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…]

Miscellaneous: books to read, a new QM journal, the imposter syndrome, the US presidential elections

While my mood of not wanting to do anything in particular still continues (and also, there is no word yet on the job-related matters, including on whether I might qualify as a Professor of Mechanical Engineering in SPPU or not), there are a few quick things that I may as well note.

Updates on 17th, 18th and 22nd Nov. 2016: See my English translation[s] of the song, at the end of the post.

Books to Read:

First, the books to read. Here are a few books on my to-read list:

  1. Sean Carroll, “The Big Picture” [^]. I have been browsing through Sean’s blog-posts since before the time the book was published, and so have grown curious. I don’t have the money to buy it, right now, but once I get the next job, I sure plan to buy it. Here is the review in NY Times [^]. And, here is a latest review, written by a software engineer (whose link appeared in Sean’s twitter feed (I don’t myself use my Twitter account, but sometimes do check out the feeds of others via browser))[^]. Judging from his posts, I do know that Sean writes really well, and I would certainly want to check out this book, eventually.
  2. Roger Penrose, “Fashion, Faith, and Fantasy in the New Physics of the Universe,” [^]. This is the latest offering by Penrose. Sometimes I simply type “quantum physics” in Google, and then, in the search results, I switch the tab over to “news.” I came to know of this book via this route, last week, when I ran into this review [^].
  3. Roger Schlafly, “How Einstein Ruined Physics: Motion, Symmetry and Revolution in Science,” [^]. Here is a review [^], though my curiosity about the book rests not on the review but on two things: (i) what I had thought of Einstein myself, as far back as in early 1990s, while at UAB (hint: Schlafly’s thesis wouldn’t be out of bounds for me), and (ii) my reading the available portions of the book at Google Books. …This book has been on my “to-read” list for quite some time, but somehow it keeps slipping off. … Anyway, to be read, soon after I land a job…

A New QM Journal:

A new journal has arrived on the QM scene: [^]. Once again, I got to know of it through the “news” tab in a Google search on “quantum physics”, when I took this link [^].

It’s an arXiv-overlay journal. What it means is that first you submit your paper to arXiv. … As you know, getting something published at arXiv carries a pretty low bar (though it is not zero, and there have been some inconsistencies rarely reported about improper rejections even at arXiv). It’s good to bring your work to the notice of your peers, but it carries no value in your academic/research publications record, because arXiv is not a proper journal as such. … Now, if your work is good, you want to keep it open-access, but you don’t want to pay for keeping it open-access, and, at the same time, you also want to have the credentials of a proper journal publication to your credit, you have a solution, in the form of this arXiv-overlay journal. You send the link to your arXiv-published paper to them. If their editorial board finds it fitting the standards and purpose of their journal, they will include it.

The concept originated, I guess, with Timothy Gowers [^] and others’ efforts, when they started a maths journal called “Discrete Analysis.” At least I do remember reading about it last year [^]. Here is Gowers’ recent blog post reflecting on the success of this arXiv-overlay journal [^]. Here is what Nature had to report about the movement a few months ago [^].

How I wish there were an arXiv for engineering sciences too.

Especially in India, there has been a proliferation of bad journals: very poor quality, but they carry an ISSN, and they are accepted as journals in the Indian academia. I don’t have to take names; just check out the record of most any engineering professor from outside the IISc/IIT system, and you will immediately come to know what I mean.

At the same time, for graduate students, especially for the good PhD students who happen to lie outside the IIT system (there are quite a few such people), and for that matter even for MTech students in IITs, finding a good publication venue sometimes is difficult. Journal publications take time—1 or 2 years is common. Despite its size, population, or GDP, India hardly has any good journals being published from here. At the same time, India has a very large, sophisticated, IT industry.

Could this idea—arXiv-overlay journal—be carried into engineering space and in India? Could the Indian IT industry help in some ways—not just technical assistance in creating and maintaining the infrastructure, but also by way of financial assistance to do that?

We know the answer already in advance. But what the hell! What is the harm in at least mentioning it on a blog?

Just an Aside (re. QM): I spent some time noting down, on my mental scratch-pad, how QM should be presented, and in doing so, ended up with some rough outlines of  a new way to do so. I will write about it once I regain enough levels of enthusiasm.

The Imposter Syndrome:

It seems to have become fashionable to talk of the imposter syndrome [^]. The first time I read the term was while going through Prof. Abinandanan’s “nanopolitan” blog [^]. Turns out that it’s a pretty widely discussed topic [^], with one write-up even offering the great insight that “true imposters don’t suffer imposter syndrome” [^]. … I had smelled, albeit mildly, something like a leftist variety of a dead rat here… Anyway, at least writing about the phenomenon does seem to be prevalent among science-writers; here is a latest (H/T Sean Carroll’s feed) [^]…

Anyway, for the record: No, I have not ever suffered from the imposter syndrome, not even once in my life, nor do I expect to do so in future.

I don’t think the matter is big enough for me to spend any significant time analyzing it, but if you must (or if you somehow do end up analyzing it, for whatever reasons), here is a hint: In your work, include the concept of “standards,” and ask yourself just one question: does the author rest his standards in reason and reality, or does he do so in some people—which, in case of the imposter syndrome, would be: the other people.

Exercise: What (all) would stand opposite in meaning to the imposter syndrome? Do you agree with the suggestion here [^]?

The US Presidential Elections: Why are they so “big”? should they be?

Recently, I made a comment at Prof. Scott Aaronson’s blog, and at that time, I had thought that I would move it here as a separate post in its own right. However, I don’t think I have the energy right now, and once it returns, I am not sure if it will not get lost in the big stack of things to do. Anyway, here is the link [^]. … As I said, I am not interested much—if at all—in the US politics, but the question I dealt with was definitely a general one.

Overall, though, my mood of boredom continues… Yaawwwnnnn….

A Song I Like:

(Hindi) “seene mein jalan…”
Lyrics: Shahryar
Music: Jaidev
Singer: Suresh Wadkar

[Pune today is comparable to the Bombay of 1979 1978—but manages to stay less magnificent.]

Update on 2016.11.17: English translation of the song:

For my English blog-readers: A pretty good translation of the lyrics is available at Atul’s site; it is done by one Sudhir; see here [^]. This translation is much better than the English sub-titles appearing in this YouTube video [^] which comes as the first result when you google for this song. …

I am not completely happy with Sudhir’s translation (on Atul’s site) either, though it is pretty good. At a couple of places or so, it gives a slightly different shade of meaning than what the original Urdu words convey.

For instance, in the first stanza, instead of

“Just for that there is a heart inside,
one searches a pretext to be alive,”

it should be something like:

“just because there is a heart,
someone searches (i.e., people search) for an excuse which can justify its beating”

Similarly, in the second stanza,  instead of:

“what is this new intensity of loneliness, my friend?”,

a more accurate translation would be:

“what kind of a station in the journey of loneliness is this, my friends?”.

The Urdu word “manzil” means: parts of the Koran, and then, it has also come to mean: a stage in a journey, a station, a destination, or even a floor in a multi-storied building. But in no case does it mean intensity, as such. The underlying thought here is something like this: “loneliness is OK, but look, what kind of a lonely place it is that I have ended up in, my friends!” And the word for “friend” appears in the plural, not the singular. The song is one of a silent/quiet reflection; it is addressed to everyone in general and none in particular.

… Just a few things like that, but yes, speaking overall, Sudhir’s translation certainly is pretty good. Much better than what I could have done purely on my own, and in any case, it is strongly recommended. … The lyrics are an indispensable part of the soul of this song—in fact, the song is so damn well-integrated, all its elements are! So, do make sure to see Sudhir’s translation, too.

Update on 2016.11.18: My own English translation:

I have managed to complete my English translation of the above song. Let me share it with you. I benefitted a great deal from Sudhir’s translation and notes about the meanings of the words, mentioned in the note above, as well as further from “ek fankaar” [^]. My translation tries to closely follow not only the original words but also their sequence. To maintain continuity, the translation is given for the entire song as a piece.

First, the original Hindi/Urdu words:

seene mein jalan aankhon mein toofaan sa kyun hai
is shehar mein har shakhs pareshaan saa kyun hai

dil hai to dhadakne ka bahaanaa koi dhoondhe
patthar ki tarah behis-o-bejaan sa kyun hai

tanahaai ki ye kaun si manzil hai rafeeqon
ta-hadd-e-nazar ek bayaabaan saa kyon hai

kyaa koi nai baat nazar aati hai ham mein
aainaa hamen dekh ke hairaan sa kyon hai

Now, my English translation, with some punctuation added by me [and with further additions in the square brackets indicating either alternative words or my own interpolations]:

Why is there jealousy in the bosom; a tempest, as it were, in the eyes?
In this city, every person—why does it seem as if he were deeply troubled [or harassed]?

[It’s as if] Someone has a heart, so he might go on looking for an alibi [or a pretext] to justify [keeping it] beating
[But] A stone, as if it were that, why is it so numb and lifeless [in the first place]?

What kind of a station in the journey of the solitude is this, [my noble] friends?
Right to the end of the sight, why is there [nothing but] a sort of a total desolation?

Is there something new that has become visible about me?
The mirror, looking at me, why does it seem so bewildered [or perplexed]?

Update on 22nd Nov. 2016: OK, just one two more iterations I must have; just a slight change in the second [and the first [, and the third]] couplet[s]. (Even if further improvements would may be possible, I am now going to stop my iterations right here.):

Why is there jealousy in the bosom; a tempest, as it were, in the eyes?
In this city, every silhouette [of a person]—why does it seem as if he were deeply troubled [or harassed]?

[It’s as if] A heart, one does have, and so, someone might go on looking for an alibi [or a pretext] to justify [keeping it] beating
[But] A stone, as if it were that, why is it so numb and lifeless [in the first place]?

What kind of a station in the journey of the solitude is this, [my noble] friends?
[That] Right to the end of the sight, why is there [nothing but] a sort of a total desolation?

Is there something new that has become visible about me?
The mirror, looking at me, why does it seem so bewildered [or perplexed]?




A nice little book on mathematics for biologists—and for the rest of us!

Have been reading QM texts and taking down my notes. Also jotting down my thoughts as they occur during these studies. There have been many threads of such thoughts, but nothing is even remotely near the completion stage. …

QM is hard—it pulls together an incredible variety of mathematical methods and tools, and each is a gem that you must spend some time on. So, the task keeps on growing…

It would have been nice to at least indicate what my thoughts are like, but it just so happens that they are rather like the margin notes. I could easily talk about them with any one (suitable) in a personal discussion, but a blog isn’t a good medium for sharing these—the ratio of the thought to the required context is too small here, and so, if these thoughts are to be converted into blog posts, too much time would be spent in just building the context.

That explains why, though I do have some free time at hand these days, I haven’t felt like blogging over the past two weeks.

But, guess I have to keep the momentum of blogging going too, and so, I will write a bit about a wonderful book on mathematics that I ran into only in the last month.

The book I am talking about packs together a lot of seeming contradictions. It is freely downloadable. It is short (130 pages). Its exercises are simple—so simple that they can’t but bolster your confidence. Yet, the text is written in such a way that it manages to pique your curiosity, so you feel like pursuing the topics covered in more depth later on. The book was written for biologists. But it has introductions to some topics that engineers wouldn’t have either understood so well, or the topics they wouldn’t have been exposed to, during their undergraduate education. By the latter, I mean topics like non-linearity and chaos. So, clearly, this book is useful also to engineers.

The book in question is:

Kirsten ten Tusscher and Alexander Panfilov (2011) “Mathematics for Biologists,” Utrecht University. [^]

A little bit more about the utility of the book to engineers, from my personal experience and viewpoint….

I have taught (simple) introductory courses on finite element method (FEM) to undergraduate students, postgraduate students, and also to practicing engineers. So let me say something with this background in mind.

If you have done a course on FEM, you know that you don’t waste your time trying to by-heart a seemingly countless number of integration formulae, as in your calculus-related mathematics courses.

Instead, you choose to play God—a rather lazy version of a God—and thereby directly assert right at the beginning of solving a(ny) problem that the unknown solution y = f(x) carries the form of only a polynomial. … I haven’t yet run into any book that pursues anything other than polynomials in more than one chapter, usually the introductory one. So, practically speaking, it’s always the polynomials.

How can you assert that it can only be a polynomial? Because, you are playing a God, that’s why. A rather lazy God.

In fact, it’s a polynomial of only a few initial terms, e.g., this one: a_0 + a_1 x + a_2 x^2 + a_3 x^3.

In other words, even if the physical situation to model is such that the actual solution would consist of, say, the first half of the sine curve (the one that runs between 0 and \pi), you still, in effect, proceed to ban the sine wave out of your universe:

“What, sine curves? Not in my universe. In my universe, sine curves are banned. They are not allowed to exist. Only the polynomials may.” That’s what you effectively say.

“But if polynomial is not the true solution, wouldn’t your `solution’ be in error?,” someone may raise this question.

When it comes to FEM, you have a ready answer: “Yes, but I can make the error as small as my computational power permits me to.”

The ability to reduce the error as small as desirable (or practically possible) is what makes numerical techniques (like FEM) “good.” (It is also what permits us to arbitrarily ban sines out of our universe; the cost to be paid is: high computational power. In short, we can play God because the machine helps us.) Let me give a concrete example of that—of the the idea of continuously reducing errors.

Remember Internet 1.0? Remember how long it took for a .JPG file to download? [Speculations on the nature of the downloaded files is left as an exercise for the reader.] It could take tens of minutes for a single image. (At least in India, it easily could.) What did your browser (say the Netscape Gold, or IE 1.0) do in the meanwhile?

The browser would initially show you a big empty rectangle in place where image was supposed to appear. Some time later, this rectangle would get filled with some big colored square boxes. As the download further progressed, once in a while, the boxes would abruptly become smaller, and thus, the nature of the picture would become somewhat easier to make out. The process of the refinement went on with the amount of the data downloaded, and a greater amount of the actually downloaded data meant: smaller colored boxes, more number of them. That is, a more refined picture. After some 30 minutes, you would have that half-MB of a file completely downloaded on to your local machine (to be shared with your friends, using 1.44 MB floppy disks (assuming these wouldn’t develop bad sectors)).

So, that was the idea: a very coarse beginning but a series of repeated refinements.

The idea works also in the non-image processing contexts. You can begin with a coarse solution and then refine it better and still better. FEM uses precisely such a principle.

So, in FEM, you can always start with a polynomial even if the true (unknown) solution happens to be a sine curve.

Yes, using a polynomial where a sine curve solution is expected, does mean that there is an error in the solution—the curve for a polynomial isn’t exactly the same as the sine curve. (Try y = x(a-x) i.e. y = ax - x^2 for the first half of \sin x.) That’s because the two aren’t one and the same function. The difference in the two functions is what we call “error.” But since as a God you have banned sine curves from your universe and allowed only the polynomials, the only thing you can do to save your Godliness is to reduce the error.

At this point, you have two choices, concerning what kind of mathematics to use for reducing errors.

(i) You can increase the solution accuracy by adding more, higher order, terms to the same polynomial. What I mean here is, a single polynomial continues to run across the entire domain all by itself, but the number of terms it carries goes on increasing [^]. Thus, for instance, you initially may use the quadratic polynomial: y = a_0 + a_1 x + a_2 x^2 (it has three terms) [^]. You then keep on adding terms for refinement:  y = a_0 + a_1 x + a_2 x^2 + a_3 x^3  (cubic polynomial, four terms) [^];, a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 (quartic, five terms) [^], etc.  That’s one way of refining the solution.

(ii) The second way is: you keep the degree of the polynomial the same (say, you use only a cubic polynomial), but you indirectly compensate for the lack of the higher order terms by increasing the number of boxes used for filling the domain. Thus, a lower-order or cruder polynomial is used for interpolations within a single box, but there are a large number of such boxes side by side to cover the entire domain. (The polynomials are arranged in such a way that at the interface between the two adjacent sub-domains, their curves touch each other, so that the solution for the entire domain remains continuous.)

In other words, the two methods of refinement are: (i) using a progressively longer but single polynomial over the entire domain, or (ii) using shorter and simpler polynomials within more and more color-boxes that cover the entire domain.

The FEM course, at this point, gets a bit more clerical—there always is a “routine” to any job. It also gets a little more difficult, for the students, from this point on. The reason is: they have forgotten some very basic maths by the time they begin to study FEM.

For instance, suppose I tell them this: Since the shape of the first half of the sine curve (i.e. the one between 0 to \pi) is like a bell or a dome—i.e. it has just a single hump—so, obviously, the simplest polynomial which reproduces this broad feature (of the first half of the sine curve) would have to be a quadratic, because a quadratic can be made to carry a bend, whereas a straight line cannot ever be made to do so. On the other hand, if we had to approximately model the full sine curve (i.e. the one between 0 to 2\pi), then, since it carries two bends—a hump in the first half and a cup in the second—therefore, the simplest polynomial that still shows this feature of a hump and a cup, will have to be a cubic.

Typically, people—even graduate engineers with fairly good proficiency in solving differential equations—tend to go (at least temporarily) blank at this point. (Yes, some of the best of them also are merely straining themselves, in my actual observation—including those at COEP.) They are unable to connect the logic of the number of the inflection points on a curve and the degree of the approximating polynomial. They can easily rattle off the relation between the number of points through which a polynomial passes and its degree, but not the number of inflection points and the degree.

It happens. Either they never looked into things like that in depth because their JEE/XII exams didn’t need them, or, in the rush of five courses a semester of new maths/application every semester, they hadn’t had the time necessary to integrate ideas. (Or, they have been slacking. Possible.)

There is no easy solution. They are supposed to have read such topics, but they don’t show any evidence of a fast enough recall. The mathematics texts they use (in engineering) don’t cover such basic topics, because these books take them for granted (e.g. check out Kreyszig, Greenberg, Barrett or Arfken). The texts which the students cursorily used in their XI/XII standards are long gone off their hands (and their heads), possibly because, I said, neither the XII boards nor the JEE asked them to do curve-tracing. Among the mathematics texts now nearby them, Wartikar (or other local books) do have this topic covered (at least cursorily), but the illustrations and the print quality in them is so bad that only a mathematician could pick up a book like that for the second time in life. [Try to believe me when I say that I say this in admiration of mathematicians.]

Curve-tracing isn’t the only elementary or basic topic on which my students habitually go blank. There also are many other similar topics.

They go blank also if I ask them what it is that they visualize for the matrix eigenvalue problem. They never have visualized anything related to matrices. (A rectangle of numbers doesn’t count as a visualization.) The very idea that an eigenvalue problem can be visualized, itself is new to them. (They, by habit, never check the Wiki for any of the topics currently ongoing in the class.) Similarly, they also go blank even if I just utter the words: “a set of coupled differential equations.” They remain “blanked out” until I drop the hint like, say, several mass-spring systems connected together via some pins. At this point the bulb lights up, but only momentarily, and then, an even thicker darkness descends on them. But even that momentary flicker is an encouraging sign, to me. That’s because, nothing at all happens if I mention “chaos”—their eyes remain glassy. Or, the precise difference between complex numbers and vectors. The eyes now begin to show a generally tired version of a confident kind of a carelessness.

They need a book. A helpfully written book. A short book. An easy to read book. The book should treat also the basic topics, but rapidly.

Engineers, that way, are well-exposed to the art of juggling through mathematics. However, even if the pace is somewhat rapid, the book should be a little more than Schaum’s series books (or a compilation of formulae)—there should be some interesting bits of conceptual explanations (or hints), some non-routine kinds of applications mentioned, too.

Further, the book should be small enough that it fits the time that the students have available. (My earlier relevant post on this problem is here [^].)

Finally, the book should not list some challenging—actually, depression-inducing—exercises at the end of a chapter; the book should not turn away the reader from mathematics on that count. (Also, it should preferably have some colorful diagrams.) And, the price should be reasonable.

There are two books that deliever in an excellent manner on all these counts. Both happen to have been originally written for biologists, not engineers.

The first book is: Edward Batschelet (1979) “Introduction to Mathematics for Life Scientists, 3/e” Springer. (Published in 1979, the only feature it misses on is: color diagrams.)

The second book is the one I am talking about, in this post.

On this blog, I had touched upon the first book a while ago, here [^].

The book now under discussion complements the first. It dwells more on the recent topics which were not covered in the first. (The third edition of the first book was in 1979.) It also is just so slightly at a more “advanced” level, though, IMO, 100% understandable to any UG student of engineering.

So, I am happy to strongly recommend the second book as well. Go ahead, check it out for yourself; the .PDF is free to download anyway!

A Song I Like:

(Western “Classical”) “Va, pensiero” (from “Nabucco”)
Original Composer: Giuseppe Verdi
Original Lyrics: Temistocle Solera

[The version which I listened to for the first time in my life was the one by James Last and his orchestra. I still continue liking that version for its own sake even today. The reason is, while the “purely” “classical” (or the classically oriented) performances naturally carry depth, they also tend to sometimes become a shade too pensive or sombre—sometimes even bringing a touch of ghoulishness into rendering.

On the other hand, the pop-instrumental versions (like those by James Last) are more lyrical—they are even lilting in a way. But this form—popular instrumental—itself is such that a performance can’t help but skim over the more serious portions. … The emotional experience of actually finding oneself in the midst of a physical enslavement—the gravitas of that situation—is made light, a bit too light by this form. And therefore, that yearning for the freedom, that soaring affirmation of freedom as a golden value by itself, also becomes that much less moving or stirring. …

As to me, there are times when I want to listen to only the rhythmic affirmation of the positive that such a piece can bring. I want to focus on the transformation that a mind undergoes in the act of even just contemplating the state of freedom. I want to directly sense that heavenly lightness of being which even just a mental contemplation of freedom is able to bring. Why, I want to even just directly sense the fleetingly light experience as was once expressed in (Marathi) “swatanatre, bhagawatee, chaanDaNee cham-cham lakhalakhashee,” or in [continuing with the same Marathi song] “gaalaawarachyaa kusumi kinvaa kusumaanchyaa gaali.” There is this kind of a lightness present also here in Verdi’s “va pensiero,” and there are times when I want to have this part stressed in my experience of the music.

And, of course, there also are other times (in my case these are somewhat more rare) when I must listen to a good “classical” rendering, for a deeper experience of all the aspects of the original music, with all its subtlety and seriousness. I thus have listened to several “classical” renderings of this piece by now, though I haven’t so far had an opportunity to sit through the entire opera. [BTW, I keep putting the scare-quotes around the word classical, because this piece is from the 19th century, and thus, it is, technically, from the Romantic era, not Classical.]

Just one more point. While the words for this song are great, I happen almost never to listen to (or look for) the words here. The music here is just too powerful for the words to matter much one way or the other, even if the words themselves happen to be as good as they are (I mean the English words [^]; I don’t know Italian). In fact, the specific historical context that the opera involves itself means almost nothing to me; only the themes in the abstract do: The interwoven themes of exilement and patriotism. Or, just plain of immigration and nostalgia. But, inescapably, above all, of enslavement and freedom. The overall theme here is complex but universal, and that’s why the specific concretes cease to matter. The words do express the theme well, but compared to the music, they, too, cease to matter…

The music… It’s just too subtle and yet too powerful, it’s too exceptional.

… This piece is supposed to be Verdi’s achievement of a lifetime. I haven’t heard a lot of Verdi, but I find it easy to believe the critic here. Seemingly very simple, it carries quite a few complex layers. It touches on many seemingly familiar musical phrases, but it still remains distinctly innovative, somehow. The music here has drama, and so, its progression does require just a bit of an emotional stamina, but it is not as much as what, say, Beethoven demands of you. The theme here is comparatively on the brighter side, and so, listening to it doesn’t exhaust you….

All in all, it’s a great piece of music! Hope you like it, too.]