Equations using the matrix notation for a simple artificial neural network

Update on 2020.01.30 16:58 IST:

I have taken the document offline. Yes, it was too incomplete, tentative, and in fact also had errors. The biggest and most obvious error was about the error vector. 🙂

No, no one pointed any of the errors or flaws to me.

Yes, I will post an expanded and revised version later, hopefully in the first week of February. (I started work on this document on 24th Jan., but also was looking into other issues.) When I am done, I will delete this entire post, and make a new entry to announce the availability of the corrected, expanded and revised document.

The original post appears below.

Go, read [^].

Let me know about the typo’s. Also, errors. [Though I don’t expect you to do that. [I will eat my estimate of your moral character, on this count, at least for the time being.]]

I can, and might, take it out of the published domain, any time I want. [Yes, I am irresponsible, careless, unreliable, etc. [Also, “imposing” type. Without “team-spirit”. One who looks down on his colleagues as being way, way, inferior to me.]]

I will also improve on it—I mean the document. In fact, I even intend to expand it, with some brief notes to be added on various activation- and loss-functions.

Eventually, I may even publish it at GitHub. Or, at arXiv. [If they let me do that. But then, another consideration: there are physicists there!]


A song I like:

(Hindi) “कभी तो मिलेगी” (kabhee to milegi)
Singer: Lata
Music: Roshan
Lyrics: Majrooh Sultanpuri

[Credits happily listed in a mostly random order. [The issue was only with the last two; it was clear which one had to appear first.]



I’m not…

“I’m not half the man I used to be.”

That’s what they all say when they become, say, mature. These days, the “they” includes me. Yep. That’s right. I have realized that I really am not half the man I used to be. … Let me count the ways…

I no longer write code every day.

I also no longer debug code every day.

I also no longer use C++ as my first programming language.

I also no longer read pop-science books on QM.

…You get the idea. …

But, actually, it’s become worse, much worse. I perhaps am no longer even a quarter of the man I used to be. …

The reason is, not only do I read through (all the pages of) the QM and maths text-books these days, I have in fact also begun going through some maths journals these days… And it doesn’t even stop there… I mean, I also have a recommendation for a maths journal, for you!

OK. Don’t be overly concerned about me. I am quite OK, that way…

About the only maths journal whose many issues I have browsed through (and have recommendation for) is the “SIAM Undergraduate Research Online (SIURO)” [^].

Do check it out. The published papers are available for free. The link to the published papers is right on the home page; it goes to here [^].

Neat… May be Americans have begun becoming smart these days or something…

[I forgot what it was that I was looking for when I stumbled across one of the papers in this journal… If I remember that paper, I will come back and mention it… [I told you, I’m not half the man I used to be…]]

A Song I Like:
(English) “Yesterday”
Band: The Beatles

[Anti-poignancy (or should it be un-poignancy?): The Beatles were in their early twenties when they penned these lyrics.]


What would you like for an undergraduate book on QM to explain to you?

1. Background:

A couple of things concerning books happened recently, in the last week or two.

(i) An acquaintance I have made at iMechanica, Dr. Biswajit Banerjee, announced last week that a new book on metamaterials and waves in composites authored by him is coming out in print within a few months.

(ii) In one of my regular visits to bookshops, I noticed a hardcover copy of Prof. Allan Bower’s book on mechanics of solids. Someone (or more likely, some institution) in Pune had ordered it, and the copy happened to be lying near the top of the stack. The book had been online for some time and I had browsed through it already. So, now, when I opened the printed version out of curiosity, I directly went to the Preface part. This is how Prof. Bower opens his Preface:

Ronald Rivlin, a pioneer in the field of nonlinear elasticity, was asked once whether he intended to write a treatise on his field. “Why should I write a book?” he replied. “People write books to learn a subject. I already know it.”

[Emphasis in bold is mine.]

2. “People write books to learn a subject:”

That line really hit me—in two different ways.

Firstly, it has always been a part of the folklore among engineering college teachers that the best way to learn a subject is to teach it. I first heard it in 1984 in the teachers’ room in Bharati Vidyapeeth’s College of Engineering. I had then realized, first-hand, how true the saying was. And, I had wondered right back then: if it is not possible to teach a course, how about writing a book on it—mainly for learning the subject.

Secondly, it occurred to me now that if I were to pick out just one subject that I wish to understand better, it would undoubtedly have to be Quantum Mechanics. And, so the thought became: why not write a book on QM—in order to learn it. The idea is not so objectionable after all. Especially if you consider that most physicists (and all mainstream physicists) assure all the rest of us (and very probably also to themselves) that nobody understands quantum mechanics.

Actually, I was planning to write 2–3 journal papers that would extend and better explain some of the fundamental features of quantum physics as using my new approach. I had been gathering my thoughts and background material together. Yet, the more I thought about it, the better it began looking to me that perhaps it makes sense to first write a book on QM before writing those articles.

3. A question for you:

So, with those thoughts in mind, I would like to raise the title question to you:

What would you like an undergraduate book on QM to explain to you? Or, better still (because it makes it more personal): What do you want me to explain to you, concerning the introductory topics of QM?

By introductory topics, I mean the topics covered in (roughly in the order of increasing depth or complexity): Eisberg and Resnick, Hameka, Phillips, Scherrer, Liboff, Gasiorowicz, etc. Also, many other books falling within this same range, e.g.: Griffiths, Zettilli, etc.

If you have any points to raise in this regard, feel free to do so. I will keep this particular post open for comments until I finish writing the book (whenever I come to do that).

Any one may write me. However, a word about the intended audience and the nature of treatment is in order.

4. The intended audience and the intended treatment of the book:

The primary intended audience is: 3rd/4th year undergraduate students in engineering and applied sciences. Especially, those who did not have a course on electromagnetic fields beforehand (such as those majoring in mechanical, materials, chemical, aeronautical, etc.).

The treatment will follow the historical order of developments. I will begin with a summary of the pre-requisites, including very rapid (and perhaps rather conceptual) surveys (done from my own viewpoint, sort of “an engineer’s viewpoint”) of such topics as: Newton’s laws; partial differential equations; The beginning of the energy- and fields-based reformulation of Newton’s mechanics by Leibniz; complex numbers and Euler’s identity; Lagrangian reformulation of Newtonian mechanics; relevant matrix theory; Fourier theory; relevant probability theory; Hamilotonian reformulation of Newtonian mechanics; electromagnetic field theory from say Coulomb to Maxwell and Hertz; the eigenvalue problem in classical physics; cavity radiation; special relativity (taken as highlighting certain features of the classical electrodynamics).

The QM proper will begin with Planck, of course, and will closely follow the historical sequence, though the notation might be modern—however, I wish to emphasize that I will not introduce Dirac’s notation until he introduces it, so to say. Similarly, I will introduce Heisenberg’s matrix mechanics before Schrodinger’s wave mechanics. I intend to leave the reader at about 1935, though an appendix or two on entanglement is possible.

I will try to keep the length to about 300 pages at the most. I would love to see if it can be done within 250 pages, but doing so appears not easily possible. I will leave out many conceptual explanations, esp. of the prior theories, primarily because that burden has already been relieved for me in the form of Manjit Kumar’s book. In a way, I do see this book as complementary to that book.

I will neither cover Feynman’s reformulation nor Bohm’s ideas.

The book will also not be a vehicle to introduce my own approach. However, it is impossible for any author to keep aside his viewpoint, while thinking or writing. In this case, I will try to restrict myself to highlighting the wonderful series of ridiculous conclusions to which the earlier theories lead (often isolated and put forth by the formulators of those theories themselves), and providing some explicit hints for getting out of them. However, in the planned book, I will not go beyond providing hints alone. … Yes, I will be willing to give out some of the material or thoughts that, properly, should have come in the research articles first. However, as far as journal articles go, frankly, I do not care!

The book will carry mathematics. (It won’t be addressed to the layman.) It will carry derivations too, but only in simple and essential terms. (By simple, I do not mean: devoting inordinate time to one-dimensional and time-independent cases. This may mean that the book ends up being suitable only to the beginning graduate students of engineering. If so, that would be OK by me.)

Further, I would often provide the derivations in an order other than what is found in the usual treatments. For instance, the prerequisites part itself will cover spherical harmonics—right in the context of classical physics. Also, the angular momentum of the EM field. (Yes, the prerequisites part will be a major part in this book, perhaps 40% of the total material.) The prerequisites part will also point out the issue of the instantaneous action-at-a-distance, right in the PDE section.

In short, it will almost be a university text-book. Except that I don’t expect any university to adopt it for their classroom usage! Therefore, there won’t be any chapter-end exercises, nor a section at the beginning of the chapter for motivating the student. However, some pointers for further thought might be provided via notes at the end of the book.

Many of the readers of this blog (and most readers at the iMechanica blog) come from engineering and applied sciences background. They are likely to have run into QM as a part of their courses on modern physics, solid physics, nanomaterials, etc. They might have had run into issues concerning the real QM. I would love it if I can provide answers to their questions. And, it goes without saying that students of “pure” sciences—physics, chemistry, etc.—would be as welcome as those of engineering sciences. This book would be directed at them, not at the layman—or at the philosophers.

Of course, as far as raising the questions go, any one may feel free to submit his query via a comment. (I will be moderating the comments here, and may not reply comments at iMechanica—where I do not moderate anything.)

All in all, it would be a book written by an engineer, and primarily for engineering/applied science undergraduates/beginning postgraduate students. There already is an excellent book in this space: Prof. Leon van Dommelen’s online book. I really like it, but thought that I would have approached many things differently, and so, thought of writing my book. Most important difference, to my mind, is that I would stick to the historical approach throughout. But, yes, as far as undertaking this huge an effort goes, Prof. van Dommelen’s book would remain a kind of an inspiration for me.

5. One final point: Sample questions:

Some time ago, I had written a list of questions that I thought UG students should ask their professors. However, there were also other topics in that post. For ease of direct referencing, here I am copy-pasting those questions below (with a bit of editing). Go through them and see if you have any other questions you wish to raise:

  • Why are quantum-mechanical forces conservative?
  • Does the usual time-dependent Schrodinger’s equation (TDSE) apply to propagation of photons? If yes, why does no textbook ever illustrate TDSE involving photons? If not, what principle goes against applicability of TDSE to photons?
  • What kind of physics would result if the QM wavefunction were not to be complex-valued but real (scalar)-valued? What if the wavefunction were to be deterministic rather than probabilistic? What contradictions would result in each case?
  • Does QM at all need an interpretation? If yes, why? Why is it that no other theory of physics seems to need special efforts at interpreting it but only QM does, esp. so if all physics theories ultimately describe the same reality? If QM does not need an interpretation, why do people talk about the phrase: “interpretation of QM”? What do they mean by that phrase?
  • What, precisely, is the physical meaning of an operator? Please don’t simply repeat for us its definition. Instead, please give us the physical meaning of the concept. Or is it the case that no physical meaning is possible for this concept and that it is doomed to remain an exclusively mathematical concept? If yes, why use it in the postulates of a physical theory—without ever taking the care to define its physical correspondents?
  • Are all quantum-theoretical operators Hermitian? If yes, why? What physical fact does this property indicate/highlight/underscore? What if they are not Hermitian?
  • Give one example of an important eigenvalue problem from classical mechanics in which the differential equation formalism is very clearly shown to be equivalent to the matrix formalism.
  • Does the QM theory necessarily require the concept of a wavepacket when it comes to detailing what a QM particle is? If yes, why? What would happen if it were not a packet of waves but instead just a monochromatic wave? If the theory does not necessarily require packets of waves, then can you suggest us any alternative treatment—if there is one?
  • In every differential equation we have studied thus far, the primary unknown always carried some or the other physical units/dimensions. For example, for mechanical waves, the primary variable would be the displacement from the equilibrium position. But the QM wavefunction seems to be a dimensionless quantity; at least, textbooks don’t seem to note down any units for it. Is it a dimensionless quantity? Why? What important things does this tell us about the nature of theorization followed in QM?
  • Is QM an action-at-a-distance theory?
  • How, precisely, does QM relate to the classical EM? Is the term V(x,y,z,t) in Schrodinger’s equation to be understood in the classical sense? If yes, why do people say that between the two, QM is more basic to EM?
  • Explain precisely how the Newtonian mechanics is implied by QM.
  • And, one question I raised yesterday, via a tweet: In the mainstream interpretation—taught to all undergraduates world-wide—it is not meaningful to speak of emission of particles. Particles are never emitted, only absorbed—because only absorption can be “observed.” The whole world is a series of absorptions, so to speak. True or false? (Hint: In Keynesian economics, there are only consumers, no producers!)

Those were the questions I thought of, some time ago. … I am sure you can do better. I now look forward to hearing from you.

* * * * *   * * * * *   * * * * *

A Song I Like:
(Hindi): “yehi woh jagah hai, yehi woh fizaayen…”
Music: O. P. Nayyar
Singer: Asha Bhosle
Lyrics: S. H. Bihari