# Wrapping up my research on QM—without having to give up on it

Guess I am more or less ready to wrap up my research on QM. Here is the exact status as of today.

1. The status today:

I have convinced myself that my approach (viz. the idea of singular potentials anchored into electronic positions, and with a $3D$ wave-field) is entirely correct, as far as QM of non-interacting particles is concerned. That is to say, as far as the abstract case of two particles in a $0$-potential $1D$ box, or a less abstract but still hypothetical case of two non-interacting electrons in the helium atom, and similar cases are concerned. (A side note: I have worked exclusively with the spinless electrons. I don’t plan to include spin right away in my development—not even in my first paper on it. Other physicists are welcome to include it, if they wish to, any time they like.)

As to the actual case of two interacting particles (i.e., the interaction term in the Hamiltonian for the helium atom), I think that my approach should come to reproduce the same results as those obtained using the perturbation theory or the variational approach. However, I need to verify this part via discussions with physicists.

All in all, I do think that the task which I had intended to complete (and to cross-check) before this month-end, is already over—and I find that I don’t have to give up on QM (as suspected earlier [^]), because I don’t have to abandon my new approach in the first place.

2. A clarification on what had to be worked out and what had to be left alone:

To me, the crucial part at this stage (i.e., for the second-half of March) was verifying whether working with the two ideas of (i) a $3D$ wavefield, and (ii) electrons as “particles” having definite positions (or more correctly, as points of singularities in the potential field), still leads to the same mathematical description as in the mainstream (linear) quantum mechanics or not.

I now find that my new approach leads to the same maths—at least for the QM of the non-interacting particles. And further, I also have very definite grounds to believe that my new approach should also work out for two interacting particles (as in the He atom).

The crucial part at this stage (i.e., for the second half of March) didn’t have so much to do with the specific non-linearity which I have proposed earlier, or the details of the measurement process which it implies. Working out the details of these ideas would have been impossible—certainly beyond the capacities of any single physicist, and over such a short period. An entire team of PhD physicists would be needed to tackle the issues arising in pursuing this new approach, and to conduct the simulations to verify it.

BTW, in this context, I do have some definite ideas regarding how to hasten this process of unraveling the many particular aspects of the measurement process. I would share them once physicists show readiness to pursue this new approach. [Just in case I forget about it in future, let me note just a single cue-word for myself: “DFT”.]

3. Regarding revising the Outline document issued earlier:

Of course, the Outline document (which was earlier uploaded at iMechanica, on 11th February 2019) [^] needs to be revised extensively. A good deal of corrections and modifications are in order, and so are quite a few additions to be made too—especially in the sections on ontology and entanglement.

However, I will edit this document at my leisure later; I will not allocate a continuous stretch of time exclusively for this task any more.

In fact, a good idea here would be to abandon that Outline document as is, and to issue a fresh document that deals with only the linear aspects of the theory—with just a sketchy conceptual idea of how the measurement process is supposed to progress in a broad background context. Such a document then could be converted as a good contribution to a good journal like Nature, Science, or PRL.

4. The initial skepticism of the physicists:

Coming to the skepticism shown by the couple of physicists (with whom I had had some discussions by emails), I think that, regardless of their objections (hollers, really speaking!), my main thesis still does hold. It’s they who don’t understand the quantum theory—and let me hasten to add that by the words “quantum theory,” here I emphatically mean the mainstream quantum theory.

It is the mainstream QM which they themselves don’t understood as well as they should. What my new approach then does is to merely uncover some of these weaknesses, that’s all. … Their weakness pertains to a lack of understanding of the $3D \Leftrightarrow 3ND$ correspondence in general, for any kind of physics: classical or quantum. … Why, I even doubt whether they understand even just the classical vibrations themselves right or not—coupled vibrations under variable potentials, that is—to the extent and depth to which they should.

In short, it is now easy for me to leave their skepticism alone, because I can now clearly see where they failed to get the physics right.

5. Next action-item:

In the near future, I would like to make short trips to some Institutes nearby (viz., in no particular order, one or more of the following: IIT Bombay, IISER Pune, IUCAA Pune, and TIFR Mumbai). I would like to have some face-to-face discussions with physicists on this one single topic: the interaction term in the Hamiltonian for the helium atom. The discussions will be held strictly in the context that is common to us, i.e., in reference to the higher-dimensional Hilbert space of the mainstream QM.

In case no one from these Institutes responds to my requests, I plan to go and see the heads of these Institutes (i.e. Deans and Directors)—in person, if necessary. I might also undertake other action items. However, I also sincerely hope and think that such things would not at all be necessary. There is a reason why I think so. Professors may or may not respond to an outsider’s emails, but they do entertain you if you just show up in their cabin—and if you yourself are smart, courteous, direct, and well… also experienced enough. And if you are capable of holding discussions on the “common” grounds alone, viz. in terms of the linear, mainstream QM as formulated in the higher-dimensional spaces (I gather it’s John von Neumann’s formulation), that is to say, the “Copenhagen interpretation.” (After doing all my studies—and, crucially, after the development of what to me is a satisfactory new approach—I now find that I no longer am as against the Copenhagen interpretation as some of the physicists seem to be.) … All in all, I do hope and think that seeing Diro’s and all won’t be necessary.

I also equally sincerely hope that my approach comes out unscathed during / after these discussions. … Though the discussions externally would be held in terms of mainstream QM, I would also be simultaneously running a second movie of my approach, in my mind alone, cross-checking whether it holds or not. (No, they wouldn’t even suspect that I was doing precisely that.)

I will be able to undertake editing of the Outline document (or leaving it as is and issuing a fresh document) only after these discussions.

6. The bottom-line:

The bottom-line is that my main conceptual development regarding QM is more or less over now, though further developments, discussions, simulations, paper-writing and all can always go on forever—there is never an end to it.

7. Data Science!

So, I now declare that I am free to turn my main focus to the other thing that interests me, viz., Data Science.

I already have a few projects in mind, and would like to initiate work on them right away. One of the “projects” I would like to undertake in the near future is: writing very brief notes, written mainly for myself, regarding the mathematical techniques used in data science. Another one is regarding applying ML techniques to NDT (nondestructive testing). Stay tuned.

A song I like:

(Western, instrumental) “Lara’s theme” (Doctor Zhivago)
Composer: Maurice Jarre

/

# Should I give up on QM?

After further and deeper studies of the Schrodinger formalism, I have now come to understand the exact position from which the physicists must be coming (I mean the couple of physicists with who I discussed the ideas of my new approach, as mentioned here [^])—why they must be raising their objections. I came to really understand their positions only now. Here is how it happened.

I was pursuing finding correspondence between the $3ND$ configuration space of the Schrodinger formalism on the one hand and the $3D$ physical space on the other, when I run into this subtle point which made everything look completely different. That point is the following:

Textbooks (or lecture notes, or lecturers) don’t ever highlight this point (in fact, indirectly, they actually obfuscate it), but I came to realize that even in the $1D$ cases like the QM harmonic oscillator (QHO), the Schrodinger formalism itself remains defined only on an abstract hyperspace—it’s just that in the case of the QHO, this hyperspace happens to be $1D$ in nature, that’s all.

I came to realize that, even in the simplest $1D$ case like the QHO the $x$ variable which appears in the Schrodinger equation does not directly refer to the physical space. In case of QHO, it refers to the change in the equilibrium separation between the centers of the two atoms.

Physicists and textbooks don’t mention this point, and in fact, the way they present QM, they make it look as if $x$ is the simple position variable. But in reality, no it is not. It can be made to look like a position variable (and not a change-in-the-interatomic-distance variable) by fixing the coordinate system to one of the two atoms (i.e. by making it a moving or Lagrangian coordinate system). But doing so leads to losing the symmetry in the motion of the two atoms, and more important, it further results in an obfuscation of the real nature of the issue. Mind you, textbook authors are trying to be helpful here. But unwittingly, they end up actually obfuscating the real story.

So, the $x$ variable whose Laplacian you take for the kinetic energy term also does not represent the physical space—not even in the simplest $1D$ cases like the QHO.

This insight, which I gained only now, has made me realize that I need to rethink through the whole thing once again.

In other words, my understanding of QM turned out to have been faulty—though the fault is much more on the part of the textbook authors (and lecturers) than on the part of someone like me—one who has learnt QM only through self-studies.

One implication of this better understanding now is that the new approach as stated in the Outline document isn’t going to work out. Even if there are a lot of good ideas in it (Only the Coulomb potentials, the specific nonlinearity proposed in the potential energy term, the ideas concerning measurements, etc.), there are several other ideas in that document which are just so weak that I will have to completely revise my entire approach once again.

Can I do that—take up a complete rethinking once again, and still hope to succeed?

Frankly, I don’t know. Not at this point of time anyway.

I still have not given up. But a sense of tiredness has crept in now. It now seems possible—very easily possible—that QM will end up defeating me, too.

But before outright leaving the fight, I would like to give it just one more try. One last try.

So, I have decided that I will “work” on this issue for just a little while more. May be a couple of weeks or so. Say until the month-end (March 2019-end). Unless I make some clearing, some breaththrough, I will not pursue QM beyond this time-frame.

What is going to be my strategy?

The only way an enterprise like mine can work out is if the connection between the $3D$ world of observations and the hyperspace formalism can be put in some kind of a valid conceptual correspondence. (That is to say, not just the measurement postulate but something deeper than that, something right at the level of the basic conceptual correspondence itself).

The only strategy that I will now pursue (before giving up on QM) is this: The Schrodinger formalism is based on the higher-dimensional configuration space not because a physicist like him would go specifically hunting for a higher-dimensional space, but primarily because the formulation of Schrodinger’s theory is based on the ideas from the energetics program, viz., the Leibniz-Lagrange-Euler-Hamilton program, their line(s) of thought.

The one possible opening I can think of as of today is this: The energetics program necessarily implies hyperspaces. However, at least in the classical mechanics, there always is a $1:1$ correspondence between such hyperspaces on the one hand and the $3D$ space on the other. Why should QM be any different? … As far as I am concerned, all the mystification they effected for QM over all these decades still does not supply any reason to believe that QM should necessarily be very different. After all, QM does make predictions about real world as described in $3D$! Why, even the position vectors that go into the potential energy operator $\hat{V}$ are defined only in the $3D$ space. …

… So, naturally, it seems that I just have to understand the nature of the correspondence between the Lagrangian mechanics and the $3D$ mechanics better. There must be some opening in there, based on this idea. In fact my suspicion is stronger: If at all there is a real opening to be found, if at all there is any real way to crack this nutty problem, then its key has to be lying somewhere in this correspondence.

So, I have decided to work on seeing if pursuing this line of thought yields something definitive or not. If it doesn’t, right within the next couple of weeks or so, I think I better throw in the towel and declare defeat.

Now, understanding the energetics program better meant opening up once again the books. But given my style, you know, it couldn’t possibly be the maths books—but only the conceptual ones.

So, this morning, I spent some time opening a couple of the movers-and-packers boxes (in which stuff was still lying as I mentioned before [^]), and also made some space in my room (somehow) by shoving the boxes a bit away to open the wall-cupboard, and brought out a few books I wanted to read  / browse through. Here they are.

The one shown opened is what I had mentioned as “the energetics book” in the background material document (see this link [^] in this post [^]). I am going to begin my last shot at QM—the understanding of the $3ND$$3D$ issue, starting with this book. The others may or may not be helpful, but I wanted to boast that they are just a part of personal library too!

Wish me luck!

(And suggest me a job in Data Science all the same! [Not having a job is the only thing that gets me (really) angry these days—and it does. So there.])

BTW, I really LOL on the Record of 17 off 71. (Just think what happened in 204!)

A song I like:

(Hindi) “O mere dil ke chain…”
Singer: Kishor Kumar
Music: R. D. Burman
Lyrics: Majrooh Sultanpuri

Minor editing to be done and a song to be added, tomorrow. But feel free to read the post right starting today.

Song added on 2019.03.10 12.09 AM IST. Subject to change if I have run it already.

/

# Absolutely Random Notings on QM—Part 3: Links to some (really) interesting material, with my comments

The “pride of place” for this post goes to a link to this book:

Norsen, Travis (2017) “Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Theory,” Springer

This book is (i) the best supplementary book for a self-study of QM, and simultaneously, also (ii) the best text-book on a supplementary course on QM, both at the better-prepared UG / beginning PG level.

A bit expensive though, but extensive preview is available on Google books, here [^]. (I plan to buy it once I land a job.)

I was interested in the material from the first three chapters only, more or less. It was a delight even just browsing through these chapters. I intend to read it more carefully soon enough. But even on the first, rapid browsing, I noticed that several pieces of understanding that I had so painstakingly come to develop (over a period of years) are given quite straight-forwardly here, as if they were a matter of well known facts—even if other QM text-books only cursorily mention them, if at all.

For instance, see the explanation of entanglement here. Norsen begins by identifying that there is a single wavefunction, always—even for a multi-particle system. Then after some explanation, he states: “But, as usual in quantum mechanics, these states do not exhaust the possibilities—instead, they merely form a basis for the space of all possible wave functions. …”… Note the emphasis on the word “basis” which Norsen helpfully puts.

Putting this point (which Norsen discusses with a concrete example), but in my words: There is always a single wavefunction, and for a multi-particle system, its basis is bigger; it consists of the components of the tensor product (formed from the components of the basis of the constituent systems). Sometimes, the single wavefunction for the multi-particle system can be expressed as a result of a single tensor-product (in which case it’s a separable state), and at all other times, only as an algebraic sum of the results of many such tensor-products (in which case they all are entangled states).

Notice how there is no false start of going from two separate systems, and then attempting to forge a single system out of them. Notice how, therefore, there is no hand-waving at one electron being in one galaxy, and another electron in another galaxy, and so on, as if to apologize for the very idea of the separable states. Norsen achieves the correct effect by beginning on the right note: the emphasis on the single wavefunction for the system as a whole to begin with, and then clarifying, at the right place, that what the tensor product gives you is only the basis set for the composite wavefunction.

There are many neat passages like this in the text.

I was about to say that Norsen’s book is the Resnick and Halliday of QM, but then came to hesitate saying so, because I noticed something odd even if my browsing of the book was rapid and brief.

Then I ran into

Ian Durham’s review of Norsen’s book, at the FQXi blog,

which is our link # 2 for this post [^].

Durham helpfully brings out the following two points (which I then verified during a second visit to Norsen’s book): (i) Norsen’s book is not exactly at the UG level, and (ii) the book is a bit partial to Bell’s characterization of the quantum riddles as well as to the Bohmian approach for their resolution.

The second point—viz., Norsen’s fascination for / inclination towards Bell and Bohm (B&B for short)—becomes important only because the book is, otherwise, so good: it carries so many points that are not even passingly mentioned in other QM books, is well written (in a conversational style, as if a speech-to-text translator were skillfully employed), easy to understand, thorough, and overall (though I haven’t read even 25% of it, from whatever I have browsed), it otherwise seems fairly well balanced.

It is precisely because of these virtues that you might come out giving more weightage to the B&B company than is actually due to them.

Keep that warning somewhere at the back of your mind, but do go through the book anyway. It’s excellent.

At Amazon, it has got 5 reader reviews, all with 5 stars. If I were to bother doing a review there, I too perhaps would give it 5 stars—despite its shortcomings/weaknesses. OK. At least 4 stars. But mostly 5 though. … I am in an indeterminate state of their superposition.

… But mark my words. This book will have come to shape (or at least to influence) every good exposition of (i.e. introduction to) the area of the Foundations of QM, in the years to come. [I say that, because I honestly don’t expect a better book on this topic to arrive on the scene all that soon.]

Which brings us to someone who wouldn’t assign the $|4\rangle + |5\rangle$ stars to this book. Namely, Lubos Motl.

If Norsen has moved in the Objectivist circles, and is partial to the B&B company, Motl has worked in the string theory, and is not just partial to it but even today defends it very vigorously—and oddly enough, also looks at that “supersymmetric world from a conservative viewpoint.” More relevant to us: Motl is not partial to the Copenhagen interpretation; he is all the way into it. … Anyway, being merely partial is something you wouldn’t expect from Motl, would you?

But, of course, Motl also has a very strong grasp of QM, and he displays it well (even powerfully) when he writes a post of the title:

“Postulates of quantum mechanics almost directly follow from experiments.” [^]

Err… Why “almost,” Lubos? 🙂

… Anyway, go through Motl’s post, even if you don’t like the author’s style or some of his expressions. It has a lot of educational material packed in it. Chances are, going through Motl’s posts (like the present one) will come to improve your understanding—even if you don’t share his position.

As to me: No, speaking from the new understanding which I have come to develop regarding the foundations of QM [^] and [^], I don’t think that all of Motl’s objections would carry. Even then, just for the sake of witnessing the tight weaving-in of the arguments, do go through Motl’s post.

Finally, a post at the SciAm blog:

“Coming to grips with the implications of quantum mechanics,” by Bernardo Kastrup, Henry P. Stapp, and Menas C. Kafatos, [^].

The authors say:

“… Taken together, these experiments [which validate the maths of QM] indicate that the everyday world we perceive does not exist until observed, which in turn suggests—as we shall argue in this essay—a primary role for mind in nature.”

No, it didn’t give me shivers or something. Hey, this is QM and its foundations, right? I am quite used to reading such declarations.

Except that, as I noted a few years ago on Scott Aaronson’s blog [I need to dig up and insert the link here], and then, recently, also at

Roger Schlafly’s blog [^],

you don’t need QM in order to commit the error of inserting consciousness into a physical theory. You can accomplish exactly the same thing also by using just the Newtonian particle mechanics in your philosophical arguments. Really.

Yes, I need to take that reply (at Schlafly’s blog), edit it a bit and post it as a separate entry at this blog. … Some other time.

For now, I have to run. I have to continue working on my approach so that I am able to answer the questions raised and discussed by people such as those mentioned in the links. But before that, let me jot down a general update.

A general update:

Oh, BTW, I have taken my previous QM-related post off the top spot.

That doesn’t mean anything. In particular, it doesn’t mean that after reading into materials such as that mentioned here, I have found some error in my approach or something like that. No. Not at all.

All it means is that I made it once again an ordinary post, not a sticky post. I am thinking of altering the layout of this blog, by creating a page that highlights that post, as well as some other posts.

But coming back to my approach: As a matter of fact, I have also written emails to a couple of physicists, one from IIT Bombay, and another from IISER Pune. However, things have not worked out yet—things like arranging for an informal seminar to be delivered by me to their students, or collaborating on some QM-related simulations together. (I could do the simulations on my own, but for the seminar, I would need an audience! One of them did reply, but we still have to shake our hands in the second round.)

In the meanwhile, I go jobless, but I keep myself busy. I am preparing a shortish set of write-ups / notes which could be used as a background material when (at some vague time in future) I go and talk to some students, say at IIT Bombay/IISER Pune. It won’t be comprehensive. It will be a little more than just a white-paper, but you couldn’t possibly call it even just the preliminary notes for my new approach. Such preliminary notes would come out only after I deliver a seminar or two, to physics professors + students.

At the time of delivering my proposed seminar, links like those I have given above, esp. Travis Norsen’s book, also should prove a lot useful.

But no, I haven’t seen something like my approach being covered anywhere, so far, not even Norsen’s book. There was a vague mention of just a preliminary part of it somewhere on Roger Schlafly’s blog several years ago, only once or so, but I can definitely say that I had already had grasped even that point on my own before Schlafly’s post came. And, as far as I know, Schlafly hasn’t come to pursue that thread at all, any time later…

But speaking overall, at least as of today, I think I am the only one who has pursued this (my) line of thought to the extent I have [^].

So, there. Bye for now.

I Song I Like:
(Hindi) “suno gajar kya gaaye…”
Singer: Geeta Dutt
Music: S. D. Burman
Lyrics: Sahir Ludhianvi
[There are two Geeta’s here, and both are very fascinating: Geeta Dutt in the audio, and Geeta Bali in the video. Go watch it; even the video is recommended.]

As usual, some editing after even posting, would be inevitable.

Some updates made and some streamlining done on 30 July 2018, 09:10 hrs IST.

Here are a few interesting links I browsed recently, listed in no particular order:

“Mathematicians Tame Turbulence in Flattened Fluids” [^].

The operative word here, of course, is: “flattened.” But even then, it’s an interesting read. Another thing: though the essay is pop-sci, the author gives the Navier-Stokes equations, complete with fairly OK explanatory remarks about each term in the equation.

(But I don’t understand why every pop-sci write-up gives the NS equations only in the Lagrangian form, never Eulerian.)

“A Twisted Path to Equation-Free Prediction” [^]. …

“Empirical dynamic modeling.” Hmmm….

“Machine Learning’s Amazing’ Ability to Predict Chaos” [^].

Click-bait: They use data science ideas to predict chaos!

8 Lyapunov times is impressive. But ignore the other, usual kind of hype: “…the computer tunes its own formulas in response to data until the formulas replicate the system’s dynamics. ” [italics added.]

“Your Simple (Yes, Simple) Guide to Quantum Entanglement” [^].

Click-bait: “Entanglement is often regarded as a uniquely quantum-mechanical phenomenon, but it is not. In fact, it is enlightening, though somewhat unconventional, to consider a simple non-quantum (or “classical”) version of entanglement first. This enables us to pry the subtlety of entanglement itself apart from the general oddity of quantum theory.”

Don’t dismiss the description in the essay as being too simplistic; the author is Frank Wilczek.

“A theoretical physics FAQ” [^].

Click-bait: Check your answers with those given by an expert! … Do spend some time here…

Tensor product versus Cartesian product.

If you are engineer and if you get interested in quantum entanglement, beware of the easily confusing terms: The tensor product and the Cartesian product.

The tensor product, you might think, is like the Cartesian product. But it is not. See mathematicians’ explanations. Essentially, the basis sets (and the operations) are different. [^] [^].

But what the mathematicians don’t do is to take some simple but non-trivial examples, and actually work everything out in detail. Instead, they just jump from this definition to that definition. For example, see: “How to conquer tensorphobia” [^] and “Tensorphobia and the outer product”[^]. Read any of these last two articles. Any one is sufficient to give you tensorphobia even if you never had it!

You will never run into a mathematician who explains the difference between the two concepts by first directly giving you a vague feel: by directly giving you a good worked out example in the context of finite sets (including enumeration of all the set elements) that illustrates the key difference, i.e. the addition vs. the multiplication of the unit vectors (aka members of basis sets).

A third-class epistemology when it comes to explaining, mathematicians typically have.

A Song I Like:

(Marathi) “he gard niLe megha…”
Music: Rushiraj
Lyrics: Muralidhar Gode

[As usual, a little streamlining may occur later on.]

# QM: A couple of defensible statements. Also, a bit on their implications for the QC.

A Special Note (added on 17th June 2018): This post is now a sticky post; it will remain, for some time, at the top of this blog.

I am likely to keep this particular post at the top of this blog, as a sticky post, for some time in the future (may be for a few months or so). So, even if posts at this blog normally appear in the reverse chronological order, any newer entries that I may post after this one would be found below this one.

[In particular, right now, I am going through a biography: “Schrodinger: Life and Thought” by Walter Moore [^]. I had bought this book way back in 2011, but had to keep it aside back then, and then, somehow, I came to forget all about it. The book surfaced during a recent move we made, and thus, I began reading it just this week. I may write a post or two about it in the near future (say within a couple of weeks or so) if something strikes me while I am at it.]

A Yawningly Long Preamble:

[Feel free to skip to the sections starting with the “Statement 1” below.]

As you know, I’ve been thinking about foundations of QM for a long, long time, a time running into decades by now.

I thought a lot about it, and then published a couple of papers during my PhD, using a new approach which I had developed. This approach was used for resolving the wave-particle duality, but only in the context of photons. However, I then got stuck when it came to extending and applying this same approach to electrons. So, I kept on browsing a lot of QM-related literature in general. Then, I ran, notably, into the Nobel laureate W. E. Lamb’s “anti-photon” paper [^], and also the related literature (use Google Scholar). I thought a lot about this paper—and also about QM. I began thinking about QM once again from the scratch, so to speak.

Eventually, I came to abandon my own PhD-time approach. At the same time, with some vague but new ideas already somewhere at the back of my mind, I once again started studying QM, once again with a fresh mind, but this time around much more systematically. …

… In the process, I came to develop a completely new understanding of QM!… It’s been at least months since I began talking about it [^]. … My confidence in this new understanding has only increased, since then.

Today’s post will be based on this new understanding. (I could call it a new theory, perhaps.)

My findings suggest a few conclusions which I think I should not hold back any longer. Hence this post.

I have been trying to locate the right words for formulating my conclusions—but without much satisfaction. Finally, I’ve decided to go ahead and post an entry here anyway, regardless of whether the output comes out as being well formulated or not.

In other words, don’t try to pin me down with the specific words I use here in this post! Instead, try to understand what I am trying to get at. In still other words: the particular words I use may change, but the intended meaning will, from now on, “always” remain the same—ummm…. more or less the same!

OK, so here are the statements I am making today. I think they are well defensible:

Notation:
QM: Quantum Mechanics, quantum mechanically, etc.
CM: Classical Mechanics
QC: Quantum Computer
QS: Quantum Supremacy ([^] and [^])

Statement 1: It is possible to explain all quantum mechanical phenomena on the basis of those principles which are already known (or have already been developed) in the context of classical mechanics.

Informal Explanation 1.1: Statement 1 holds true. It’s just that when it comes to explaining the QM phenomena (i.e., when it comes to supplying a physical mechanism for QM), even if the principles do remain the same, the way they are to be combined and applied is different. These differences basically arise because of a reason mentioned in the next Informal Explanation.

Informal Explanation 1.2: Yes, the tradition of 80+ years, involving an illustrious string of Nobel laureates and others, is, in a way, “wrong.” The QM principles are not, fundamentally speaking, very different from those encountered in the CM. It’s just that some of the objects that QM assumes and talks about are different (only partly different) from those assumed in the CM.

Corollary 1 of Statement 1: A quantum computer could “in principle” be built as an “application layer” on top of the “OS platform” supplied by the classical mechanics.

Informal Explanation 1.C1.1: Hierarchically speaking, QM remains the most fundamental or the “ground” layer. The aspects of the physical reality that CM refers to, therefore, indeed are at a layer lying on top of QM. This part does continue to remain the same.

However, what the Corollary 1 now says is that you can also completely explain the workings of QM in terms of a virtual QM machine that is built on top of the well-known principles of CM.

If someone builds a QC on such a basis (which would be a virtual QC on top of CM), then it would be just a classical mechanically functioning simulator—an analog simulator, I should add—that simulates the QM phenomena.

Informal Explanation 1.C1.2: The phrase “in principle” does not always translate into “easily.” In this case, it in factt is very easily possible that building a big enough a QC of this kind (i.e. the simulating QC) may very well turn out to be an enterprise that is too difficult to be practically feasible.

Corollary 2 of Statement 1: A classical system can be designed in such a way that it shows all the features of the phenomenon of quantum entanglement (when the classical system is seen from an appropriately high-level viewpoint).

Informal Explanation 1.C2.1: There is nothing “inherently quantum-mechanical” about entanglement. The well-known principles of CM are enough to explain the phenomena of entanglement.

Informal Explanation 1.C2.2: We use our own terms. In particular, when we say “classical mechanics,” we do not mean these words in the same sense in which a casual reader of the QM literature, e.g. of Bell’s writings, may read them.

What we mean by “classical mechanics” is the same as what an engineer who has never studied QM proper means, when he says “classical mechanics” (i.e., the Newtonian mechanics + the Lagrangian and Hamiltonian reformulations including variational principles, as well as the more modern developments such as studies of nonlinear systems and the catastrophe theory).

Statement 2: It can be shown that even if the Corollary 1 above does hold true, the kind of quantum computer it refers to would be such that it will not be able to break a sufficiently high-end RSA encryption (such as what is used in practice today, at the high-end).

Aside 2.1: I wouldn’t have announced Statement 1 unless I was sure—absolutely goddamn sure, in fact—about the Statement 2. In fact, I must have waited for at least half a year just to make sure about this aspect, looking at these things from this PoV, then from that PoV, etc.

Statement 3: Inasmuch as the RSA-beating QC requires a controlled entanglement over thousands of qubits, it can be said, on the basis of the new understanding (the one which lies behind the Statement 1 above), that the goal of achieving even “just” the quantum supremacy seems highly improbable, at least in any foreseeable future, let alone achieving the goal of breaking the high-end RSA encryption currently in use. However, proving these points, esp. that the currently employed higher-end RSA cannot be broken, will require further development of the new theory, particularly a quantitative theory for the mechanism(s) involved in the quantum mechanical measurements.

Informal Explanation 3.1: A lot of funding has already gone into attempts to build a QC. Now, it seems that the US government, too, is considering throwing some funds at it.

The two obvious goal-posts for a proper QC are: (i) first gaining enough computational power to run past the capabilities of the classical digital computers, i.e., achieving the so-called “quantum supremacy,” and then, (ii) breaking the RSA encryption as is currently used in the real-world at the high-end.

The question of whether the QC-related researches will be able to achieve these two goals or not depends on the question of whether there are natural reasons/causes which might make it highly improbable (if not outright impossible) to achieve these two goals.

We have already mentioned that it can be shown that it will not be possible for a classical (analog) quantum simulator (of the kind we have in mind) to break the RSA encryption.

• Combination 1: CM-based QC Simulator that is able to break the RSA encryption.

We have said that it can be shown (i.e. proved) that the above combination would be impossible to have. (The combination is that extreme.)

However, it still leaves open 3 more combinations of a QC and a goal-post:

• Combination 2: CM-based QC Simulator that exceeds the classical digital computer
• Combination 3: Proper QC (working directly off the QM platform) that exceeds the classical digital computer
• Combination 4: Proper QC (working directly off the QM platform) that is able to break the RSA encryption.

As of today, a conclusive statement cannot be made regarding the last three combinations, not even on the basis of my newest approach to the quantum phenomena, because the mathematical aspects which will help settle questions of this kind, have not yet been developed (by me).

Chances are good that such a theory could be developed, at least in somewhat partly-qualitative-and-partly-quantitative terms, or in terms of some quantitative models that are based on some good analogies, sometime in the future (say within a decade or so). It is only when such developments do occur that we will be able to conclusively state something one way or the other in respect of the last three combinations.

However, relying on my own judgment, I think that I can safely state this much right away: The remaining three combinations would be tough, very tough, to achieve. The last combination, in particular, is best left aside, because the combination is far too complex that it can pose any real threat, at least as of today. I can say this much confidently—based on my new approach. (If you have some other basis to feel confident one way or the other, kindly supply the physical mechanism for the same, please, not just “math.”)

So, as of today, the completely defensible statements are the Statement No. 1 and 2 (with all their corollaries), but not the Statement 3. However, a probabilistic judgment for the Statement 3 has also been given.

A short (say, abstract-like) version:

A physical mechanism to explain QM phenomena has been developed, at least in the bare essential terms. It may perhaps become possible to use such a knowledge to build an analog simulator of a quantum computer. Such a simulator would be a machine based only on the well-known principles of classical mechanics, and using the kind of physical objects that the classical mechanics studies.

However, it can also be easily shown that such a simulator will not be able to break the RSA encryption using algorithm such as Shor’s. The proof rests on an idealized abstraction of classical objects (just the way the ideal fluid is an abstraction of real fluids).

On the basis of the new understanding, it becomes clear that trying to break RSA encryption using a QC proper (i.e. a computer that’s not just a simulator, but is a QC proper that directly operates at the level of the QM platform itself) would be a goal that is next to impossible to achieve. In fact, even achieving just the “quantum supremacy” (i.e., beating the best classical digital computer) itself can be anticipated, on the basis of the new understanding, as a goal that would be very tough to achieve, if at all.

Researches that attempt to build a proper QC may be able to bring about some developments in various related areas such as condensed matter physics, cryogenics, electronics, etc. But it is very highly unlikely that they would succeed in achieving the goal of quantum supremacy itself, let alone the goal of breaking the RSA encryption as it is deployed at the high-end today.

A Song I Like:

(Hindi) “dilbar jaani, chali hawaa mastaanee…”
Music: Laxmikant Pyarelal
Singers: Kishore Kumar, Lata Mangeshkar
Lyrics: Anand Bakshi

PS: Note that, as is usual at this blog, an iterative improvement of the draft is always a possibility. Done.

Revision History:

1. First posted on 2018.06.15, about 12:35 hrs IST.
2. Considerably revised the contents on 2018.06.15, 18:41 hrs IST.
3. Edited to make the contents better on 2018.06.16, 15:30 hrs IST. Now, am mostly done with this post except, may be, for a minor typo or so, here or there.
4. Edited (notably, changed the order of the Combinations) on 2018.06.17, 23:50 hrs IST. Also corrected some typos and streamlined the content. Now, I am going to leave this post in the shape it is. If you find some inconsistency or so, simple! Just write a comment or shoot me an email.
5. 2018.06.27 02:07 hrs IST. Changed the song section.

# Yes I know it!—Part 2

This post directly continues from my last post. The content here was meant to be an update to my last post, but it grew, and so, I am noting it down as a separate post in its own right.

Thought about it [I mean my last post] a lot last night and this morning. I think here is a plan of action I can propose:

I can deliver a smallish, informally conducted, and yet, “official” sort of a seminar/talk/guest lecture, preferably at an IIT/IISER/IISc/similar institute. No honorarium is expected; just arrange for my stay. (That too is not necessary if it will be IIT Bombay; I can then stay with my friend; he is a professor in an engineering department there.)

Once arranged by mutual convenience, I will prepare some lecture notes (mostly hand-written), and deliver the content. (I guess at this stage, I will not prepare Beamer slides, though I might include some audio-visual content such as simulations etc.)

Questions will be OK, even encouraged, but the format will be that of a typical engineering class-room lecture. Discussions would be perfectly OK, but only after I finish talking about the “syllabus” first.

The talk should preferably be attended also by a couple of PhD students or so (of physics/engineering physics/any really relevant discipline, whether it’s acknowledged as such by UGC/AICTE or not). They should separately take down their notes and show me these later. This will help me understand where and how I should modify my notes. I will then myself finalize my notes, perhaps a few days after the talk, and send these by email. At that stage, I wouldn’t mind posting the notes getting posted on the ‘net.

Guess I will think a bit more about it, and note about my willingness to deliver the talk also at iMechanica. The bottom-line is that I am serious about this whole thing.

A few anticipated questions and their answers (or clarifications):

1. What I have right now is, I guess, sufficient to stake a claim. But I have not taken the research to sufficiently advanced stage that I can say that I have all the clarifications worked out. It’s far more than just a sketchy conceptual idea, and does have a lot of maths too, but it’s less than, say, a completely worked out (or series of) mathematical theory. (My own anticipation is that if I can do just a series of smaller but connected mathematical models/simulations, it should be enough as my personal contribution to this new approach.)
2. No, as far as QM is concerned, the approach I took in my PhD time publications is not at all relevant. I have completely abandoned that track (I mean to say as far as QM is concerned).
3. However, my PhD time research on the diffusion equation has been continuing, and I am happy to announce that it has by now reached such a certain stage of maturation/completion that I should be writing another paper(s) on it any time now. I am happy that something new has come out of some 10+ years of thought on that issue, after my PhD-time work. Guess I could now send the PhD time conference paper to a journal, and then cover the new developments in this line in continuation with that one.
4. Coming back to QM: Any one else could have easily got to the answers I have. But no, to the best of my knowledge, none else actually has. However, it does seem to me now that time is becoming ripe, and not to stake a claim at least now could be tantamount to carelessness on my part.
5. Yes, my studies of philosophy, especially Ayn Rand’s ITOE (and Peikoff’s explanations of that material in PO and UO) did help me a lot, but all that is in a more general sense. Let me put it this way: I don’t think that I would have had to know (or even plain be conversant with) ITOE to be able to formulate these new answers to the QM riddles. And certainly, ITOE wouldn’t at all be necessary to understand my answers; the general level of working epistemology still is sufficiently good in physics (and more so, in engineering) even today.  At the same time, let me tell you one thing: QM is very vast, general, fundamental, and abstract. I guess you would have to be a “philosophizing” sort of a guy. Only then could you find this continuous and long preoccupation with so many deep and varied abstractions, interesting enough. Only then could the foundations of QM interest you. Not otherwise.
6. To formulate answers, my natural proclivity to have to keep on looking for “physical” processes/mechanisms/objects for every mathematical idea I encounter, did help. But you wouldn’t have to have the same proclivity, let alone share my broad convictions, to be able to understand my answers. In other words, you could be a mathematical Platonist, and yet very easily come to understand the nature of my answers (and perhaps even come to agree with my positions)!
7. To arrange for my proposed seminar/talk is to agree to be counted as a witness (for any future issues related to priority). But that’s strictly in the usual, routine, day-to-day academic sense of the term. (To wit, see how people interact with each other at a journal club in a university, or, say, at iMechanica.)
8. But, to arrange for my talk is not to be willing to certify or validate its content. Not at all.
9. With that being said, since this is India, let me also state a relevant concern. Don’t call me over just to show me down or ridicule me either. (It doesn’t happen in seminar talks, but it does happen during job interviews in Pune. It did happen to me in my COEP interview. It got repeated, in a milder way, in other engineering colleges in SPPU (the Pune University). So I have no choice but to note this part separately.)
10. Once again, the issue is best clarified by giving the example. Check out how people treated me at iMechanica. If you are at an IIT/IISc/similar institute/university and are willing to treat me similarly, then do think of calling me over.

More, may be later. I will sure note my willingness to deliver a seminar at an IIT (or at a good University department) or so, at iMechanica also, soon enough. But right now I don’t have the time, and have to rush out. So let me stop here. Bye for now, and take care… (I would add a few more tags to the post-categories later on.)

# Yes I know it!

Note: A long update was posted on 12th December 2017, 11:35 IST.

This post is spurred by my browsing of certain twitter feeds of certain pop-sci. writers.

The URL being highlighted—and it would be, say, “negligible,” but for the reputation of the Web domain name on which it appears—is this: [^].

I want to remind you that I know the answers to all the essential quantum mysteries.

Not only that, I also want to remind you that I can discuss about them, in person.

It’s just that my circumstances—past, and present (though I don’t know about future)—which compel me to say, definitely, that I am not available for writing it down for you (i.e. for the layman) whether here or elsewhere, as of now. Neither am I available for discussions on Skype, or via video conferencing, or with whatever “remoting” mode you have in mind. Uh… Yes… WhatsApp? Include it, too. Or something—anything—like that. Whether such requests come from some millionaire Indian in USA (and there are tons of them out there), or otherwise. Nope. A flat no is the answer for all such requests. They are out of question, bounds… At least for now.

… Things may change in future, but at least for the time being, the discussions would have to be with those who already have studied (the non-relativistic) quantum physics as it is taught in universities, up to graduate (PhD) level.

And, you have to have discussions in person. That’s the firm condition being set (for the gain of their knowledge 🙂 ).

Just wanted to remind you, that’s all!

Update on 12th December 2017, 11:35 AM IST:

I have moved the update to a new post.

A Song I Like:

(Western, Instrumental) “Berlin Melody”
Credits: Billy Vaughn

[The same 45 RPM thingie [as in here [^], and here [^]] . … I was always unsure whether I liked this one better or the “Come September” one. … Guess, after the n-th thought, that it was this one. There is an odd-even thing about it. For odd ‘n” I think this one is better. For even ‘n’, I think the “Come September” is better.

… And then, there also are a few more musical goodies which came my way during that vacation, and I will make sure that they find their way to you too….

Actually, it’s not the simple odd-even thing. The maths here is more complicated than just the binary logic. It’s an n-ary logic. And, I am “equally” divided among them all. (4+ decades later, I still remain divided.)… (But perhaps the “best” of them was a Marathi one, though it clearly showed a best sort of a learning coming from also the Western music. I will share it the next time.)]

[As usual, may be, another revision [?]… Is it due? Yes, one was due. Have edited streamlined the main post, and then, also added a long update on 12th December 2017, as noted above.]

# Blog-Filling—Part 3

Note: A long Update was added on 23 November 2017, at the end of the post.

Today I got just a little bit of respite from what has been a very tight schedule, which has been running into my weekends, too.

But at least for today, I do have a bit of a respite. So, I could at least think of posting something.

But for precisely the same reason, I don’t have any blogging material ready in the mind. So, I will just note something interesting that passed by me recently:

1. Catastrophe Theory: Check out Prof. Zhigang Suo’s recent blog post at iMechanica on catastrophe theory, here [^]; it’s marked by Suo’s trademark simplicity. He also helpfully provides a copy of Zeeman’s 1976 SciAm article, too. Regular readers of this blog will know that I am a big fan of the catastrophe theory; see, for instance, my last post mentioning the topic, here [^].
2. Computational Science and Engineering, and Python: If you are into computational science and engineering (which is The Proper And The Only Proper long-form of “CSE”), and wish to have fun with Python, then check out Prof. Hans Petter Langtangen’s excellent books, all under Open Source. Especially recommended is his “Finite Difference Computing with PDEs—A Modern Software Approach” [^]. What impressed me immediately was the way the author begins this book with the wave equation, and not with the diffusion or potential equation as is the routine practice in the FDM (or CSE) books. He also provides the detailed mathematical reason for his unusual choice of ordering the material, but apart from his reason(s), let me add in a comment here: wave $\Rightarrow$ diffusion $\Rightarrow$ potential (Poisson-Laplace) precisely was the historical order in which the maths of PDEs (by which I mean both the formulations of the equations and the techniques for their solutions) got developed—even though the modern trend is to reverse this order in the name of “simplicity.” The book comes with Python scripts; you don’t have to copy-paste code from the PDF (and then keep correcting the errors of characters or indentations). And, the book covers nonlinearity too.
3. Good Notes/Teachings/Explanations of UG Quantum Physics: I ran across Dan Schroeder’s “Entanglement isn’t just for spin.” Very true. And it needed to be said [^]. BTW, if you want a more gentle introduction to the UG-level QM than is presented in Allan Adam (et al)’s MIT OCW 8.04–8.06 [^], then make sure to check out Schroeder’s course at Weber [^] too. … Personally, though, I keep on fantasizing about going through all the videos of Adam’s course and taking out notes and posting them at my Web site. [… sigh]
4. The Supposed Spirituality of the “Quantum Information” Stored in the “Protein-Based Micro-Tubules”: OTOH, if you are more into philosophy of quantum mechanics, then do check out Roger Schlafly’s latest post, not to mention my comment on it, here [^].

The point no. 4. above was added in lieu of the usual “A Song I Like” section. The reason is, though I could squeeze in the time to write this post, I still remain far too rushed to think of a song—and to think/check if I have already run it here or not. But I will try add one later on, either to this post, or, if there is a big delay, then as the next “blog filler” post, the next time round.

[Update on 23 Nov. 2017 09:25 AM IST: Added the Song I Like section; see below]

OK, that’s it! … Will catch you at some indefinite time in future here, bye for now and take care…

A Song I Like:

(Western, Instrumental) “Theme from ‘Come September'”
Credits: Bobby Darin (?) [+ Billy Vaughn (?)]

[I grew up in what were absolutely rural areas in Maharashtra, India. All my initial years till my 9th standard were limited, at its upper end in the continuum of urbanity, to Shirpur, which still is only a taluka place. And, back then, it was a decidedly far more of a backward + adivasi region. The population of the main town itself hadn’t reached more than 15,000 or so by the time I left it in my X standard; the town didn’t have a single traffic light; most of the houses including the one we lived in) were load-bearing structures, not RCC; all the roads in the town were of single lanes; etc.

Even that being the case, I happened to listen to this song—a Western song—right when I was in Shirpur, in my 2nd/3rd standard. I first heard the song at my Mama’s place (an engineer, he was back then posted in the “big city” of the nearby Jalgaon, a district place).

As to this song, as soon as I listened to it, I was “into it.” I remained so for all the days of that vacation at Mama’s place. Yes, it was a 45 RPM record, and the permission to put the record on the player and even to play it, entirely on my own, was hard won after a determined and tedious effort to show all the elders that I was able to put the pin on to the record very carefully. And, every one in the house was an elder to me: my siblings, cousins, uncle, his wife, not to mention my parents (who were the last ones to be satisfied). But once the recognition arrived, I used it to the hilt; I must have ended up playing this record for at least 5 times for every remaining day of the vacation back then.

As far as I am concerned, I am entirely positive that appreciation for a certain style or kind of music isn’t determined by your environment or the specific culture in which you grow up.

As far as songs like these are concerned, today I am able to discern that what I had immediately though indirectly grasped, even as a 6–7 year old child, was what I today would describe as a certain kind of an “epistemological cleanliness.” There was a clear adherence to certain definitive, delimited kind of specifics, whether in terms of tones or rhythm. Now, it sure did help that this tune was happy. But frankly, I am certain, I would’ve liked a “clean” song like this one—one with very definite “separations”/”delineations” in its phrases, in its parts—even if the song itself weren’t to be so directly evocative of such frankly happy a mood. Indian music, in contrast, tends to keep “continuity” for its own sake, even when it’s not called for, and the certain downside of that style is that it leads to a badly mixed up “curry” of indefinitely stretched out weilings, even noise, very proudly passing as “music”. (In evidence: pick up any traditional “royal palace”/”kothaa” music.) … Yes, of course, there is a symmetrical downside to the specific “separated” style carried by the Western music too; the specific style of noise it can easily slip into is a disjointed kind of a noise. (In evidence, I offer 90% of Western classical music, and 99.99% of Western popular “music”. As to which 90%, well, we have to meet in person, and listen to select pieces of music on the fly.)

Anyway, coming back to the present song, today I searched for the original soundtrack of “Come September”, and got, say, this one [^]. However, I am not too sure that the version I heard back then was this one. Chances are much brighter that the version I first listened to was Billy Vaughn’s, as in here [^].

… A wonderful tune, and, as an added bonus, it never does fail to take me back to my “salad days.” …

… Oh yes, as another fond memory: that vacation also was the very first time that I came to wear a T-shirt; my Mama had gifted it to me in that vacation. The actual choice to buy a T-shirt rather than a shirt (+shorts, of course) was that of my cousin sister (who unfortunately is no more). But I distinctly remember she being surprised to learn that I was in no mood to have a T-shirt when I didn’t know what the word meant… I also distinctly remember her assuring me using sweet tones that a T-shirt would look good on me! … You see, in rural India, at least back then, T-shirts weren’t heard of; for years later on, may be until I went to Nasik in my 10th standard, it would be the only T-shirt I had ever worn. … But, anyway, as far as T-shirts go… well, as you know, I was into software engineering, and so….

Bye [really] for now and take care…]

# What am I thinking about? …and what should it be?

It’s the “derivation” of the Schrodinger equation. Here’s how a simplest presentation of it goes:

The kinetic energy $T$ of a massive particle is given, in classical mechanics, as
$T = \dfrac{1}{2}mv^2 = \dfrac{p^2}{2m}$
where $v$ is the velocity, $m$ is the mass, and $p$ is the momentum. (We deal with only the scalar magnitudes, in this rough-and-ready “analysis.”)

If the motion of the particle occurs additionally also under the influence of a potential field $V$, then its total energy $E$ is given by:
$E = T + V = \dfrac{p^2}{2m} + V$

In classical electrodynamics, it can be shown that for a light wave, the following relation holds:
$E = pc$
where $E$ is the energy of light, $p$ is its momentum, and $c$ is its speed. Further, for light in vacuum:
$\omega = ck$
where $k = \frac{2\pi}{\lambda}$ is the wavevector.

Planck hypothesized that in the problem of the cavity radiation, the energy-levels of the electromagnetic oscillators in the metallic cavity walls maintained at thermal equilibrium are quantized, somehow:
$E = h \nu = \hbar \omega$
where $\hbar = \frac{h}{2\pi}$  and $\omega = 2 \pi \nu$ is the angular frequency. Making this vital hypothesis, he could successfully predict the power spectrum of the cavity radiation (getting rid of the ultraviolet catastrophe).

In explaining the photoelectric effect, Einstein hypothesized that lights consists of massless particles. He took Planck’s relation $E = \hbar \omega$ as is, and then, substituted on its left hand-side the classical expression for the energy of the radiation $E = pc$. On the right hand-side he substituted the relation which holds for light in vacuum, viz. $\omega = c k$. He thus arrived at the expression for the quantized momentum for the hypothetical particles of light:
$p = \hbar k$
With the hypothesis of the quanta of light, he successfully explained all the known experimentally determined features of the photoelectric effect.

Whereas Planck had quantized the equilibrium energy of the charged oscillators in the metallic cavity wall, Einstein quantized the electromagnetic radiation within the cavity itself, via spatially discrete particles of light—an assumption that remains questionable till this day (see “Anti-photon”).

Bohr hypothesized a planetary model of the atom. It had negatively charged and massive point particles of electrons orbiting around the positively charged and massive, point-particles of the nucleus. The model carried a physically unexplained feature of the stationary of the electronic orbits—i.e. the orbits travelling in which an electron, somehow, does not emit/absorb any radiation, in contradiction to the classical electrodynamics. However, this way, Bohr could successfully predict the hydrogen atom spectra. (Later, Sommerfeld made some minor corrections to Bohr’s model.)

de Broglie hypothesized that the relations $E = \hbar \omega$ and $p = \hbar k$ hold not only just for the massless particles of light as proposed by Einstein, but, by analogy, also for the massive particles like electrons. Since light had both wave and particle characters, so must, by analogy, the electrons. He hypothesized that the stationarity of the Bohr orbits (and the quantization of the angular momentum for the Bohr electron) may be explained by assuming that matter waves associated with the electrons somehow form a standing-wave pattern for the stationary orbits.

Schrodinger assumed that de Broglie’s hypothesis for massive particles holds true. He generalized de Broglie’s model by recasting the problem from that of the standing waves in the (more or less planar) Bohr orbits, to an eigenvalue problem of a differential equation over the entirety of space.

The scheme of  the “derivation” of Schrodinger’s differential equation is “simple” enough. First assuming that the electron is a complex-valued wave, we work out the expressions for its partial differentiations in space and time. Then, assuming that the electron is a particle, we invoke the classical expression for the total energy of a classical massive particle, for it. Finally, we mathematically relate the two—somehow.

Assume that the electron’s state is given by a complex-valued wavefunction having the complex-exponential form:
$\Psi(x,t) = A e^{i(kx -\omega t)}$

Partially differentiating twice w.r.t. space, we get:
$\dfrac{\partial^2 \Psi}{\partial x^2} = -k^2 \Psi$
Partially differentiating once w.r.t. time, we get:
$\dfrac{\partial \Psi}{\partial t} = -i \omega \Psi$

Assume a time-independent potential. Then, the classical expression for the total energy of a massive particle like the electron is:
$E = T + V = \dfrac{p^2}{2m} + V$
Note, this is not a statement of conservation of energy. It is merely a statement that the total energy has two and only two components: kinetic energy, and potential energy.

Now in this—classical—equation for the total energy of a massive particle of matter, we substitute the de Broglie relations for the matter-wave, viz. the relations $E = \hbar \omega$ and $p = \hbar k$. We thus obtain:
$\hbar \omega = \dfrac{\hbar^2 k^2}{2m} + V$
which is the new, hybrid form of the equation for the total energy. (It’s hybrid, because we have used de Broglie’s matter-wave postulates in a classical expression for the energy of a classical particle.)

Multiply both sides by $\Psi(x,t)$ to get:
$\hbar \omega \Psi(x,t) = \dfrac{\hbar^2 k^2}{2m}\Psi(x,t) + V(x)\Psi(x,t)$

Now using the implications for $\Psi$ obtained via its partial differentiations, namely:
$k^2 \Psi = - \dfrac{\partial^2 \Psi}{\partial x^2}$
and
$\omega \Psi = i \dfrac{\partial \Psi}{\partial t}$
and substituting them into the hybrid equation for the total energy, we get:
$i \hbar \dfrac{\partial \Psi(x,t)}{\partial t} = - \dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi(x,t)}{\partial x^2} + V(x)\Psi(x,t)$

That’s what the time-dependent Schrodinger equation is.

And that—the “derivation” of the Schrodinger equation thus presented—is what I have been thinking of.

Apart from the peculiar mixture of the wave and particle paradigms followed in this “derivation,” the other few points, to my naive mind, seem to be: (i) the use of a complex-valued wavefunction, (ii) the step of multiplying the hybrid equation for the total energy, by this wavefunction, and (iii) the step of replacing $\omega \Psi(x,t)$ by $i \dfrac{\partial \Psi}{\partial t}$, and also replacing $k^2 \Psi$ by $- \dfrac{\partial^2 \Psi}{\partial x^2}$. Pretty rare, that step seems like, doesn’t it? I mean to say, just because it is multiplied by a variable, you are replacing a good and honest field variable by a partial time-derivative (or a partial space-derivative) of that same field variable! Pretty rare, a step like that is, in physics or engineering, don’t you think? Do you remember any other place in physics or engineering where we do something like that?

Is there is any mechanical engineering topic that you want me to explain to you?

If so, send me your suggestions. If I find them suitable, I will begin thinking about them. May be, I will even answer them for you, here on this blog.

If not…

If not, there is always this one, involving the calculus of variations, again!:

Derbes, David (1996) “Feynman’s derivation of the Schrodinger equation,” Am. J. Phys., vol. 64, no. 7, July 1996, pp. 881–884

I’ve already found that I don’t agree with how Derbes uses the term “local”, in this article. His article makes it seem as if the local is nothing but a smallish segment on what essentially is a globally determined path. I don’t agree with that implication. …

However, here, although this issue is of relevance to the mechanical engineering proper, in the absence of a proper job (an Officially Approved Full Professor in Mechanical Engineering’s job), I don’t feel motivated to explain myself.

Instead, I find the following article by a Mechanical Engineering professor interesting: [^]

And, oh, BTW, if you are a blind follower of Feynman’s, do check out this one:

Briggs, John S. and Rost, Jan M. (2001) “On the derivation of the time-dependent equation of Schrodinger,” Foundations of Physics, vol. 31, no. 4, pp. 693–712.

I was delighted to find a mention of a system and an environment (so close to the heart of an engineer), even in this article on physics. (I have not yet finished reading it. But, yes, it too invokes the variational principles.)

OK then, bye for now.

[As usual, may be I will come back tomorrow and correct the write-up or streamline it a bit, though not a lot. Done on 2017.01.19.]

[E&OE]

# NASA’s EM drive, and the nature of the quantum theory

NASA’s EM drive has made it to the Forbes. Brian Koberlein, an astrophysicist who teaches at the Rochester Institute of Technology, provides a decent coverage; see, here [^].

First things first. I hardly know anything about the EM drive. Yes, I did go through the news reports about it a week ago or so, but about the only salient thing I noticed was that it was a replication of a result. The original result itself was found by the physicists community to be, to make an understatement, something like absolutely enormously incredible. … Given NASA’s reputation (at least among the physicists community), therefore, the scene would be ripe for quite some energetic speculations—at least discussions. Newsworthy.

But still, I myself don’t know much about the experiment. Not even a schematic sketch of the apparatus was provided in the general news coverage about the experiment so far, and I didn’t look into the paper itself because I knew it would be beyond me.

But since it was the Forbes where Koberlein’s coverage appeared, I decided to go through it. The description would be dumbed down enough that even I could get something out of it, I thought.

Well, even in this Forbes piece, there was no discussion of the actual apparatus, but the author did discuss the issue in terms of the Copenhagen interpretation, and that’s where the story became interesting to me. Koberlein writes:

In the usual Copenhagen interpretation of quantum theory, an object is defined by its wavefunction. The wavefunction describes the probability of finding a particle in a particular location. The object is in an indefinite, probabilistic state described by the wavefunction until it is observed. When it is observed, the wavefunction collapses, and the object becomes a definite particle with a definite location.

I am not an expert on the Copenhagen interpretation. However, I can tell that most popular science books would present the Copenhagen interpretation exactly in this manner. So, you can’t say that the author was presenting the Copenhagen interpretation in a misleading way. (Why, I even remember John Gribbin (Schrodinger’s Cat, and later, … Kitten), and Alastair Rae (Illusion or Reality) presenting these matters more or less precisely this way about a quarter of a century ago, if not earlier.)

Still, I did have an issue here. It is in the very last sentence in the quoted passage.

As you know, I have been writing and re-writing, and arranging and re-arranging the “syllabus” for my planned “book” on QM. In particular, these past few days, I have been doing exactly that. Since the subject matter thus was fresh in my mind, I could see that the way that the QM was developed by the original masters (Heisenberg, Schrodinger, Pauli, …), the spirit of their actual theorization was such that the last sentence in the quoted passage could not actually be justified.

Even though the usual mainstream QM presentation proceeds precisely along those lines, the actual spirit of the theorization by the original founders, has begun looking different to me.

I have a very difficult position to state here, so let me try to put it using some other words:

I am not saying that Koberlein’s last sentence is not a part of the Copenhagen interpretation. I am also not saying that Heisenberg did not have the Copenhagen interpretation in his mind, whenever he spoke about QM (as in contrast to discovering and working on QM). I am also aware that Schrodinger wanted to get rid of the quantum jumps—and could find no way to do so.

Yet, what I am saying is this: Given my self-study of QM using university text-books (like McQuarry, Resnick, Griffiths, Gasiorowicz, …), esp. over the last year, I can now clearly see that the collapse postulate wasn’t—or shouldn’t have been—a part of the spirit of the original theory-building.

Since I am dwelling on the spirit of the original (non-relativistic) QM, it is relevant to point out to you to someone who has putting up a particularly spirited defence of it over a period of time. I mean the Czeck physicist Lubos Motl. See, for example his post: “Stupidity of the pop science consensus about many worlds’ ”  [^]. Do go through it. Highly recommended. I know that Motl often is found involved in controversies. However, in this particular post (and the related and similar posts he has been making for quite some time), he remains fairly well-focused on the QM itself. He also happens to be extraordinarily lucid and clear in this post; see his discussion of the logical OR vs. the logical AND, for instance.

Even though Motl seems to have been arguing for the original founders, if you think through his writings, it also seems as if he does not place too much of an emphasis on the collapse postulate either—even though they did. He in fact seems to think that QM needs no interpretation at all, and as I suppose, this position would mean that QM does not need the Copenhagen interpretation (complete with the collapse postulate) either.

No, considering all his relevant posts about QM over time, I don’t think that I can agree with Motl; my position is that QM is incomplete, whereas he has strongly argued that it is complete. (I will come to show you how QM is incomplete, but first, I have to complete writing the necessary pre-requisites in the form of my book). Yet, I have found his writings (esp. those from 2015-end) quite helpful.

The detour to Motl’s blog was not so much of a detour at all. Here is another post by Motl, “Droplets and pilot waves vs. quantum mechanics” [^], done in 2014. This post apparently was in response to Prof. Bush (MIT) et al’s droplets experiment, and Koberlein, in his Forbes story today, does touch upon the droplets experiment and the Bohm interpretation, even if only in the passing. As to me, well, I have written about both the droplets experiment as well as Bohm’s theory in the past, so let me not go there once again. [I will add links to my past posts here, in the revision tomorrow.] As a matter of fact, I sometimes wonder whether it wouldn’t be a good idea to stop commenting on QM until my book is in at least version 0.5.

Anyway, coming back to Koberlein’s piece, I really liked the way he contrasts Bohm’s theory from Copenhagen interpretation:

The pilot wave model handles quantum indeterminacy a different way. Rather than a single wavefunction, quanta consist of a particle that is guided by a corresponding wave (the pilot wave). Since the position of the particle is determined by the pilot wave, it can exhibit the wavelike behavior we see experimentally. In pilot wave theory, objects are definite, but nonlocal. Since the pilot wave model gives the same predictions as the Copenhagen approach, you might think it’s just a matter of personal preference. Either maintain locality at the cost of definiteness, or keep things definite by allowing nonlocality. But there’s a catch.

Although the two approaches seem the same, they have very different assumptions about the nature of reality.

No, Brian, they are the same—inasmuch as they both are essentially non-local, and give rise to exactly the same quantitative predictions. If so, it’s just us who don’t understand how their seemingly different assumptions mean the same underlying physics, that’s all.

That’s why, I will go out on a limb and say that if the new paper about NASA’s EM drive has successfully used the Bohmian mechanics, and if it does predict the experimental outcome correctly, then it’s nothing but some Bohmian faithfuls looking for a “killer app” for their interpretation, that’s all. If what I understand about QM is right, and if the Bohmian mechanics predicts something, it’s just a matter of time before the mainstream formalism of QM (roughly, the Copenhagen interpretation) would also begin to predict exactly the same thing. (In the past, I had made a statement in the reverse way: whether Bohmian mechanics is developed enough to give the same predictions as the mainstream QM, you can always expect that it would get developed soon enough.)

As to my own writings on QM (I mean presenting QM the way I would like to do), as I told you, I have been working on it in recent times, even if only in an off-and-on manner. Yet, by now, I am done through more than half of the phase of finalizing the “syllabus” topics and sequence. (Believe me, this was a major challenge. For a book on QM, deciding what thesis you have for your book, and finalizing the order in which the presentation should be made, is more difficult—far more difficult—than writing down the specific contents of the individual sections and the equations in them.)

Writing the book itself can start any time now, though by now I clearly know that it’s going to be a marathon project. Months, in the least, it will take for me to finish.

Also, don’t wait for me to put up parts of it on the Web, any time soon. … It is a fact that I don’t have any problem sharing my drafts before the publication of the book as such. Yet, it also is a fact that if every page is going to be changing every day, I am not going to share such premature “editions” publicly either. After all, sharing also means inviting comments, and if you yourself haven’t firmed up your writing, comments and all are likely to make it even more difficult to finish the task of writing.

But yes, after thinking off-and-on about it for years (may be 5+ years), and after undergoing at least two false starts (which are all gone in the HDD crashes I had), I am now happy about the shape that the contents are going to take.

More, may be later. As to the Song I Like section, I don’t have anything playing at the back of my mind right away, so let me see if something strikes me by the time I come back tomorrow to give a final editing touch to this post. In that case, I will add this section; else, not!

[E&OE]