this particular book, take a small (40 page) notebook, and write down (in the long hand) just the titles of the sections of each chapter of this book, followed by a listing of the important concepts / equations / proofs introduced in it. … You see, the section titles of this book themselves are complete sentences that encapsulate very neat nuggets. Here are a couple of examples: “5.6: The harmonic oscillator accounts for the infrared spectrum of a diatomic molecule.” Yes, that’s a section title! Here is another: “6.2: If a Hamiltonian is separable, then its eigenfunctions are products of simpler eigenfunctions.” See why I recommend this book? And this (40 page notebook) way of studying it?
  • “Quantum physics of atoms, molecules, solids, nuclei, and particles” (yes, that’s the title of this single volume!) by Eisberg and Resnick. This Resnick is the same one as that of Resnick and Halliday. Going through the same topics via yet another thick book (almost 850 pages) can get exasperating, at least at times. But guess if you show some patience here, it should simplify things later. …. Confession: I was too busy with teaching and learning engineering topics like FEM, CFD, and also with many other things in between. So, I could not find the time to read this book the way I would have liked to. But from whatever I did read (and I did go over a fairly good portion of it), I can tell you that not finishing this book was a mistake on my part. Don’t repeat my mistake. Further, I do keep going back to it, and may be as a result, I would one day have finished it! One more point. This book is more than quantum chemistry; it does discuss the time-dependent parts too. The only reason I include it in this sub-section (chemistry) rather than the next (physics) is because the emphasis here is much more on TISE than TDSE.

II.4. Quantum physics level (includes TDSE):

  • “Quantum physics” by Alastair I. M. Rae. Hands down, the best book in its class. To my mind, it easily beats all of the following: Griffiths, Gasiorowicz, Feynman, Susskind, … .
    Oh, BTW, this is the only book I have ever come across which does not put scare-quotes around the word “derivation,” while describing the original development of the Schrodinger equation. In fact, this text goes one step ahead and explicitly notes the right idea, viz., that Schrodinger’s development is a derivation, but it is an inductive derivation, not deductive. (… Oh God, these modern American professors of physics!)
    But even leaving this one (arguably “small”) detail aside, the book has excellence written all over it. Far better than the competition.
    Another attraction: The author touches upon all the standard topics within just about 225 pages. (He also has further 3 chapters, one each on relativity and QM, quantum information, and conceptual problems with QM. However, I have mostly ignored these.) When a book is of manageable size, it by itself is an overload reducer. (This post is not a portion from a text-book!)
    The only “drawback” of this book is that, like many British authors, Rae has a tendency to seamlessly bunch together a lot of different points into a single, bigger, paragraph. He does not isolate the points sufficiently well. So, you have to write a lot of margin notes identifying those distinct, sub-paragraph level, points. (But one advantage here is that this procedure is very effective in keeping you glued to the book!)
  • “Quantum physics” by Griffiths. Oh yes, Griffiths is on my list too. It’s just that I find it far better to go through Rae first, and only then come to going through Griffiths.
  • … Also, avoid the temptation to read both these books side-by-side. You will soon find that you can’t do that. And so, driven by what other people say, you will soon end up ditching Rae—which would be a grave mistake. Since you can keep going through only one of them, you have to jettison the other. Here, I would advise you to first complete Rae. It’s indispensable. Griffiths is good too. But it is not indispensable. And as always, if you find the time and the inclination, you can always come back to Griffiths.

II.5. Side reading:

Starting sometime after finishing the initial UG quantum chemistry level books, but preferably after the quantum physics books, use the following two:

  • “Foundations of quantum mechanics” by Travis Norsen. Very, very good. See my “review” here [^]
  • “Foundations of quantum mechanics: from photons to quantum computers” by Reinhold Blumel.
    Just because people don’t rave a lot about this book doesn’t mean that it is average. This book is peculiar. It does look very average if you flip through all its pages within, say, 2–3 minutes. But it turns out to be an extraordinarily well written book once you begin to actually read through its contents. The coverage here is concise, accurate, fairly comprehensive, and, as a distinctive feature, it also is fairly up-to-date.
    Unlike the other text-books, Blumel gives you a good background in the specifics of the modern topics as well. So, once you complete this book, you should find it easy (to very easy) to understand today’s pop-sci articles, say those on quantum computers. To my knowledge, this is the only text-book which does this job (of introducing you to the topics that are relevant to today’s research), and it does this job exceedingly well.
  • Use Blumel to understand the specifics, and use Norsen to understand their conceptual and the philosophical underpinnings.

II.Appendix: Miscellaneous—no levels specified; figure out as you go along:

  • “Schrodinger’s cat” by John Gribbin. Unquestionably, the best pop-sci book on QM. Lights your fire.
  • “Quantum” by Manjit Kumar. Helps keep the fire going.
  • Kreyszig or equivalent. You need to master the basic ideas of the Fourier theory, and of solutions of PDEs via the separation ansatz.
  • However, for many other topics like spherical harmonics or calculus of variations, you have to go hunting for explanations in some additional books. I “learnt” the spherical harmonics mostly through some online notes (esp. those by Michael Fowler of Univ. of Virginia) and QM textbooks, but I guess that a neat exposition of the topic, couched in contexts other than QM, would have been helpful. May be there is some ancient acoustics book that is really helpful. Anyway, I didn’t pursue this topic to any great depth (in fact I more or less skipped over it) because as it so happens, analytical methods fall short for anything more complex than the hydrogenic atoms.
  • As to the variational calculus, avoid all the physics and maths books like a plague! Instead, learn the topic through the FEM books. Introductory FEM books have become vastly (i.e. categorically) better over the course of my generation. Today’s FEM text-books do provide a clear evidence that the authors themselves know what they are talking about! Among these books, just for learning the variational calculus aspects, I would advise going through Seshu or Fish and Belytschko first, and then through the relevant chapter from Reddy‘s book on FEM. In any case, avoid Bathe, Zienkiewicz, etc.; they are too heavily engineering-oriented, and often, in general, un-necessarily heavy-duty (though not as heavy-duty as Lancosz). Not very suitable for learning the basics of CoV as is required in the UG QM. A good supplementary book covering CoV is noted next.
  • “From calculus to chaos: an introduction to dynamics” by David Acheson. A gem of a book. Small (just about 260 pages, including program listings—and just about 190 pages if you ignore them.) Excellent, even if, somehow, it does not appear on people’s lists. But if you ask me, this book is a must read for any one who has anything to do with physics or engineering. Useful chapters exist also on variational calculus and chaos. Comes with easy to understand QBasic programs (and their updated versions, ready to run on today’s computers, are available via the author’s Web site). Wish it also had chapters, say one each, on the mechanics of materials, and on fracture mechanics.
  • Linear algebra. Here, keep your focus on understanding just the two concepts: (i) vector spaces, and (ii) eigen-vectors and -values. Don’t worry about other topics (like LU decomposition or the power method). If you understand these two topics right, the rest will follow “automatically,” more or less. To learn these two topics, however, don’t refer to text-books (not even those by Gilbert Strang or so). Instead, google on the online tutorials on computer games programming. This way, you will come to develop a far better (even robust) understanding of these concepts. … Yes, that’s right. One or two games programmers, I very definitely remember, actually did a much superior job of explaining these ideas (with all their complexity) than what any textbook by any university professor does. (iii) Oh yes, BTW, there is yet another concept which you should learn: “tensor product”. For this topic, I recommend Prof. Zhigang Suo‘s notes on linear algebra, available off iMechanica. These notes are a work in progress, but they are already excellent even in their present form.
  • Probability. Contrary to a wide-spread impression (and to what one group of QM interpreters say), you actually don’t need much of statistics or probability in order to get the essence of QM right. Whatever you need has already been taught to you in your UG engineering/physics courses.Personally, though I haven’t yet gone through them, the two books on my radar (more from the data science angle) are: “Elementary probability” by Stirzaker, and “All of statistics” by Wasserman. But, frankly speaking, as far as QM itself is concerned, your intuitive understanding of probability as developed through your routine UG courses should be enough, IMHO.
  • As to AJP type of articles, go through Dan Styer‘s paper on the nine formulations (doi:10.1119/1.1445404). But treat his paper on the common misconceptions (10.1119/1.18288) with a bit of caution; some of the ideas he lists as “misconceptions” are not necessarily so.
  • arXiv tutorials/articles: Sometime after finishing quantum chemistry and before beginning quantum physics, go through the tutorial on QM by Bram Gaasbeek [^]. Neat, small, and really helpful for self-studies of QM. (It was written when the author was still a student himself.) Also, see the article on the postulates by Dorabantu [^]. Definitely helpful. Finally, let me pick up just one more arXiv article: “Entanglement isn’t just for spin” by Dan Schroeder [^]. Comes with neat visualizations, and helps demystify entanglement.
  • Computational physics: Several good resources are available. One easy to recommend text-book is the one by Landau, Perez and Bordeianu. Among the online resources, the best collection I found was the one by Ian Cooper (of Univ. of Sydney) [^]. He has only MatLab scripts, not Python, but they all are very well documented (in an exemplary manner) via accompanying PDF files. It should be easy to port these programs to the Python eco-system.

Yes, we (finally) are near the end of this post, so let me add the mandatory catch-all clauses: This list is by no means comprehensive! This list supersedes any other list I may have put out in the past. This list may undergo changes in future.


OK. A couple of last minute addenda: For contrast, see the article “What is the best textbook for self-studying quantum mechanics?” which has appeared, of all places, on the Forbes!  [^]. (Looks like the QC-related hype has found its way into the business circles as well!) Also see the list at BookScrolling.com: “The best books to learn about quantum physics” [^].

OK. Now, I am really done.

A song I like:
(Marathi) “kiteedaa navyaane tulaa aaThavaave”
Music: Mandar Apte
Singer: Mandar Apte. Also, a separate female version by Arya Ambekar
Lyrics: Devayani Karve-Kothari

[Arya Ambekar’s version is great too, but somehow, I like Mandar Apte’s version better. Of course, I do often listen to both the versions. Excellent.]

[Almost 5000 More than 5,500 words! Give me a longer break for this time around, a much longer one, in fact… In the meanwhile, take care and bye until then…]


See, how hard I am trying to become an Approved (Full) Professor of Mechanical Engineering in SPPU?—3

I was looking for a certain book on heat transfer which I had (as usual) misplaced somewhere, and while searching for that book at home, I accidentally ran into another book I had—the one on Classical Mechanics by Rana and Joag [^].

After dusting this book a bit, I spent some time in one typical way, viz. by going over some fond memories associated with a suddenly re-found book…. The memories of how enthusiastic I once was when I had bought that book; how I had decided to finish that book right within weeks of buying it several years ago; the number of times I might have picked it up, and soon later on, kept it back aside somewhere, etc.  …

Yes, that’s right. I have not yet managed to finish this book. Why, I have not even managed to begin reading this book the way it should be read—with a paper and pencil at hand to work through the equations and the problems. That was the reason why, I now felt a bit guilty. … It just so happened that it was just the other day (or so) when I was happily mentioning the Poisson brackets on Prof. Scott Aaronson’s blog, at this thread [^]. … To remove (at least some part of) my sense of guilt, I then decided to browse at least through this part (viz., Poisson’s brackets) in this book. … Then, reading a little through this chapter, I decided to browse through the preceding chapters from the Lagrangian mechanics on which it depends, and then, in general, also on the calculus of variations.

It was at this point that I suddenly happened to remember the reason why I had never been able to finish (even the portions relevant to engineering from) this book.

The thing was, the explanation of the \delta—the delta of the variational calculus.

The explanation of what the \delta basically means, I had found right back then (many, many years ago), was not satisfactorily given in this book. The book did talk of all those things like the holonomic constraints vs. the nonholonomic constraints, the functionals, integration by parts, etc. etc. etc. But without ever really telling me, in a forth-right and explicit manner, what the hell this \delta was basically supposed to mean! How this \delta y was different from the finite changes (\Delta y) and the infinitesimal changes (\text{d}y) of the usual calculus, for instance. In terms of its physical meaning, that is. (Hell, this book was supposed to be on physics, wasn’t it?)

Here, I of course fully realize that describing Rana and Joag’s book as “unsatisfactory” is making a rather bold statement, a very courageous one, in fact. This book is extraordinarily well-written. And yet, there I was, many, many years ago, trying to understand the delta, and not getting anywhere, not even with this book in my hand. (OK, a confession. The current copy which I have is not all that old. My old copy is gone by now (i.e., permanently misplaced or so), and so, the current copy is the one which I had bought once again, in 2009. As to my old copy, I think, I had bought it sometime in the mid-1990s.)

It was many years later, guess some time while teaching FEM to the undergraduates in Mumbai, that the concept had finally become clear enough to me. Most especially, while I was going through P. Seshu’s and J. N. Reddy’s books. [Reflected Glory Alert! Professor P. Seshu was my class-mate for a few courses at IIT Madras!] However, even then, even at that time, I remember, I still had this odd feeling that the physical meaning was still not clear to me—not as as clear as it should be. The matter eventually became “fully” clear to me only later on, while musing about the differences between the perspective of Thermodynamics on the one hand and that of Heat Transfer on the other. That was some time last year, while teaching Thermodynamics to the PG students here in Pune.

Thermodynamics deals with systems at equilibria, primarily. Yes, its methods can be extended to handle also the non-equilibrium situations. However, even then, the basis of the approach summarily lies only in the equilibrium states. Heat Transfer, on the other hand, necessarily deals with the non-equilibrium situations. Remove the temperature gradient, and there is no more heat left to speak of. There does remain the thermal energy (as a form of the internal energy), but not heat. (Remember, heat is the thermal energy in transit that appears on a system boundary.) Heat transfer necessarily requires an absence of thermal equilibrium. … Anyway, it was while teaching thermodynamics last year, and only incidentally pondering about its differences from heat transfer, that the idea of the variations (of Cov) had finally become (conceptually) clear to me. (No, CoV does not necessarily deal only with the equilibrium states; it’s just that it was while thinking about the equilibrium vs. the transient that the matter about CoV had suddenly “clicked” to me.)

In this post, let me now note down something on the concept of the variation, i.e., towards understanding the physical meaning of the symbol \delta.

Please note, I have made an inline update on 26th December 2016. It makes the presentation of the calculus of variations a bit less dumbed down. The updated portion is clearly marked as such, in the text.

The Problem Description:

The concept of variations is abstract. We would be better off considering a simple, concrete, physical situation first, and only then try to understand the meaning of this abstract concept.

Accordingly, consider a certain idealized system. See its schematic diagram below:






There is a long, rigid cylinder made from some transparent material like glass. The left hand-side end of the cylinder is hermetically sealed with a rigid seal. At the other end of the cylinder, there is a friction-less piston which can be driven by some external means.

Further, there also are a couple of thin, circular, piston-like disks (D_1 and D_2) placed inside the cylinder, at some x_1 and x_2 positions along its length. These disks thus divide the cylindrical cavity into three distinct compartments. The disks are assumed to be impermeable, and fitting snugly, they in general permit no movement of gas across their plane. However, they also are assumed to be able to move without any friction.

Initially, all the three compartments are filled with a compressible fluid to the same pressure in each compartment, say 1 atm. Since all the three compartments are at the same pressure, the disks stay stationary.

Then, suppose that the piston on the extreme right end is moved, say from position P_1 to P_2. The final position P_2 may be to the left or to the right of the initial position P_1; it doesn’t matter. For the current description, however, let’s suppose that the position P_2 is to the left of P_1. The effect of the piston movement thus is to increase the pressure inside the system.

The problem is to determine the nature of the resulting displacements that the two disks undergo as measured from their respective initial positions.

There are essentially two entirely different paradigms for conducting an analysis of this problem.

The “Vector Mechanics” Paradigm:

The first paradigm is based on an approach that was put to use so successfully by Newton. Usually, it is called the paradigm of vector analysis.

In this paradigm, we focus on the fact that the forced displacement of the piston with time, x(t), may be described using some function of time that is defined over the interval lying between two instants t_i and t_f.

For example, suppose the function is:
x(t) = x_0 + v t,
where v is a constant. In other words, the motion of the piston is steady, with a constant velocity, between the initial and final instants. Since the velocity is constant, there is no acceleration over the open interval (t_i, t_f).

However, notice that before the instant t_i, the piston velocity was zero. Then, the velocity suddenly became a finite (constant) value. Therefore, if you extend the interval to include the end-instants as well, i.e., if you consider the semi-closed interval [t_i, t_f), then there is an acceleration at the instant t_i. Similarly, since the piston comes to a position of rest at t = t_f, there also is another acceleration, equal in magnitude and opposite in direction, which appears at the instant t_f.

The existence of these two instantaneous accelerations implies that jerks or pressure waves are sent through the system. We may model them as vector quantities, as impulses. [Side Exercise: Work out what happens if we consider only the open interval (t_i, t_f).]

We can now apply Newton’s 3 laws, based on the idea that shock-waves must have begun at the piston at the instant t = t_i. They must have got transmitted through the gas kept under pressure, and they must have affected the disk D_1 lying closest to the piston, thereby setting this disk into motion. This motion must have passed through the gas in the middle compartment of the system as another pulse in the pressure (generated at the disk D_1), thereby setting also the disk D_2 in a state of motion a little while later. Finally, the pulse must have got bounced off the seal on the left hand side, and in turn, come back to affect the motion of the disk D_2, and then of the disk D_1. Continuing their travels to and fro, the pulses, and hence the disks, would thus be put in a back and forth motion.

After a while, these transients would move forth and back, superpose, and some of their constituent frequencies would get cancelled out, leaving only those frequencies operative such that the three compartments are put under some kind of stationary states.

In case the gas is not ideal, there would be damping anyway, and after a sufficiently long while, the disks would move through such small displacements that we could easily ignore the ever-decreasing displacements in a limiting argument.

Thus, assume that, after an elapse of a sufficiently long time, the disks become stationary. Of course, their new positions are not the same as their original positions.

The problem thus can be modeled as basically a transient one. The state of the new equilibrium state is thus primarily seen as an effect or an end-result of a couple of transient processes which occur in the forward and backward directions. The equilibrium is seen as not a primarily existing state, but as a result of two equal and opposite transient causes.

Notice that throughout this process, Newton’s laws can be applied directly. The nature of the analysis is such that the quantities in question—viz. the displacements of the disks—always are real, i.e., they correspond to what actually is supposed to exist in the reality out there.

The (values of) displacements are real in the sense that the mathematical analysis procedure itself involves only those (values of) displacements which can actually occur in reality. The analysis does not concern itself with some other displacements that might have been possible but don’t actually occur. The analysis begins with the forced displacement condition, translates it into pressure waves, which in turn are used in order to derive the predicted displacements in the gas in the system, at each instant. Thus, at any arbitrary instant of time t > t_i (in fact, the analysis here runs for times t \gg t_f), the analysis remains concerned only with those displacements that are actually taking place at that instant.

The Method of Calculus of Variations:

The second paradigm follows the energetics program. This program was initiated by Newton himself as well as by Leibnitz. However, it was pursued vigorously not by Newton but rather by Leibnitz, and then by a series of gifted mathematicians-physicists: the Bernoulli brothers, Euler, Lagrange, Hamilton, and others. This paradigm is essentially based on the calculus of variations. The idea here is something like the following.

We do not care for a local description at all. Thus, we do not analyze the situation in terms of the local pressure pulses, their momenta/forces, etc. All that we focus on are just two sets of quantities: the initial positions of the disks, and their final positions.

For instance, focus on the disk D_1. It initially is at the position x_{1_i}. It is found, after a long elapse of time (i.e., at the next equilibrium state), to have moved to x_{1_f}. The question is: how to relate this change in x_1 on the one hand, to the displacement that the piston itself undergoes from P_{x_i} to P_{x_f}.

To analyze this question, the energetics program (i.e., the calculus of variations) adopts a seemingly strange methodology.

It begins by saying that there is nothing unique to the specific value of the position x_{1_f} as assumed by the disk D_1. The disk could have come to a halt at any other (nearby) position, e.g., at some other point x_{1_1}, or x_{1_2}, or x_{1_3}, … etc. In fact, since there are an infinity of points lying in a finite segment of line, there could have been an infinity of positions where the disk could have come to a rest, when the new equilibrium was reached.

Of course, in reality, the disk D_1 comes to a halt at none of these other positions; it comes to a halt only at x_{1_f}.

Yet, the theory says, we need to be “all-inclusive,” in a way. We need not, just for the aforementioned reason, deny a place in our analysis to these other positions. The analysis must include all such possible positions—even if they be purely hypothetical, imaginary, or unreal. What we do in the analysis, this paradigm says, is to initially include these merely hypothetical, unrealistic positions too on exactly the same footing as that enjoyed by that one position which is realistic, which is given by x_{1_f}.

Thus, we take a set of all possible positions for each disk. Then, for each such a position, we calculate the “impact” it would make on the energy of the system taken as a whole.

The energy of the system can be additively decomposed into the energies carried by each of its sub-parts. Thus, focusing on disk D_1, for each one of its possible (hypothetical) final position, we should calculate the energies carried by both its adjacent compartments. Since a change in D_1‘s position does not affect the compartment 3, we need not include it. However, for the disk D_1, we do need to include the energies carried by both the compartments 1 and 2. Similarly, for each of the possible positions occupied by the disk D_2, it should include the energies of the compartments 2 and 3, but not of 1.

At this point, to bring simplicity (and thereby better) clarity to this entire procedure, let us further assume that the possible positions of each disk forms a finite set. For instance, each disk can occupy only one of the positions that is some -5, -4, -3, -2, -1, 0, +1, +2, +3, +4 or +5 distance-units away from its initial position. Thus, a disk is not allowed to come to a rest at, say, 2.3 units; it must do so either at 2 or at 3 units. (We will thus perform the initial analysis in terms of only the integer positions, and only later on extend it to any real-valued positions.) (If you are a mechanical engineering student, suggest a suitable mechanism that can ensure only integer relative displacements.)

The change in energy E of a compartment is given by
\Delta E = P A \Delta x,
where P is the pressure, A is the cross-sectional area of the cylinder, and \Delta x is the change in the length of the compartment.

Now, observe that the energy of the middle compartment depends on the relative distance between the two disks lying on its sides. Yet, for the same reason, the energy of the middle compartment does depend on both these positions. Hence, we must take a Cartesian product of the relative displacements undergone by both the disks, and only then calculate the system energy for each such a permutation (i.e. the ordered pair) of their positions. Let us go over the details of the Cartesian product.

The Cartesian product of the two positions may be stated as a row-by-row listing of ordered pairs of the relative positions of D_1 and D_2, e.g., as follows: the ordered pair (-5, +2) means that the disk D_1 is 5 units to the left of its initial position, and the disk D_2 is +2 units to the right of its initial position. Since each of the two positions forming an ordered pair can range over any of the above-mentioned 11 number of different values, there are, in all, 11 \times 11 = 121 number of such possible ordered pairs in the Cartesian product.

For each one of these 121 different pairs, we use the above-given formula to determine what the energy of each compartment is like. Then, we add the three energies (of the three compartments) together to get the value of the energy of the system as a whole.

In short, we get a set of 121 possible values for the energy of the system.

You must have noticed that we have admitted every possible permutation into analysis—all the 121 number of them.

Of course, out of all these 121 number of permutations of positions, it should turn out that 120 number of them have to be discarded because they would be merely hypothetical, i.e. unreal. That, in turn, is because, the relative positions of the disks contained in one and only one ordered pair would actually correspond to the final, equilibrium position. After all, if you conduct this experiment in reality, you would always get a very definite pair of the disk-positions, and it this same pair of relative positions that would be observed every time you conducted the experiment (for the same piston displacement). Real experiments are reproducible, and give rise to the same, unique result. (Even if the system were to be probabilistic, it would have to give rise to an exactly identical probability distribution function.) It can’t be this result today and that result tomorrow, or this result in this lab and that result in some other lab. That simply isn’t science.

Thus, out of all those 121 different ordered-pairs, one and only one ordered-pair would actually correspond to reality; the rest all would be merely hypothetical.

The question now is, which particular pair corresponds to reality, and which ones are unreal. How to tell the real from the unreal. That is the question.

Here, the variational principle says that the pair of relative positions that actually occurs in reality carries a certain definite, distinguishing attribute.

The system-energy calculated for this pair (of relative displacements) happens to carry the lowest magnitude from among all possible 121 number of pairs. In other words, any hypothetical or unreal pair has a higher amount of system energy associated with it. (If two pairs give rise to the same lowest value, both would be equally likely to occur. However, that is not what provably happens in the current example, so let us leave this kind of a “degeneracy” aside for the purposes of this post.)

(The update on 26 December 2016 begins here:)

Actually, the description  given in the immediately preceding paragraph was a bit too dumbed down. The variational principle is more subtle than that. Explaining it makes this post even longer, but let me give it a shot anyway, at least today.

To follow the actual idea of the variational principle (in a not dumbed-down manner), the procedure you have to follow is this.

First, make a table of all possible relative-position pairs, and their associated energies. The table has the following columns: a relative-position pair, the associated energy E as calculated above, and one more column which for the time being would be empty. The table may look something like what the following (partial) listing shows:

(0,0) -> say, 115 Joules
(-1,0) -> say, 101 Joules
(-2,0) -> say, 110 Joules

(2,2) -> say, 102 Joules
(2,3) -> say, 100 Joules
(2,4) -> say, 101 Joules
(2,5) -> say, 120 Joules

(5,0) -> say, 135 Joules

(5,5) -> say 117 Joules.

Having created this table (of 121 rows), you then pick each row one by and one, and for the picked up n-th row, you ask a question: What all other row(s) from this table have their relative distance pairs such that these pairs lie closest to the relative distance pair of this given row. Let me illustrate this question with a concrete example. Consider the row which has the relative-distance pair given as (2,3). Then, the relative distance pairs closest to this one would be obtained by adding or subtracting a distance of 1 to each in the pair. Thus, the relative distance pairs closest to this one would be: (3,3), (1,3), (2,4), and (2,2). So, you have to pick up those rows which have these four entries in the relative-distance pairs column. Each of these four pairs represents a variation \delta on the chosen state, viz. the state (2,3).

In symbolic terms, suppose for the n-th row being considered, the rows closest to it in terms of the differences in their relative distance pairs, are the a-th, b-th, c-th and d-th rows. (Notice that the rows which are closest to a given row in this sense, would not necessarily be found listed just above or below that given row, because the scheme followed while creating the list or the vector that is the table would not necessarily honor the closest-lying criterion (which necessarily involves two numbers)—not at least for all rows in the table.

OK. Then, in the next step, you find the differences in the energies of the n-th row from each of these closest rows, viz., the a-th, b-th, c-th and c-th rows. That is to say, you find the absolute magnitudes of the energy differences. Let us denote these magnitudes as: \delta E_{na} = |E_n - E_a|\delta E_{nb} = |E_n - E_b|\delta E_{nc} = |E_n - E_c| and \delta E_{nd} = |E_n - E_d|.  Suppose the minimum among these values is \delta E_{nc}. So, against the n-th row, in the last column of the table, you write the value \delta E_{nc}.

Having done this exercise separately for each row in the table, you then ask: Which row has the smallest entry in the last column (the one for \delta E), and you pick that up. That is the distinguished (or the physically occurring) state.

In other words, the variational principle asks you to select not the row with the lowest absolute value of energy, but that row which shows the smallest difference of energy from one of its closest neighbours—and these closest neighbours are to be selected according to the differences in each number appearing in the relative-distance pair, and not according to the vertical place of rows in the tabular listing. (It so turns out that in this example, the row thus selected following both criteria—lowest energy as well as lowest variation in energy—are identical, though it would not necessarily always be the case. In short, we can’t always get away with the first, too dumbed down, version.)

Thus, the variational principle is about that change in the relative positions for which the corresponding change in the energy vanishes (or has the minimum possible absolute magnitude, in case the positions form a discretely varying, finite set).

(The update on 26th December 2016 gets over here.)

And, it turns out that this approach, too, is indeed able to perfectly predict the final disk-positions—precisely as they actually are observed in reality.

If you allow a continuum of positions (instead of the discrete set of only the 11 number of different final positions for one disk, or 121 number of ordered pairs), then instead of taking a Cartesian product of positions, what you have to do is take into account a tensor product of the position functions. The maths involved is a little more advanced, but the underlying algebraic structure—and the predictive principle which is fundamentally involved in the procedure—remains essentially the same. This principle—the variational principle—says:

Among all possible variations in the system configurations, that system configuration corresponds to reality which has the least variation in energy associated with it.

(This is a very rough statement, but it will do for this post and for a general audience. In particular, we don’t look into the issues of what constitute the kinematically admissible constraints, why the configurations must satisfy the field boundary conditions, the idea of the stationarity vs. of a minimum or a maximum, i.e., the issue of convexity-vs.-concavity, etc. The purpose of this post—and our example here—are both simple enough that we need not get into the whole she-bang of the variational theory as such.)

Notice that in this second paradigm, (i) we did not restrict the analysis to only those quantities that are actually taking place in reality; we also included a host (possibly an infinity) of purely hypothetical combinations of quantities too; (ii) we worked with energy, a scalar quantity, rather than with momentum, a vector quantity; and finally, (iii) in the variational method, we didn’t bother about the local details. We took into account the displacements of the disks, but not any displacement at any other point, say in the gas. We did not look into presence or absence of a pulse at one point in the gas as contrasted from any other point in it. In short, we did not discuss the details local to the system either in space or in time. We did not follow the system evolution, at all—not at least in a detailed, local way. If we were to do that, we would be concerned about what happens in the system at the instants and at spatial points other than the initial and final disk positions. Instead, we looked only at a global property—viz. the energy—whether at the sub-system level of the individual compartments, or at the level of the overall system.

The Two Paradigms Contrasted from Each Other:

If we were to follow Newton’s method, it would be impossible—impossible in principle—to be able to predict the final disk positions unless all their motions over all the intermediate transient dynamics (occurring over each moment of time and at each place of the system) were not be traced. Newton’s (or vectorial) method would require us to follow all the details of the entire evolution of all parts of the system at each point on its evolution path. In the variational approach, the latter is not of any primary concern.

Yet, in following the energetics program, we are able to predict the final disk positions. We are able to do that without worrying about what all happened before the equilibrium gets established. We remain concerned only with certain global quantities (here, system-energy) at each of the hypothetical positions.

The upside of the energetics program, as just noted, is that we don’t have to look into every detail at every stage of the entire transient dynamics.

Its downside is that we are able to talk only of the differences between certain isolated (hypothetical) configurations or states. The formalism is unable to say anything at all about any of the intermediate states—even if these do actually occur in reality. This is a very, very important point to keep in mind.

The Question:

Now, the question with which we began this post. Namely, what does the delta of the variational calculus mean?

Referring to the above discussion, note that the delta of the variational calculus is, here, nothing but a change in the position-pair, and also the corresponding change in the energy.

Thus, in the above example, the difference of the state (2,3) from the other close states such as (3,3), (1,3), (2,4), and (2,2) represents a variation in the system configuration (or state), and for each such a variation in the system configuration (or state), there is a corresponding variation in the energy \delta E_{ni} of the system. That is what the delta refers to, in this example.

Now, with all this discussion and clarification, would it be possible for you to clearly state what the physical meaning of the delta is? To what precisely does the concept refer? How does the variation in energy \delta E differ from both the finite changes (\Delta E) as well as the infinitesimal changes (\text{d}E) of the usual calculus?

Note, the question is conceptual in nature. And, no, not a single one of the very best books on classical mechanics manages to give a very succinct and accurate answer to it. Not even Rana and Joag (or Goldstein, or Feynman, or…)

I will give my answer in my next post, next year. I will also try to apply it to a couple of more interesting (and somewhat more complicated) physical situations—one from engineering sciences, and another from quantum mechanics!

In the meanwhile, think about it—the delta—the concept itself, its (conceptual) meaning. (If you already know the calculus of variations, note that in my above write-up, I have already supplied the answer, in a way. You just have to think a bit about it, that’s all!)

An Important Note: Do bring this post to the notice of the Officially Approved Full Professors of Mechanical Engineering in SPPU, and the SPPU authorities. I would like to know if the former would be able to state the meaning—at least now that I have already given the necessary context in such great detail.

Ditto, to the Officially Approved Full Professors of Mechanical Engineering at COEP, esp. D. W. Pande, and others like them.

After all, this topic—Lagrangian mechanics—is at the core of Mechanical Engineering, even they would agree. In fact, it comes from a subject that is not taught to the metallurgical engineers, viz., the topic of Theory of Machines. But it is taught to the Mechanical Engineers. That’s why, they should be able to crack it, in no time.

(Let me continue to be honest. I do not expect them to be able to crack it. But I do wish to know if they are able at least to give a try that is good enough!)

Even though I am jobless (and also nearly bank balance-less, and also cashless), what the hell! …

…Season’s greetings and best wishes for a happy new year!

A Song I Like:

[With jobless-ness and all, my mood isn’t likely to stay this upbeat, but anyway, while it lasts, listen to this song… And, yes, this song is like, it’s like, slightly more than 60 years old!]

(Hindi) “yeh raat bhigee bhigee”
Music: Shankar-Jaikishan
Singers: Manna De and Lata Mangeshkar
Lyrics: Shailendra


The anti-, an anti-anti-, my negativism, and miscellaneous


A better title could very well have been “What I am up against.” However, that title, I thought, would be misleading. … I really am up against many things which I am going to touch on, in this post. But the point is, these are not the only things that I am up against, and so, that title would therefore be too general.

Part I: The Anti-

First, of course, comes the anti.

I stumbled across W. E. Lamb, Jr.’s excellent paper: “Anti-photon” (1995) Appl. Phys. B, vol. 60, p. 77–84. Here is the abstract:

“It should be apparent from the title of this article that the author does not like the use of the word “photon”, which dates from 1926. In his view, there is no such thing as a photon. Only a comedy of errors and historical accidents led to its popularity among physicists and optical scientists. I admit that the word is short and convenient. Its use is also habit forming. Similarly, one might find it convenient to speak of the “aether” or “vacuum” to stand for empty space, even if no such thing existed. There are very good substitute words for “photon”, (e.g., “radiation” or “light”), and for “photonics” (e.g., “optics” or “quantum optics”). Similar objections are possible to use of the word “phonon”, which dates from 1932. Objects like electrons, neutrinos of finite rest mass, or helium atoms can, under suitable conditions, be considered to be particles, since their theories then have viable non-relativistic and non-quantum limits. This paper outlines the main features of the quantum theory of radiation and indicates how they can be used to treat problems in quantum optics.”

BTW, in case you don’t know, W. E. Lamb, Jr., was an American, who received a Nobel in physics, for his work related to the fine structure of hydrogen [^].

So, that’s the first bit of what I am up against.

Also in case you didn’t notice, the initials are important; this isn’t (Sir) Horace Lamb (who, in case you don’t know, was that late 19th–early 20th century British guy who wrote books on hydrodynamics and acoustics that people like me still occasionally refer to [^]. (Lamb and Love continue to remain in circulation (even if a low circulation) among mechanicians even today. (Love, who? … That’s an exercise left for the reader…)))

Oh, BTW, talking of very good books that now have come in the public domain, and (the preparation required for) QM, and all the anti- and un- things, note that Professor Howard Georgi [^]’s excellent book on waves has by now come in the public domain [^].

(Even if only parenthetically, I have to note: I am anti-diversity, too. … This anti thing simply doesn’t leave me alone, though I will try to minimize its usage. Starting right now. … Georgi was born in California. He also maintains a page about women in physics [^].)

… Ummm, I’d better wrap up this part, and so…

… All in all, you can see that I don’t seem to be taking my opposition very seriously, though I admit I should start doing so some day. But the paper is great. (We were talking about the anti-photon paper, remember?) Here is an excerpt in case I haven’t already succeeded in persuading you to go through it, immediately:

“During my eight years in Berkeley, I had just one conversation with Lewis, in 1937, when he called me into his office to give some advice. It was: “When a theorist does not know what to do next, he is useless. An experimental scientist can always go into his laboratory and “polish up the brass”.”

This is the same Lewis who coined the word “photon.” … Now it convinces you to go through the paper, doesn’t it? (The paper is by Lamb; W. E. Lamb.)

[… On a more serious note, this paper has very good notings regarding the history of the idea of the photon.]

Part II: The Anti- Equals the Anti-Anti-

There is no typo here.

Even as I was recoiling off the glow (I won’t use “radiation” or “light”) of [the physics Nobel laureate] Lamb’s reputation, I began wondering precisely how I would counter his anti-photon argument. I even thought of doing a blog post about it. (After all, recently, Roger Schlafly has been hinting at that same idea, too. [May be TBD: insert links])

However, a better sense prevailed, and I did a Google search. I found a good blog post that gives a good rejoinder to the anti-photon arguments. The post is written in simple enough language that any one could understand. … But should I recommend it to you?… The thing is: It comes from a physicist who is reputed to have attempted teaching quantum physics to dogs. Or, at least, teaching people how to teach quantum physics, to dogs.

But of course, in physics, personalities don’t count, and neither do, you know, sort of like, “insults.” [I am also anti-animal rights, BTW [though all in favor of dogs].] And so, let me lead you to the relevant post.

The quantum physics-loving folks would have guessed the man by now (and every one, the fact that the author must be a man, not a woman). So the only remaining part would be which post by Chad Orzel. Here it is [^]. Once you finish reading it (including the comments on the post), then, also go through these couple of others posts by him touching on the same topic [^] [^] (and their blog comments). And, a great post (at wired.com!) by Rhett Allain [^] on the anti-photon side, to which Orzel makes a reference.

Orzel’s basic argument is that anti-bunching equals anti-anti-photon.

That explains the second part of the title.

But, before wrapping up this part, just a word on the PhD guides on the “polishing brass” side, and Indians. The anti-bunching experiments were done by Leonard Mandel [^], who among other things also guided Rupamanjari Ghosh’s PhD thesis. … Rupa…, who? I will save you the trouble of googling; see here: [^ (I am anti-government in education and science, too)] and here [^ (oh well, this post is getting just too long)].

Part III: My Negativism

Roger Schlafly has just recently written an interestingly long post on quantum entanglement. (Very long, by his standards.) In that post [^], he identifies himself as a logical positivist. This isn’t the first time that he has attributed logical positivism to his intellectual positions. Schlafly’s recent post is written, as usual, with good/great clarity

Now consider the premises, this time three, instead of the usual two: (i) Schlafly identifies himself as a logical positivist, (ii) I don’t agree with some part of his positions, and (iii) logic is logic—it cares for completeness.

Ergo, I must be a logical negativist.

That explains the third part of the title.

Some day I plan to write a post on the triplet and singlet states, and quantum entanglement.

Some still later day, I plan to explain how QM is incomplete, by pointing out how it can be made complete. … That is too big a goal to keep, you say?

Well, I do plan to at least explain in simpler terms the phenomenon of quantum entanglement, but only in reference to the text-book treatments. … That should be doable, what say?

… Don’t hold me responsible etc. on this promise; I am careless etc.; and so,  it might very well be in mid-2016 when I might actually deliver on it. … So, for the time being, make do with my logical negativism.

Part IV: Miscellaneous

M1: The preface to Georgi’s book notes the help he received while writing the book inter alia from (the same) Griffiths (as the one who has written very popular undergraduate text-books on electrodynamics and QM). (Griffiths studied at Harvard where Georgi has been a professor, though chances are they were contemporaries.) (No, this Griffiths isn’t the same as the Griffiths of fracture mechanics [^].) (Yes, this Georgi is the same as the one who has advocated the unparticle mechanics [^]. (But why didn’t he use the anti- prefix here?))

M2: The most succinct (and as far as I can make out, correct) treatment of the meaning of “hidden variables” has been not in the recent Internet writings on Bell’s inequalities but in Griffith’s undergraduate text-book on QM.

Why I mention this bit… That’s because, recently, the MIT professor Scott Aaronson had a field day about hidden variables (notably with Travis Norsen) [^], though since then he seems to have moved on to some other things related to theoretical computational complexity, e.g. this graph isomorphism-related thingie [^].

But, no, if you want to know about the so-called hidden variables well (and don’t have my “approach” or at least my “confidence”), then don’t look up the material on the ‘net or blog posts, esp. those by CS folks or complexity theorists. Instead, hit Griffith’s (text-)book.

M3: However, I am unhappy about Griffith’s treatment of the quantum postulates—he (like QChem and most all UG QM books) has only the usual \Psi and doesn’t include the spinor function right while discussing the state definition. Indeed, he continues implicitly treating the two in a somewhat disjoint manner even afterwards (exactly like all UG text-books do). Separable doesn’t mean disjointed.

I am also unhappy about Griffith’s (and every other QM text-book’s) treatment of the basic ideas of identical particles and their states—the treatments are just not conceptually clarifying enough. … May I assist you rewriting this topic, Professor Griffiths? … Oh well… Before I actually make that offer to him, I will try my hand at the task, at this blog…. Sometime in/after mid-2016. (Hopefully earlier.)

But, yes, if you ask me, it’s only the spin and identical particles that still remain truly nebulous topics for the student, today. With single-particle interference experiments and the ubiquity of simulations, one wouldn’t think that people would have too much difficulty with wave-particle duality or interference etc.

Contrast staring at one or two manually drawn static graphs in a book/paper, and imagining how things would change with time, under different governing equations and different boundary conditions, vs. going through simulations on your smartphone, adjusting FPS, changing boundary conditions with the flick of a button… Students (like me) must be having it exponentially easier to learn QM these days, as compared to those hapless 20th century guys.

The points where today’s students are likely to falter would be a bit more advanced ones, like angular momentum. In fact, today’s students don’t know angular momentum well even in the classical mechanics settings. (Ask yourself: how clear and confident are you about, say, Coriolis forces, say, as covered in Shames, or in Timoshenko and Young?).

So, to wrap up, it has to be identical particles and spin that still remain the really difficult topics. Now, it so happens that it is these concepts that underlie popular expositions of entanglement. Little surprise that people never get the confidence that they would be able to deal with entanglement right.

(Focusing on “just” two states of the spin up- and down-, and therefore treating the phenomenon via an abstract two component vector, and then thinking that starting a discussion with this “simple” vector, is a very bad idea, epistemologically speaking. … Yes, I am anti-Susskind’s “theoretical minimum,” too. And yes, Griffiths is right in choosing the traditional way (of the sequence in which to present the QM spin). It’s just that he needs to explain it in (even) better manner, that’s all….)

M4: The day before yesterday was the first time this year that I happened to finally sense that wonderful winter-time air of Pune’s, while returning in the evening from our college. (Monday was a working day for us; no continuous 9-day patch of a vacation.)

It still doesn’t feel like the Diwali air this year in Pune, but it’s getting close: I spotted some nice fog/mist on the nearby nallah (i.e. a small stream) and a nearby canal, a couple of times. …

This has been a year of (heavy) drought. And anyway, these days, there is virtually no difference between the Diwali days and the rest of the year. … Shopping malls are fully Diwali-like at any time of the year for those who have the money, and most women—whether working or otherwise—these days outsource their (Marathi) “chakalee”-making anyway—even during Diwali. So, there isn’t much of a difference between the Diwali days and the other days. Except for the weather. Weather still continues to change in a distinctly perceptible way sometime around Diwali. … So, that’s about all what Diwali means to me, this year.

And, of course, some memories of the magical Diwalis that I have spent in my childhood… Many of these were spent (at least for the (Marathi) “bhau-beej” day and a couple of days more) at my maternal uncle’s place (a very small town, a sub taluka-level place). … As far as I am concerned, those Diwali’s are still real; they would easily remain that way throughout my life.

PS: Having written the post, I just stepped into the kitchen to make me a cup of tea, and that’s when father told me that home-made (Marathi) “chakalee”s had arrived from our family friends just last evening; I didn’t know about it.

Instantaneously, my song-selection collapsed into an anti-previously measured state. (It happens. Real life is more weird than QM.)


Happy Diwali!

PS (also) to Epilogue:

Excuse me for a couple of weeks now. I will continue studying QM (from text-books), but I will also have to be taking out my notes for an undergraduate course on CFD (computational fluid dynamics, in case you didn’t know) that I should be teaching the next semester—which begins right in mid-December. (In India, we don’t always follow the Christmas–New Year’s–Next Term sequence.) I anyway will also be traveling a bit (just short distances like Mumbai and Nasik or so) over the next couple of weeks. So, I don’t think I will have the time to write a post. (That, in fact, was the reason why I threw in a lot of stuff right in this post.)

… So, there… Take care, and best wishes, once again, for a bright and happy Diwali (and to those of you who start a new year in Diwali, best wishes for a happy and prosperous new year too.)

A Song I Like

(Marathi) “tabakaamadhye ithe tevatee…”  (search on the transcriptionally incorrect “divya divyanchi jyot”)
Singer: Asha Bhosale
Lyrics: Ravindra Bhat
Music: Sudhir Phadake

[PS: I kept on adding material after publication of post, and now it has become some 1.5 times the original one. Sorry about that (though I did all the revisions right within 18 hours of publication), but now I am going stop editing any further. Put up with my grammatical mistakes and awkward constructions, as usual. And, if in doubt, ask me! Bye for now.]


“They don’t even touch a good text-book!”

“They don’t even touch a good text-book!”

This line is a very common refrain that one often hears in faculty rooms or professors’ cabins, in engineering colleges in India.

Speaking in factual terms, there is a lot of truth to it. The assertion itself is overwhelmingly true. The fact that the student has never looked into a good (or “reference” or “foreign authors'”) text is immediately plain and clear to anyone who has ever graded their examination papers, or worked as an examiner on the oral/viva voce examinations.

The undergraduate Indian students these days, esp. those in Pune and Mumbai, and esp. those in the private engineering colleges, always refer to only a locally published text for all their studies.

These texts are published by a few local publishers well known to the students (and their professors). I wouldn’t mind dropping a few names: Nirali, Pragati, TechMax, etc. The books are published at almost throw-away prices (e.g. Rs. 200–300). (There also exists a highly organized market for the second-hand books. No name written, no pencil marks? Some 75% of the cost returned. Etc. There is a bold print, too—provided, the syllabus hasn’t changed in the meanwhile. In that case, there is no resale value whatsoever!)

The authors of these texts themselves are professors in these same private engineering colleges. They know the system in and out. No, I am not even hinting at any deliberate fraud or malpractice here. Quite on the contrary.

The professors who write these local text-books often are enthusiastic teachers themselves. You would have to be very enthusiastic, because the royalties they “command” could be as low as a one-time payment of Rs. 50,000/- or so. The payment is always only a one-time payment (meaning, there are no recurring royalties even if a text book becomes a “hit”), and it never exceeds Rs. 1.5 lakhs lump-sum or so. (My figures are about 5 year old.) Even if each line is copied verbatim from other books, the sheer act of having to write down (and then proof-read) some 200 to 350 pages requires for the author to invest, I have been told, between 2 to 4 months, working overtime, neglecting family and all. The monthly salary of these professors these days can easily approach or exceed Rs. 1 lakh. So, clearly, money is not the prime motivation here. It has to be something else: Enthusiasm, love of teaching, or even just the respect or reputation that an author hopes to derive in the sub-community of these local engineering colleges!

These professors—the authors—also often are well experienced (15–40 years of teaching experience is common), and they know enough to know what kind of examination questions are likely to come up on the university examinations. (They themselves have gone through the same universities.) They write these books targeting only task: writing the marks-scoring answers on those university examinations. Thus, these “text” books are more or less nothing but a student aid (or what earlier used to be called the “guide” books).

It in fact has evolved into a separate genre by itself. Contrary to an impression wide-spread among professors of private engineering colleges in India, there in fact are somewhat similar books also used heavily by the students in the USA. Thus, these local Indian books are nothing but an improvised version of the Schaums’ series in science and engineering (or the Sparks Notes in the humanities, in the US schools).

But there is a further feature here. There is a total customization thrown in here. These local books are now-a-days written (or at least adapted) to exactly match the detailed syllabus of each university separately. So, there are different books, by the same author and for the same subject, but one for Mumbai University, and the other for Pune University, etc. Students never mix up the universities.

The syllabus for each university is followed literally, down to dividing the text into chapters as per the headings of the modules mentioned in the syllabus (usually six per course), and dividing each chapter into sections, with the headings and order of these sections strictly following the order and the letter of the syllabus. The text in each section is followed by a compilation of the past university examination questions (of that same university) pertaining to that particular section alone. Most of these past examination questions are solved in the text—that’s the bulk of the book. When the opening page of a chapter lists the sections in it, the list also carries, in the parentheses, whether this section is “theory” or “numericals”.

Overall, the idea is, even just looking at the “text” book, a student can easily anticipate whether a question is likely to be asked on a given section or not, and if yes, of what kind. The students also work out many logics: “Every semester, they have asked a question on this section. So we have to mug it up well.” Or, playing the “contra”: “Last three semesters, not a single question here? It’s going to come this time round.” Etc. (Yes, I followed this practice in my lectures, too—I did want my students to score well on the final university examinations, after all!)

The customization, for each revision of the syllabus of each university, is done down to that level of detail. So, for the first year course on electrical engineering, you have one text-book of title, say, Electrical Engg. (FE), Pune University, 2012 course, and another text book, now of the title, say, Basic Electrical Technology (FE), Mumbai University, 2011 course. Etc.

That’s what I mean, when I use the phrase the “local” text-books.

I certainly don’t mean the SI Units editions of American texts, or the Indian Standards-adapted editions of reputed texts (such as, say, Shigley’s on design or Thomson and Dahleh’s on vibrations). I don’t mean the inexpensive Indian editions of foreign texts (such as those by Pearson, Wiley, ELBS, etc.) I also don’t mean the text-books written by the well-known Indian authors working right in India (such as those by IIT professors, and published by, say, Universities Press, Narosa, or PHI). I don’t even mean the more general text-books written for Indian universities and/or the AMIE examinations (such as those by S. Chand, Khanna, CBS, etc.). When I say “local” text-books, I specifically mean the books of the kind mentioned above.

Undergraduate students in Pune and Mumbai these days refer only to these local books.

They (really) don’t even bother to touch a good reference text, even if it’s available on the college library shelf.

In contrast, in our times, the problem was, we simply didn’t have the “foreign authors'” texts available to us—not always even in the COEP library. In those days, sometimes, such books happened to be too expensive, even for COEP’s library. And, even back then, Shahani’s text-books anyway were available. But at least, they didn’t cater to only the Pune university (they would list problems from universities as far flung as Madras, Gorakhpur, Agra, Allahabad, etc.) And, in fact, these books were generally looked down upon. Even by the students themselves.

The contrast to today’s situation is too glaring. Naturally, professors sometimes do end up saying the title line with a tone of exasperation.

Yes, I used to sometimes say that line myself, of course with sarcasm, when I taught in the late ’80s in the Pune of those days. (The situation back then was not so acute.) Almost as if by habit, I also repeated the line when I more recently taught a course at COEP (2009, FEM). However, observing students, somehow, my line had somehow begun to lose that cutting edge it once had. First, at COEP, I had the freedom to design this course (on FEM), and they did buy at least Logan and/or Cook. (Even if I was distributing my PDF notes.) And, there was something else to it, too. I somehow got a vague feel that it somehow wouldn’t be fully right to blame students (I mean COEP students in general). However, my COEP stint was only for one semester, only for one course, and only as a visiting faculty. So, the vague feel simply remained what it was—just a vague feel.

Then, recently in 2014, when I began teaching at a private engineering college in Mumbai, I once again heard this line from the other professors. And, I used it myself too. With the usual sarcasm. I did that perhaps for the most part of my first semester there.

However, some way down the line, I once again got that vague feel that, may be, something was “wrong” somewhere, even here, in Mumbai: these kids really were trying to be sincere, and yet, for some reason unknown to me, they still wouldn’t at all refer to good texts.

This is an aside, but I can tell you that it’s very easy to read the faces of the insincere people, esp. when they are young. There are some insincere students too. But, at least going by my own experience, they are in a minority. (It is a headache-some minority. Yet, by numerical magnitude alone, it certainly is in a small minority.) I am not saying this to be politically correct, or to win points from students. What I said is the factual case. In fact, my experience is that when it comes to in-sincerity, parents easily outperform their children. May be because, the specific parents that we mostly end up seeing in college are those whose kids have some problem—low attendance, fee payments, other issues, etc. The parents with whom we get to interact really well, thus, happens to be a self-selected sub-group. They aren’t necessarily representative of all parents… Yet, I am also sure that that’s not the real reason why I think parents can easily be more insincere. I think the real reason is that, at their age, the kids are actually unable to fake too much. It’s far easier for them to be sincere than to be a fake and still get away with it. They just can’t manage it, regardless of their desire. And, looking at it in a better light, I here remember what Ayn Rand had once said in a somewhat similar context, “one doesn’t start out in life by spitting on one’s own face—it’s not in the essential nature of life” or something like that. (Off-hand, I think, it was in the preface to the 25th anniversity edition of The Fountainhead.) So, the kids, by and large, are sincere. … By the time they themselves become parents—well, let’s leave that story right here. (We need them to make all those fee payments, anyway…)

So, coming back to the main thread, I would anyway generally chat with the students, and so, I started asking, esp. some of the more talkative students, the reason why they might not be referring to good texts. After all, in my lectures, I would try to provide very specific references: specific section numbers or even page numbers, in a specific edition of a specific reference text. (And these texts were available in the college library.) Why, I once had even distributed an original research paper. (It was Griffth’s seminal 1920 paper starting the field of fracture mechanics. Griffith’s argument here is rather conceptual, and the paper has surprisingly very little maths. Whatever the maths there is, it is very easily accessible to the SE students, too.)

The result of my initial attempts to understand the reason (why students don’t read good texts) was not so encouraging. The talkative students began dropping by my cabin once in a while, asking which section to use while answering a certain assignment question or so. However, they still only rarely used those better texts, when it came to actually completing their assignments. And, in the unit tests (and in the final end-sem examination), they invariably ended up quoting only the local text books (whether verbatim or not).

The exercise was, thus, futile. And yet, the students’ sincerity—at least the sincerity of their desire, as in contrast to their actions—could not be put in doubt.

So, I took it as a challenge. I set this as a problem for myself: To discover the main reason(s) why my students don’t refer to good text-books. The real underlying reason(s), regardless of whatever they otherwise did to impress me.

It took a while for me to crack the problem. I would anyway generally chat with them, enquiring where they lived, what their parents did, about their friends and brothers and sisters, etc. In addition, I would also observe, now with this new challenge somewhere at the back of my mind, how they behaved (or rushed around) in college: in hallways, labs, canteen, college ground, even at the bus-stop just outside the college, etc.

…Finally, I got it! At least one reason, a main reason, a systemic reason that applied even to those who otherwise were good, talented, curious, or just plain sincere.

As soon as I discovered the reason, I shared it with every one. In fact, I first shared it with my students, before I did with my colleagues or superiors. The answer lies in an Excel spreadsheet, here [^]. (It actually was created in OpenOffice Calc, on Windows 7.)  Go ahead, download it, and play with it a bit. The embedded formulae should be self-explanatory.

The numbers used in the spreadsheet may differ. The specific numbers I have used in the spreadsheet refer to my estimates while working at a college in Mumbai, in particular, in Navi Mumbai. In Mumbai, the time lost commuting is really an issue. If a student lives in Thane or Andheri and attends a college in Navi Mumbai, he easily spends about 3–4 hours in the daily commute (home->bus->railway station/second bus/metro–>another bus or six-seater, all of it taking about 1.5 hours one way, or more). In Pune, the situation is much more heterogeneous. One student could be spending 3 hours commuting both ways (think: from Nigdi to VIT) whereas some other student could be just happily walking to the college campus (think: Paud Phata residents, and MIT). It all depends. In Pune, many students would be using two-wheelers. In any case, for a professor, the only practical guideline for the entire class that he can at all use, would have to be statistical in nature. So, it’s the class average for the daily commute time that matters. For Mumbai in general, it could be 2–3 hours, for Pune students, it could be, say, between 1 to 2 hours (both ways put together).

So Pune is a bit easier on students. In contrast, for many of my Mumbai students, the situation was bad (or even very bad), and they were trying hard (or very hard) to make the best of it. It must have been at least a bit frustrating to them when professors like me, on the top of everything, were demanding making references to good foreign texts, and openly using a sarcastic tone—even if generously laced with humor—if they didn’t. It must have been frustrating to at least 40–60% of them. (The number is my estimate of those who were genuinely interested in referring to good books, even if only for the better-drawn and colorful diagrams, photographs, and also mathematical proofs that came without errors or without arbitrary replacement of \partial by d.)

And why do I say that it must have been frustrating? Why didn’t I say it might have been frustrating?

Because, I cannot ever forget that look of that incredibly honest appreciation which slowly appeared on all their faces (including the faces of the “back-benchers”), as I shared my discovery in detail with them.

* * * * *   * * * * *   * * * * *

How about your college? Your case?

Do you have the time to read good, lengthy, or conceptually clarifying “reference” texts? Say, Timoshenko (app. mech. and strength of materials); Shames, or Popov (strength of materials); White, or Fox & McDonald, or Som & Biswas (Fluid Mech.); Holman, or Nag, or Sukhatme (Heat Transfer)?

And, if you do, do you spend time reading these texts? If yes, did you complete them (I mean only the portion relevant to the syllabus) in the same semester that you were learning or teaching the subject for the first time? Could you have?

And yes, in my last sentence, I have included “teaching” too. My questions are directed to the professors too. In fact, my questions are directed, first and foremost, only at them.

After all, it is the professors—or at least some of us—who are in the driver’s seat here; the students never are. It is the professors who (i) design the syllabii as well as the examination schemes (including the number of tests to have and their nature), (ii) decide on the number of assignments (and leave no opportunity to level criticism in our capacity as External Examiners, if the length or difficulty of an assignment falls short), (iii) decide on the course text-books (and take due care to list more than 5 prescribed text-books, and more than 10 reference books per course) (iv) decide on the student attendance criteria in detail, up to the individual course level, and report on the defaulting students (and follow through with the meetings with their parents) every two weeks or at least once a month, (v) set the examination papers according to the established pattern—after all, it’s only us who is going to check the papers!, (vi) sometimes, write those local text-books!, and (vii) also keep the expectation that students should somehow show in their final university examination answer books, some evidence of having gone through some good, thick, reference texts, too. Whether we ourselves had managed to do that during our own UG years or not!

And, yes, I also want the IIX professors to ponder over these matters. All their students enjoy a fully residential program; these kids from these private engineering colleges mostly don’t. They at IIXs always get to design all their course syllabii and decide on the examination patterns, and they even get to enjoy the sole responsibility to grade their students. The possibility of adopting a marks normalization scheme, after the examination, always lies at hand, with them, just in case a topic took too long with a certain class or so… Are they then being reasonable in their request demand that the students of these “other” engineering colleges in India be well-read enough, at least by the time the students join them at IIXs for ME/MTech studies?

As to me, no, as I indicated in my earlier posts, while being a professor, I could not always find the time to do that—referring to good text-books. I tried, but basically my situation wasn’t much different from that of my students—we both were short on the available time. So, I didn’t always succeed.

[As to my own UG years, it was mixed: I did hunt for months, and got my hands on, the books like Reed-Hill, White, Holman, etc. However, I would be dishonest if I claimed that it was right during my UG years that I had got whatever I did, from books like these. In my case, the learning continued for years. Yes, I even bought and religiously studied once again even Thomas & Finney’s calculus, when I was in my PhD program at UAB. Despite my attempts during the UG years, I really cannot ascribe a large part, or even a significant part of my current understanding to my UG years. Your case may be different; I was just narrating my own experience.]

… And, as far making references to good books goes, now that I do have time at my hand these days, there is another problem: I don’t know what course in particular I will be teaching the next semester, and where—or for that matter, whether some college will even hire me in the first place, or not.

So, I end up “wasting” my time writing blog posts like this one. Thus, I, too, end up not touching a good reference text!

* * * * *   * * * * *   * * * * *

A Song I Like:

(Hindi) “aane waalaa pal, jaane waalaa hai…”
Lyrics: Gulzaar
Singer: Kishor Kumar
Music: R. D. Burman

[I will go over this post once again, editing it, and may be adding a bit here and there. Done. This post is already too long. So, I will write another post—a brief one—to jot down some tips to make the best possible use of the student’s time—including my suggestions to the engineering colleges as to what they can do to help the students. Also, my take on whether the system as noted above has diluted the quality of education or not—esp. as in contrast to what we had as UG students at COEP more than three decades ago.]



Hey physicists! Revise your books to follow this particular order!!

If you are a working/budding physicist, in all likelihood, you have been taught your quantum physics using books that mangle the historical order of development. Indeed, even McQuarrie’s book on quantum chemistry gives only a sketchy idea about the actual order in which the subject actually got developed. (IMO, the quantum chemistry books are better suited for a self-study of quantum physics. Among them, McQuarrie’s is the easiest to follow, though Levine’s has some topics better covered.)

The conceptual confusion that results out of abandoning the proper hierarchical order is just too huge. For instance, here is a quick question: Every one knows that Bohr’s model came on the scene before the real QM did. But the question is: When did the correspondence principle arrive? During those Bohr-Einstein debates? Or earlier?

Answer: Earlier. Right in 1913, when Bohr put forth his model. The Bohr-Einstein debates, in contrast, came much later, around the 1927 times, i.e., after all the essential principles of QM had already been discovered. And, BTW, it’s the complementarity principle which came during these latter times of the Bohr-Einstein debates.

Interesting? Ready for another question? Ok. Here we go.

Identify which development came first: (i) Dirac’s use of the Poisson brackets in quantum theory, (ii) The application of the matrix mechanics to the calculation of the hydrogen atom spectrum, (iii) The probability interpretation of the wave function?

If you are like 99% of others, you will say: In the order: (iii), (ii) and (i). The correct answer is: (i), (ii) and (iii), precisely in that order!

Don’t let yourself think that such questions are good for those fun quiz competitions or for the generally satisfying trivia. There is a very simple but very profound truth hidden in here: If Pauli could work out the hydrogen atom spectrum before anyone had even an inkling of a probability interpretation, what it obviously means is that there is some way that Pauli used, which is (implicitly or explicitly) more fundamental than some formal system that posits “probability currents” as the first axiom of QM.

More generally, if the historically less-progressed context (i.e. knowledge available by a certain year X) was factually enough (or sufficient) for someone to think of a great new idea, then, among all the conceivable or proposed ordering of topics or contexts that can be taken as foundational to explain that novel idea, the historically least progressed context is the only one that is necessary.

All the rest of the conceivable schemes are either after-thoughts, or organizational devices like mnemonics, or mere deductive tricks, or worse: mere cognitive burden on anyone who takes them seriously as a hierarchically proper scheme.

Having said that, now, pick up any of the introductory textbooks on quantum theory, carefully check out the order in which the topics are progressed in that book, and then ask yourself: How much of an unnecessary, useless cognitive burden is this particular author (i.e. an influential physicist) thrusting on your mind? How much lighter, better, would you feel if the order were something like the following? (The dates in parentheses follow the YYYY/MM format):

  • Planck (1900/10): The quantization of energy of the electromagnetic oscillators in the walls of a light-radiating cavity
  • Einstein(1905/06): The explanation of the photoelectric effect by quantizing the light radiation itself
  • Einstein(1906/12): The first quantum theory of the specific heat of solids
  • Bohr(1913/02–09): An explanation of the pattern of the discrete lines in the atomic spectra
  • Bohr(1913/02–09): The correspondence principle
  • Sommerfeld (1916–1920): Corrections to the Bohr model, introducing additional quantum numbers
  • Compton (1923/05): A light scattering experiment, which confirms the quantum nature of light
  • de Broglie (1923/09): The hypothesis of the matter waves, with a view to extend the wave-particle duality of light to matter as well
  • Pauli (1925/01): The discovery of the exclusion principle, for the electrons in atoms
  • Heisenberg (1925/06): The invention of the arrays of observables, to explain the atomic spectra
  • Born and Jordon (1925/09): The first physical law stated using non-commuting symbols: pq - qp = i\hbar I
  • Goudsmit and Uhlenbeck (1925/10): The experimental discovery of the electron spin
  • Pauli (1925/11): The first success in applying the matrix mechanics to the line spectrum of hydrogen, including the Stark effect
  • Dirac (1925/11): The identification of a Poisson brackets structure in Heisenberg’s analysis.
  • Born, Jordon and Heisenberg (1925/11): The “three-man paper” on the mathematics of matrix mechanics submitted for publication (9 days after Dirac’s above paper)
  • Schrodinger (1925/12): Formulation of the first ideas of his wave mechanics
  • Schrodinger (1926/01): Successful application of his wave equation to the hydrogen atom
  • Schrodinger (1926/03): Demonstration of the mathematical equivalence of the matrix mechanics and the wave mechanics
  • Born (1926/07): The probability interpretation of the wave function
  • Dirac (1926/09): The transformation theory—the wave and matrix mechanics as special cases
  • Davisson and Germer (1927/01): Experimental confirmation of diffraction of electrons by a crystal lattice
  • Heisenberg (1927/02): Formulation of the uncertainty principle
  • Bohr (1927/09): Formulation of the complementarity principle and the Copenhagen interpretation
  • Thomson (1927/11): Another experiment which confirms that matter diffracts
  • Dirac (1930/05): Publication of the first edition of his book (having its “first chapter missing”)
  • Dirac (1939): The third edition of his book introduces the bra-ket notation—the starting point for today’s “Alice and Bob”-obsessed idiots

As you probably know, I have been trying to follow the historical sequence in writing my book. So, in a way, I have been looking a bit carefully into the historical order in which things happened. Still, I had a few surprises in store even for me when I really sat down to compile the above list. Here they are: (i) Einstein’s 1906 paper (which I used to put somewhere in the late teens), and, (ii) Dirac’s 1925 paper.

I am sure that things like the following would come as surprises to many of you: (i) Dirac’s transformation theory being formulated before either the uncertainty principle or the complementarity principle was, (ii) Pauli working out the hydrogen atom line spectrum using the matrix mechanics barely within 2 months of the writing of Heisenberg’s first paper, and in fact before Heisenberg himself could succeed doing so, (iii) Born identifying the matrix nature of the Heisenberg’s non-commutative arrays and Jordon working out the derivations of the mathematics involved in it.

I also think that the help that was both required and received by Heisenberg, might have come as a surprise to many of you, esp. when contrasted with Schrodinger’s single-handed development of all the fundamentals of the wave mechanics—except, of course, the probability interpretation of the wave function, which was supplied by Born.

Most importantly, I think, quite a few must have been shocked to find that Dirac could work out his theory, even predict the existence of anti-matter, without explicitly using the bra-ket notation itself. It has become a fashion to explain this notation right in chapter 1 (though, thankfully, not right in the preface—not yet, anyway).

While writing on the heuristics that he follows while deciding whether a paper on the P-vs-NP issue is worth reading or not, Prof. Scott Aaronson has indicated that any paper not written in LaTeX is suspect.

I have a similar test for books, papers, tutorials etc. on quantum physics, especially the introductory or foundational ones. (Seriously. I have actually followed it over quite a few years in the recent past, and very successfully, too.)

I don’t take any paper/notes/tutorials/book on the foundations of quantum physics for a serious consideration (i.e., I don’t even browse or flip through its abstract) if it has “Alice” and “Bob” written anywhere within it.

Ditto, for any textbook on quantum physics, if it has those two words appearing within the first 90% of the real text matter.

So, hey physicists! Revise your books to follow the kind of an order I have given above!!


Because, I say so. That’s why.

* * * * *   * * * * *   * * * * *

No “A Song I Like” section, once again. I still go jobless. Keep that in mind.

[This is initial draft, published on September 26, 2012, 8:07 PM, IST. May be I will make some minor corrections/updates later on.]