A wish. Looking for a couple of suitable post-docs. Covid-19 in India.

1. A wish…

I wish there were a neat, scholarly, account of the early development regarding the relativity theory. …

… There are tons of material on the topic, but not a single one seems to be to my liking…. I mean, even while rapidly browsing through so many of them, they all seem to fall short—rather, so awfully short. Reason: Most, if not all of them, seem intent on deifying Einstein, and / or, the primacy of maths over physics. [Did I cover all the eigenbases? May be not. So, let me add.] … The worst of them spend themselves out on promoting the idea that coming up with good but radical ideas in physics is all about getting lucky in some day-dreaming involving some mathematical ideas. 

OTOH, The “model” for the book which I have in mind here is something like what Prof. Malcolm Longair has done for QM; see his book: “Quantum concepts in physics: an alternative approach to the understanding of quantum mechanics.” [^].

… High time someone should have undertaken a similar effort. But unfortunately, it’s entirely lacking.

… The wish isn’t without purpose. The more I study the quantum mechanical spin, the more I realize the handicap which I have of not having already studied the relativity theory.

I can always postpone the fully consistent description, following my new approach, for the QM spin. [No one / no organization has ever sponsored my research. [Though, they all are hell bent on “following up” on me.]]

However, now that I have developed (what I believe to be) a good, basic, ontology for the QM phenomena, I have begun to see a promising pathway, at least from the viewpoint of a basic ontology, from a non-relativistic description of QM to a relativistic one—I mean the special relativistic one.


2. Looking for a couple of suitable post-docs…

Another possibility I am toying with, currently, is this:

Over a considerable period of time, say over a year or so, to build a series of Python/C++ scripts/programs that illustrate the classical EM in action, but following my new ontological ideas. These ideas are for the Maxwell-Lorentz EM, but I do anticipate that these would provide the easiest pathway to integrating the Special Relativity with the non-relativistic QM.

The trouble is: I will have to get into the FDTD algorithmics too, and I don’t have the time to do it. (In case you didn’t know, when it comes to EM, the best technique is FDTD.)

Wish I had a competent post-doc—actually two—working simultaneously with me! Right now!!

One could build the above-mentioned FDTD applets, but following the way I want them to be built.

The other one could work on “FEM-ization” of my FDM code (i.e., for the He atom, done with my new approach, and yet to be published). Once he is done, he could explore doing the same with FDTD (yes of course!), and compare and contrast the two. The FEM-ization of my FDM code won’t be very highly demanding, in the sense, people have done the finite elements formulation for the helium atom, and also have implemented it in code—decades ago… But of course, they did so following the mainstream QM. It would be a fun for the post-doc to implement it using the ideas I will be proposing—shortly.

Then, both could work on the ideas for the relativistic QM. … The pace of the work would depend on what they bring to the table, and how they perform.

Fallout? If you are a smart PhD in the concerned areas, I need not provide even a hint about it…


3. Status update on my QM research:

Currently, I am typing a set of notes on the topic of the quantum mechanical angular momentum, including the spin. For the time being, I am mostly following Dan Schroeder’s notes (which I mentioned in the post before the last, here [^]). Once done, I don’t mind uploading these notes—for proofreading by you the potential post-docs. [Who else?]

While typing these notes, it has become once again very clear to me—crystal clear, in fact—as to how my “theory” for the QM spin (following my new approach) falls short. … So short, in fact. … My “theory” doesn’t just look awfully arbitrary; it is so.

All in all, don’t expect the same kind of confidence from me for the spin-related aspects as for the spin-less ones. I mean, in the upcoming document on my new approach.


4. Back to the potential post-docs:

Exciting enough?

If yes, drop me a line. Especially, if you are working with Google / similar company. I gather they officially allow you some fraction of your official time for your own hobby projects too…


5. If you are someone young and enthusiastic, say from the Pune city and all (and in general, from India):

They have relaxed the curbs. However, I have a word of advice for you.

Don’t step out unless absolutely necessary, and if so doing, take all the precautions.

It’s just a matter of a few months now…

…BTW, I am also thinking whether the government shouldn’t relax the enforced gap of three months in between the first and the second dose for the jabs. … There are circumstantial matters which indicate that a gap in between two to three months might be ideal; that three months might be too long a period. (Actually, this matter had struck me right on the day that the Central Government increased the gap from 6 weeks to 12 weeks in one, single, move. …However, at that time, I had thought it prudent to wait and watch. Now, I think I can—nay, should—share my opinon. … I also have some other points about these matters, but these are not so important. I will sure mention these as and when it becomes necessary to do so.)


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


A song I like:

(Hindi) ज़िंदगी आ रहा हूँ मैं… (“zindagi aa rahaa hoon main…”)
Lyrics: Javed Akhtar
Music: Hridaynath Mangeshkar
Singer: Kishore Kumar

[

Credits happily listed in a random order. A good quality audio is here [^]. … Although I haven’t seen this movie, recently I watched the video for this song, and found that I enjoyed it too. A good quality video is here [^].

… I always loved this song, esp. the tune and the arrangement / orchestration. … And of course, Javed Akhtar’s awesome lyrics. … Well, yes, Kishore does sound, at places in this song, just a slight bit… and how shall I put it?… He doesn’t sound as if he were in his best frame of singing, here. His voice sounds a bit too “broad”, and perhaps heavy, and even a bit “tired” perhaps? as if he were straining a bit?…  Even then, of course, being Kishore, he does manage to pull a great job. [It’s just that, knowing Kishore, one wants to note this aside… I know, hair-splitting, it is. … Can’t help. … Sometimes.]

… [BTW, if you are young and dynamic and result-oriented etc.: The guy in this video is Sonam Kapoor’s dad. He used to be young. Once upon time. Me too. [Though I never ever had the hair-style he displays here. A lot of my class-mates did, mostly following The “Bachchan”. Not me. […Yeah, I know.]]

… All the same, all that you’ve to do now is to wait for just a few more months, that’s all… 2021 isn’t a meme on Twitter the way 2020 was. Nevertheless, in India, we have to wait. So, just listen to songs like this for just a wee bit more. … I can tell you, from experience: The scenery, esp. the Sahyaadri’s, does stay great also well until January / February next year. (And if you really love Sahyaadri’s, well, they are there, forever…)

…So there.]

…And if you are new to this song, see if you too like it…

Take care and bye for now…

]

 

Still loitering around…

As noted in the last post, I’ve been browsing a lot. However, I find that the signal-to-noise ratio is, in a way, too low. There are too few things worth writing home about. Of course, OTOH, some of these things are so deep that they can keep one occupied for a long time.

Anyway, let me give many (almost all?) of the interesting links I found since my last post. These are being noted in no particular order. In most cases, the sub-title says it all, and so, I need not add comments. However, for a couple of videos related to QM, I do add significant amount of comments. … BTW, too many hats to do the tipping to. So let me skip that part and directly give you the URLs…


“A `digital alchemist’ unravels the mysteries of complexity”:

“Computational physicist Sharon Glotzer is uncovering the rules by which complex collective phenomena emerge from simple building blocks.” [^]


“Up and down the ladder of abstraction. A systematic approach to interactive visualization.” [^]

The tweet that pointed to this URL had this preface: “One particular talent stands out among the world-class programmers I’ve known—namely, an ability to move effortlessly between different levels of abstraction.”—Donald Knuth.

My own thinking processes are such that I use visualization a lot. Nay, I must. That’s the reason I appreciated this link. Incidentally, it also is the reason why I did not play a lot with the interactions here! (I put it in the TBD / Some other day / Etc. category.)


“The 2021 AI index: major growth despite the pandemic.”

“This year’s report shows a maturing industry, significant private investment, and rising competition between China and the U.S.” [^]


“Science relies on constructive criticism. Here’s how to keep it useful and respectful.” [^]

The working researcher, esp. the one who blogs / interacts a lot, probably already knows most all this stuff. But for students, it might be useful to have such tips collected in one place.


“How to criticize with kindness: Philosopher Daniel Dennett on the four steps to arguing intelligently.” [^].

Ummm… Why four, Dan? Why not, say, twelve? … Also, what if one honestly thinks that retards aren’t ever going to get any part of it?… Oh well, let me turn to the next link though…


“Susan Sontag on censorship and the three steps to refuting any argument” [^]

I just asked about four steps, and now comes Sontag. She comes down to just three steps, and also generalizes the applicability of the advice to any argument… But yes, she mentions a good point about censorship. Nice.


“The needless complexity of modern calculus: How 18th century mathematicians complicated calculus to avoid the criticisms of a bishop.” [^]

Well, the article does have a point, but if you ask me, there’s no alternative to plain hard work. No alternative to taking a good text-book or two (like Thomas and Finney, as also Resnick and Halliday (yes, for maths)), paper and pen / pencil, and working your way through. No alternative to that… But if you do that once for some idea, then every idea which depends on it, does become so simple—for your entire life. A hint or a quick reference is all you need, then. [Hints for the specific topic of this piece: the Taylor series, and truncation thereof.] But yes, the article is worth a fast read (if you haven’t read / used calculus in a while). … Also, Twitterati who mentioned this article also recommended the wonderful book from the next link (which I had forgotten)…


“Calculus made easy” [^].

The above link is to the Wiki article, which in turn gives the link to the PDF of the book. Check out the preface of the book, first thing.


“The first paper published in the first overlay journal (JTCAM) in Solid Mechanics” [^]

It’s too late for me (I have left mechanics as a full-time field quite a while ago) but I do welcome this development. … A few years ago, Prof. Timothy Gowers had begun an overlay journal in maths, and then, there also was an overlay journal for QM, and I had welcomed both these developments back then; see my blog post here [^].


“The only two equations that you should know: Part 1” [^].

Dr. Joglekar makes many good points, but I am not sure if my choice for the two equations is going to be the same.

[In fact, I don’t even like the restriction that there should be just two equations. …And, what’s happenning? Four steps. Then, three steps. Now, two equations… How long before we summarily turn negative, any idea?]

But yes, a counter-balance like the one in this article is absolutely necessary. The author touches on E = mc^2 and Newton’s laws, but I will go ahead and add topics like the following too: Big Bang, Standard Model, (and, Quantum Computers, String Theory, Multiverses, …).


“Turing award goes to creators of computer programming building blocks” [^] “Jeffrey Ullman and Alfred Aho developed many of the fundamental concepts that researchers use when they build new software.”

Somehow, there wasn’t as much of excitement this year as the Turing award usually generates.

Personally, though, I could see why the committee might have decided to recognize Aho and Ullman’s work. I had once built a “yacc”-like tool that would generate the tables for a table-driver parser, given the abstract grammar specification in the extended Backus-Noor form (EBNF). I did it as a matter of hobby, working in the evenings. The only resource I used was the “dragon book”, which was written by Profs. Aho, Sethi, and Ullman. It was a challenging but neat book. (I am not sure why they left Sethi out. However, my knowledge of the history of development of this area is minimal. So, take it as an idle wondering…)

Congratulations to Profs. Aho and Ullman.


“Stop calling everything AI, machine-learning pioneer says” [^] “Michael I. Jordan explains why today’s artificial-intelligence systems aren’t actually intelligent”

Well, “every one” knows that, but the fact is, it still needs to be said (and even explained!)


“How a gene for fair skin spread across India” [^] “A study of skin color in the Indian subcontinent shows the complex movements of populations there.”

No, the interesting thing about this article, IMO, was not that it highlighted Indians’ fascination / obsession for fairness—the article actually doesn’t even passingly mention this part. The real interesting thing, to me, was: the direct visual depiction, as it were, of Indian Indologists’ obsession with just one geographical region of India, viz., the Saraswati / Ghaggar / Mohan Ja Daro / Dwaarkaa / Pakistan / Etc. And, also the European obsession with the same region! … I mean check out how big India actually is, you know…

H/W for those interested: Consult good Sanskrit dictionaries and figure out the difference between निल (“nila”) and नील (“neela”). Hint: One of the translations for one of these two words is “black” in the sense “dark”, but not “blue”, and vice-versa for the other. You only have to determine which one stands for what meaning.

Want some more H/W? OK… Find out the most ancient painting of कृष्ण (“kRSNa”) or even राम (“raama”) that is still extant. What is the colour of the skin as shown in the painting? Why? Has the painting been dated to the times before the Europeans (Portugese, Dutch, French, Brits, …) arrived in India (say in the second millennium AD)?


“Six lessons from the biotech startup world” [^]

Dr. Joglekar again… Here, I think every one (whether connected with a start-up or not) should go through the first point: “It’s about the problem, not about the technology”.

Too many engineers commit this mistake, and I guess this point can be amplified further—the tools vs. the problem. …It’s but one variant of the “looking under the lamp” fallacy, but it’s an important one. (Let me confess: I tend to repeat the same error too, though with experience, one does also learn to catch the drift in time.)


“The principle of least action—why it works.” [^].

Neat article.

I haven’t read the related book [“The lazy universe: an introduction to the principle of least action”] , but looking at the portions available at Google [^], even though I might have objections to raise (or at least comments to make) on the positions taken by the author in the book, I am definitely going to add it to the list of books I recommend [^].

Let me mention the position from which I will be raising my objections (if any), in the briefest (and completely on-the-fly) words:

The principle of the least action (PLA) is a principle that brings out what is common to calculations in a mind-bogglingly large variety of theoretical contexts in physics. These are the contexts which involve either the concept of energy, or some suitable mathematical “generalizations” of the same concept.

As such, PLA can be regarded as a principle for a possible organization of our knowledge from a certain theoretical viewpoint.

However, PLA itself has no definite ontological content; whatever ontological content you might associate with PLA would go on changing as per the theoretical context in which it is used. Consequently, PLA cannot be seen as capturing an actual physical characteristic existing in the world out there; it is not a “thing” or “property” that is shared in common by the objects, facts or phenomena in the physical world.

Let me give you an example. The differential equation for heat conduction has exactly the same form as that for diffusion of chemical species. Both are solved using exactly the same technique, viz., the Fourier theory. Both involve a physical flux which is related to the gradient vector of some physically existing scalar quantity. However, this does not mean that both phenomena are produced by the same physical characteristic or property of the physical objects. The fact that both are parabolic PDEs can be used to organize our knowledge of the physical world, but such organization proceeds by making appeal to what is common to methods of calculations, and not in reference to some ontological or physical facts that are in common to both.

Further, it must also be noted, PLA does not apply to all of physics, but only to the more fundamental theories in it. In particular, try applying it to situations where the governing differential equation is not of the second-order, but is of the first- or the third-order [^]. Also, think about the applicability of PLA for dissipative / path-dependent processes.

… I don’t know whether the author (Dr. Jennifer Coopersmith) covers points like these in the book or not… But even if she doesn’t (and despite any differences I anticipate as of now, and indeed might come to keep also after reading the book), I am sure, the book is going to be extraordinarily enlightening in respect of an array of topics. … Strongly recommended.


Muon g-2.

I will give some the links I found useful. (Not listed in any particular order)

  • Dennis Overbye covers it for the NYT [^],
  • Natalie Wolchoever for the Quanta Mag [^],
  • Dr. Luboš Motl for his blog [^],
  • Dr. Peter Woit for his blog [^],
  • Dr. Adam Falkowski (“Jester”) for his blog [^],
  • Dr. Ethan Siegel for the Forbes [^], and,
  • Dr. Sabine Hossenfelder for Sci-Am [^].

If you don’t want to go through all these blog-posts, and only are looking for the right outlook to adopt, then check out the concluding parts of Hossenfelder’s and Siegel’s pieces (which conveniently happen to be the last two in the above list).

As to the discussions: The Best Comment Prize is hereby being awarded, after splitting it equally into two halves, to “Manuel Gnida” for this comment [^], and to “Unknown” for this comment [^].


The five-man quantum mechanics (aka “super-determinism”):

By which, I refer to this video on YouTube: “Warsaw Spacetime Colloquium #11 – Sabine Hossenfelder (2021/03/26)” [^].

In this video, Dr. Hossenfelder talks about… “super-determinism.”

Incidentally, this idea (of super-determinism) had generated a lot of comments at Prof. Dr. Scott Aaronson’s blog. See the reader comments following this post: [^]. In fact, Aaronson had to say in the end: “I’m closing this thread tonight, honestly because I’m tired of superdeterminism discussion.” [^].

Hossenfelder hasn’t yet posted this video at her own blog.

There are five people in the entire world who do research in super-determinism, Hossenfelder seems to indicate. [I know, I know, not all of them are men. But I still chose to say the five-man QM. It has a nice ring to it—if you know a certain bit from the history of QM.]

Given the topic, I expected to browse through the video really rapidly, like a stone that goes on skipping on the surface of water [^], and thus, being done with it right within 5–10 minutes or so.

Instead, I found myself listening to it attentively, not skipping even a single frame, and finishing the video in the sequence presented. Also, going back over some portions for the second time…. And that’s because Hossenfelder’s presentation is so well thought out. [But where is the PDF of the slides?]

It’s only after going through this video that I got to understand what the idea of “super-determinism” is supposed to be like, and how it differs from the ordinary “determinism”. Spoiler: Think “hidden variables”.

My take on the video:

No, the idea (of super-determinism) isn’t at all necessary to explain QM.

However, it still was neat to get to know what (those five) people mean by it, and also, more important: why these people take it seriously.

In fact, given Hossenfelder’s sober (and intelligent!) presentation of it, I am willing to give them a bit of a rope too. …No, not so long that they can hang themselves with it, but long enough that they can perform some more detailed simulations. … I anticipate that when they conduct their simulations, they themselves are going to understand the query regarding the backward causation (raised by a philosopher during the interactive part of the video) in a much better manner. That’s what I anticipate.

Another point. Actually, “super-determinism” is supposed to be “just” a theory of physics, and hence, it should not have any thing to say about topics like consciousness, free-will, etc. But I gather that at least some of them (out of the five) do seem to think that the free-will would have to be denied, may be as a consequence of super-determinism. Taken in this sense, my mind has classified “super-determinism” as being the perfect foil to (or the other side of) panpsychism. … As to panpsychism, if interested, check out my take on it, here [^].

All along, I had always thought that super-determinism is going to turn out to be a wrong idea. Now, after watching this video, I know that it is a wrong idea.

However, precisely for the same reason (i.e., coming to know what they actually have in mind, and also, how they are going about it), I am not going to attack them, their research program. … Not necessary… I am sure that they would want to give up their program on their own, once (actually, some time after) I publish my ideas. I think so. … So, there…


“Video: Quantum mechanics isn’t weird, we’re just too big” YouTube video at: [^]

The speaker is Dr. Phillip Ball; the host is Dr. Zlatko Minev. Let me give some highlights of their bio’s: Ball has a bachelor’s in chemistry from Oxford and a PhD in physics from Bristol. He was an editor at Nature for two decades. Minev has a BS in physics from Berkeley and a PhD in applied physics from Yale. He works in the field of QC at IBM (which used to be the greatest company in the computers industry (including software)).

The abstract given at the YouTube page is somewhat misleading. Ignore it, and head towards the video itself.

The video can be divided into two parts: (i) the first part, ~47 minutes long, is a presentation by Ball; (ii) the second part is a chat between the host (Minev) and the guest (Ball). IMO, if you are in a hurry, you may ignore the second part (the chat).

The first two-third portion of the first part (the presentation) is absolutely excellent. I mean the first 37 minutes. This good portion (actually excellent) gets over once Ball goes to the slide which says “Reconstructing quantum mechanics from informational rules”, which occurs at around 37 minutes. From this point onward, Ball’s rigour dilutes a bit, though he does recover by the 40:00 minutes mark or so. But from ~45:00 to the end (~47:00), it’s all down-hill (IMO). May be Ball was making a small little concession to his compatriots.

However, the first 37 minutes are excellent (or super-excellent).

But even if you are absolutely super-pressed for time, then I would still say: Check out at least the first 10 odd minutes. … Yes, I agree 101 percent with Ball, when it comes to the portion from ~5:00 through 06:44 through 07:40.

Now, a word about the mistakes / mis-takes:

Ball says, in a sentence that begins at 08:10 that Schrodinger devised the equation 1924. This is a mistake / slip of the tongue. Schrodinger developed his equation in late 1925, and published it in 1926, certainly not in 1924. I wonder how come it slipped past Ball.

Also, the title of the video is somewhat misleading. “Bigness” isn’t really the distinguishing criterion in all situations. Large-distance QM entanglements have been demonstrated; in particular, photons are (relativistic) QM phenomena. So, size isn’t necessarily always the issue (even if the ideas of multi-scaling must be used for bridging between “classical” mechanics and QM).

And, oh yes, one last point… People five-and-a-half feet tall also are big enough, Phil! Even the new-borns, for that matter…

A personal aside: Listening to Ball, somehow, I got reminded of some old English English movies I had seen long back, may be while in college. Somehow, my registration of the British accent seems to have improved a lot. (Or may be the Brits these days speak with a more easily understandable accent.)


Status of my research on QM:

If I have something to note about my research, especially that related to the QM spin and all, then I will come back a while later and note something—may be after a week or two. …

As of today, I still haven’t finished taking notes and thinking about it. In fact, the status actually is that I am kindaa “lost”, in the sense: (i) I cannot stop browsing so as to return to the study / research, and (ii) even when I do return to the study, I find that I am unable to “zoom in” and “zoom out” of the topic (by which, I mean, switching the contexts at will, in between all: the classical ideas, the mainstream QM ideas, and the ideas from my own approach). Indeed (ii) is the reason for (i). …

If the same thing continues for a while, I will have to rethink whether I want to address the QM spin right at this stage or not…

You know, there is a very good reason for omitting the QM spin. The fact of the matter is, in the non-relativistic QM, the spin can only be introduced on an ad-hoc basis. … It’s only in the relativistic QM that the spin comes out as a necessary consequence of certain more basic considerations (just the way in the non-relativistic QM, the ground-state energy comes out as a consequence of the eigenvalue nature of the problem; you don’t have to postulate a stable orbit for it as in the Bohr theory). …

So, it’s entirely possible that my current efforts to figure out a way to relate the ideas from my own approach to the mainstream QM treatment of the spin are, after all, a basically pointless exercise. Even if I do think hard and figure out some good and original ideas / path-ways, they aren’t going to be enough, because they aren’t going to be general enough anyway.

At the same time, I know that I am not going to get into the relativistic QM, because it has to be a completely distinct development—and it’s going require a further huge effort, perhaps a humongous effort. And, it’s not necessary for solving the measurement problem anyway—which was my goal!

That’s why, I have to really give it a good thought—whether I should be spending so much time on the QM spin or not. May giving some sketchy ideas (rather, making some conceptual-level statements) is really enough… No one throws so much material in just one paper, anyway! Even the founders of QM didn’t! … So, that’s another line of thought that often pops up in my mind. …

My current plan, however, is to finish taking the notes on the mainstream QM treatment of the spin anyway—at least to the level of Eisberg and Resnick, though I can’t finish it, because this desire to connect my approach to the mainstream idea also keeps on interfering…

All in all, it’s a weird state to be in! … And, that’s what the status looks like, as of now…


… Anyway, take care and bye for now…


A song I, ahem, like:

It was while browsing that I gathered, a little while ago, that there is some “research” which “explains why” some people “like” certain songs (like the one listed below) “so much”.

The research in question was this paper [^]; it was mentioned on Twitter (where else?). Someone else, soon thereafter, also pointed out a c. 2014 pop-sci level coverage [^] of a book published even earlier [c. 2007].

From the way this entire matter was now being discussed, it was plain and obvious that the song had been soul-informing for some, not just soul-satisfying. The song in question is the following:

(Hindi) सुन रुबिया तुम से प्यार हो गया (“sun rubiyaa tum se pyaar ho gayaa”)
Music: Anu Malik
Lyrics: Prayag Raj
Singers: S. Jaanaki, Shabbir Kumar

Given the nature of this song, it would be OK to list the credits in any order, I guess. … But if you ask me why I too, ahem, like this song, then recourse must be made not just to the audio of this song [^] but also to its video. Not any random video but the one that covers the initial sequence of the song to an adequate extent; e.g., as in here [^].


History:
2021.04.09 19:22 IST: Originally published.
2021.04.10 20:47 IST: Revised considerably, especially in the section related to the principle of the least action (PLA), and the section on the current status of my research on QM. Also minor corrections and streamlining. Guess now I am done with this post.

Finding a cozy n comfy enough a spot…

Update on 2021.02.02:

I have made a couple of inline updates in the sections 5. and 7. below.


I have already begun cleaning up and reorganizing the code, and writing the boring document. The work so far has come along pretty well. I have by now achieved a satisfactory level of consistency in the numerical results for the hydrogen atom in a 3D box.

As indicated in my last post here, I had found that it’s more useful to focus on the cell-side of the mesh rather than on the physical size of the box or the number of nodes per side of the cube.

Now, given below are some of the details of certain further, systematic, trials which I conducted, in order to arrive at optimum ranges for numerical parameters.

Since the analytical solution is available only for the hydrogenic atoms (i.e. systems with a positively charged nucleus and just one electron, e.g. the hydrogen atom and the He+ ion), these systematic studies were conducted only for them.

If you came here expecting that I might have something to say about the reproducibility for the helium atom, then well, you will have to wait for some 2–3 weeks. The nature of the issues themselves is like that. You can’t hurry things like these too much, as the studies below sure bring out.

So, anyway, here are the highlights of the systematic studies I conducted and some of the representative results.

All the results reported in this post are in the atomic units.


1. Finitization policy to replace the singularity of the Coulomb field:

In my last post here, I had mentioned that in FDM, we have to use a finite value in place of the -\infty at the nucleus. As a necessary consequence, we have to adopt some policy for this finitization.

While conducting my studies, now I found that it is better to frame this policy not in terms of a chosen fraction of the cell-side, but in terms of a certain relevant datum and a multiplying factor. The better procedure is this:

Whatever be the degree of mesh refinement, having constructed the mesh, calculate the always-finite PE value at the FDM node right next to the singularity. Then, multiply this PE value by a certain factor, and use it in place of the theoretically -\infty value at the singular node. Let’s give a name to this multiplying factor; let’s call it the Coulomb Field’s Singularity Finitization Factor (“CFSF” for short).

Notice that using this terminology, a CFSF value of 2.0 turns out to be exactly the same as using the half cell-side rule. However, framing the finitization policy in terms of the CFSF factor has the advantage that it makes it easier to compare the differences in the relative sharpness of the potential well at different mesh-refinement levels.

OK.

I then found that while using FDM, the eigenvalue solver is very sensitive to even small variations to the value of CFSF.

If you use a CFSF value of 1.8, then it turns out that the PE well does not go down enough in the neighbourhood of the singularity, and therefore, the reported ground-state eigenvalues can easily go to about -0.45, -0.2, or even worse. Refining the mesh doesn’t help—within the domain and mesh sizes which are both practicable on my laptop and relevant for the H and He atom modelling. Note, the analytical solution is: -0.5, exactly.

Conclusion: Using even a slightly lower CFSF spoils the results.

OTOH, if you use a CFSF of 2.2 to 2.5, then the ground-state energy can go lower than the exact value of -0.5. Now, this is a sure-shot indication that your numerical modelling has gone wrong.

In general, with FDM, you would expect that with mesh refinement, the convergence in the energy values would be not just monotonic but also one-sided, and also that the convergence would occur from “above” (because the energy values here are negative). In other words, if my understanding of the theory of numerical analysis is correct, then a properly done (meaningful) numerical simulation cannot produce energies below the analytical solution of -0.5.

So, clearly, using a CFSF value even slightly greater than 2.0 is bad for the health of the numerical simulations.

In the earlier trials reported in the last post, I had simply guessed that the value of 2.0 might be good enough for my initial trials. Now it turns out that my computational modeller’s intuition was pretty much on target—or at least, that I was plain lucky! The CFSF value of 2.0 indeed happens to be quite the best value to choose, given the rest of the parameters like the cell-side, the domain size, and the rest of the details of this problem (viz., the strength of the Coulomb singularity, the nature of the Schrodinger equation, the use of uniform and structured meshes, the FDM discretization, etc.).


2. Simple-minded mesh refinement doesn’t produce consistent results:

Suppose you keep the domain size fixed at, say, 20.0, and vary the mesh refinement levels.

Now, your naive expectation might be that as you refine the mesh by increasing the number of nodes per side of the cube, you should get more and more accurate results. That’s our usual experience with problems like diffusion in continua, and even for problems like convection in fluids.

However, the category of the QM problems is different! Here, we have an eigenvalue problem that must be solved with a singular potential field. The naive expectations built on simple problems like the Poisson-Laplace equation or the diffusion equation, go for a toss. Harmonic analysis might still apply in some form (frankly I don’t even know if it does!), but the singularity sure plays tricks!

This is an ugly fact, and frankly, I had not foreseen it. But it’s there. I had to keep myself reminding of the different nature of the eigenvalue problem together with the singular fields.

As you refine the mesh too much, then the absolute value of the PE at a node right next to the point of singularity increases without bound! This fact mandates that the finite value we (more or less) arbitrarily chose to use in place of the actually infinite value for the singular point, has itself to increase further too.

But, for some reasons not known to me (but which by now do feel vaguely reasonable!) the eigenvalue solver begins to experience difficulties with such increases in the absolute value of the PE value at the singularity. Roughly, the trouble begins to happen as the minimum potential energy (at the singular node) goes below -20 or so. In fact, I even found that a highly refined mesh might actually report a positive value for the ground-state energy—no bonding but, on the contrary, a repulsion of the electron!


3. Wavefunction fields are important in my new approach, but they don’t always converge to the analytical solution very well!:

With a reasonable level of mesh refinement, the ground-state energy does monotonically approach the exact figure of -0.5 However, I’ve found that a convergence in energy is not necessarily accompanied also by a good convergence trend in the absolute values of the wavefunction!

In the H-atom, for the ground-state analytical solution, the absolute value of the wavefunction has its maximum right at the nucleus; the wavefunction field forms a cusp at the nucleus, in fact. The analytical value for \psi(x)-max goes like: 0.564189584\dots. (That’s because in the atomic units, the Bohr radius a_0 is chosen to be exactly equal to 1, and so, at r = 0, the ground-state wavefunction for the H-atom becomes \psi(x_{\text{at nucleus}}) = 1/\sqrt{\pi}.)

Now the trouble I found was this:

With mesh refinement, even as the energy is nicely converging to something like -0.4938884 (against -0.5), the \psi-max might still be lingering around a lower figure like 0.516189. The \psi-max values converge more slowly, and their convergence shows opposite trend!

For relatively coarse meshes (i.e. high \Delta x of the FDM mesh), the \psi-max value is actually way much higher than the analytical solution; it even becomes as bad as 3.276834 or 1.393707. As you refine the mesh, they do begin to fall down and approach the analytical solution.

However, with further mesh refinement, the \psi-max values continue to fall down! They cross the analytical solution level of 0.564189584 too, and still continue to fall further! And, this behaviour occurs even as energy result is still approaching the exact solution in a nice-and-expected monotonic manner.

So, the trouble is: Using the right mesh size is actually a trade-off! You have to sacrifice some convergence on the energy number, so as to have a good (reliable) value for the \psi-max measure.

The trouble doesn’t stop there; see the next section.


4. Energies for the excited-states don’t always come out very well:

With appropriately high levels of mesh-refinement, the ground-state energy might be showing good convergence trends. Even the \psi-max values might be good enough (like 0.52 or so). But the energy and/or \psi-max for the first excited state still easily give trouble.

The energy for the first excited state for the hydrogen atom is, by analytical solution, -0.125, exactly.

The numerical values, when the simulation is working right, could be like -0.11, or even better, say -0.123, or thereabout. But that happens only when the mesh is of the intermediate refinement (the cell-side is neither too small nor too large).

However, with a more refined mesh (smaller cell-sides), the PE well can remain more or less rightly shaped for the ground-state energy, but it can still become too deep for the first-excited state energy! The first excited state energy can suddenly get degraded to a value like -0.04471003.

Indeed, there seems to be some kind of a numerical compensation going on in between the \psi-max values and the energy values, especially for the first-excited state energies. The ground-state energies remain much better, in relative terms. (If the mesh refinement is very high, even the ground-state energy goes off the track to something like -0.2692952 or even positive values. That’s what I meant by “appropriately” high levels of mesh refinement.)

I didn’t compare the numerical results with the analytical solutions for energies or \psi-max values for second-excited states or higher. Computation of the bonding energy makes reference only to the ground state, and so, I stopped my exploration of this side of the FDM + eigenvalue solver behaviour at this stage.


5. Atomic sizes reported by the numerical modeling show very good trends:

Another important consideration in my new approach has to do with the atomic radius of the atoms being modelled (hydrogen and helium).

After optimizing the mesh refinement (i.e., effectively, the cell-side), I conducted a series of numerical trials using different domain sizes (from 7.5 through 40.0 ), and implemented a rough-and-ready code to estimate the following measure:

The side of the nucleus-centered sub-cube in which roughly 95 \% (or 99 \%) of the probability cloud is contained.

This size can be taken as a good measure for the atomic size.

In the above working definition, I say roughly 95 \%, because I didn’t care to interpolate the wavefunction fields in between their nodal values. What this means is that the side of the sub-cube effectively changes only in the integer steps, and therefore, the percentage of the sub-cube contained may not be exactly 95 \%; it could be 97.5 \% for one domain size, and 95.3 \% for another domain size, just to pick up some numbers.

But even while using this rough and ready measure (and implementation), I found that the results were quite meaningfully consistent.

But why conduct these trials?

Well, realize that (1) the simulation box has a finite size, and (2) the Dirichlet conditions are being imposed at all the boundary nodes. Given these two constraints, the solution is going to show boundary-/edge-effects, i.e., the solution is going to depend on the domain size.

Now, in my approach, the spread of the probability cloud enters the calculations in a crucial manner. Numerically “extracting” the size of the simulated atom was, therefore, an important part of optimizing the simulations.

The expected behaviour of the above mentioned “size effect” was that as the domain size increases, the calculated atomic size, too, should increase. The question was: Were these differences in the numerically determined sizes important enough? did they vary too much? if yes, how much? The following is what I found:

First, I fixed the domain size (cube side) at 10.0, and varied the mesh refinement (from roughly 41 nodes per side to 121 and 131). I found that the calculated atomic sizes for the hydrogen atom varied but in a relatively small range—which was a big and happy surprise to me. The calculated size went from 5.60 while using a coarse mesh (requiring eigenvalue computation time of about 10 seconds) to a value like 5.25 for an intermediate refinement of the mesh (exe. time 2 min. 32 seconds i.e. 152 seconds), to 5.23 for as fine a mesh as my machine can handle (131 \times 131 \times 131, which required an exe. time of about 20 minutes i.e. 1200 seconds, for each eigenvalue computation call). Remember, all these results were for a domain size of 10.0.

Next, I changed the domain cube side to 15.0, and repeated the trials, for various levels of mesh refinements. Then, ditto, for the domain side of 20.0 and 40.0.

Collecting the results together:

  • 10.0
    • coarse: 5.60
    • intermediate: 5.25
    • fine: 5.23
  • 15.0
    • coarse: 6.0
    • intermediate: 5.62
  • 20.0
    • coarse: 6.40
    • intermediate: 5.50
    • fine: 5.67
  • 40.0
    • coarse: 4.0
    • intermediate: 5.5
    • fine: 6.0
    • very fine: 6.15

You might be expecting very clear-cut trends and it’s not the case here. However, remember, due to the trickiness of the eigenvalue solver in the presence of a “singular” PE well, not to mention the roughness of the size-estimation procedure (only integer-sized sub-cubes considered, and no interpolations of \psi to internodal values), a monotonic sort of behaviour is simply not to be expected here.

Indeed, if you ask me, these are pretty good trends, even if they are only for the hydrogen atom.

Note, for the helium atom, my new approach would require giving eigenvalue computation calls thousands of times. So, at least on this count of atomic radius computation, the fact that even the coarse or mid-level mesh refinement results didn’t vary much (they were in the range of 5.25 to 5.6) was very good. Meaning, I don’t have to sacrifice a lot of accuracy due to this one factor taken by itself.

For comparison, the atomic size (diameter) for the hydrogen atom is given in the literature (Wiki), when translated into atomic units, comes out variously as: (1) 0.94486306 using some “empirical” curve-fitting to some indirect properties of gases; (2) 4.5353427 while using the van der Waal criterion, and (3) 2.0031097 using “calculations” (whose basis or criteria I do not know in detail).

Realize, the van der Waal measure is closest to the criterion used by me above. Also, it is only expected that when using FDM, due to the numerical approximations, just the way the FDM ground-state energy values should come out algebraically greater (they do, say -0.49 vs. the exact datum of -0.5), the FDM \psi-max measure should come out smaller (it does, say 0.52 vs. the analytical solution of \approx 0.56), similarly, for the same reasons, the rough-and-ready estimated atomic size should come out as greater (it does, say 5.25 to 5.67 as the domain size increases from 10.0 to 40.0, the. van der Waal value being 4.54 ).

Inline update on 2021.02.02 19:26 IST: After the publication of this post, I compared the above-mentioned results with the analytical solution. I now find that the sizes of the sub-cubes found using FDM, and using the analytical solution for the hydrogen atom, come out as identical!  This is a very happy news. In other words, making comparisons with the van der Waal size and the other measure was not so relevant anyway; I should have compared the atomic sizes (found using the sub-cubes method) with the best datum, which is, the analytical solution! To put this finding in some perspective, realize that the FDM-computed wavefunctions still do differ a good deal from the analytical solution, but the volume integral for an easy measure like 95 \% does turn out be the same. The following proviso’s apply for this finding: The good match between the analytical solution and the FDM solution are valid only for (i) the range of the domain sizes considered here (roughly, 10 to 40), not for the smaller box sizes (though the two solution would match even better for bigger boxes), and (ii) only when using the Simpson procedure for numerically evaluating the volume integrals. I might as well also note that the Simpson procedure is, relatively, pretty crude. As the sizes of the sub-cubes go on increasing, the Simpson procedure can give volume integrals in excess of 1.0 for both the FDM and the analytical solutions. Inline update over.

These results are important because now I can safely use even a small sized domain like a 10.0-side cube, which implies that I can use a relatively crude mesh of just 51 nodes per side too—which means a sufficiently small run-time for each eigenvalue function call. Even then, I would still remain within a fairly good range on all the important parameters.

Of course, it is already known with certainty that the accuracy for the bonding energy for the helium atom is thereby going to get affected adversely. The accuracy will suffer, but the numerical results would be on the basis of a sweet-zone of all the numerical parameters of relevance—when validated against hydrogen atom. So, the numerical results, even for the helium atom, should have greater reliability.

Considerations like conformance to expected behaviour in convergence, stability, and reliability are far more important considerations in numerical work of this nature. As to sheer accuracy itself, see the next section too.


6. Putting the above results in perspective:

All in all, for the convergence behaviour for this problem (eigenvalue-eigenvector with singular potentials) there are no easy answers. Not even for just the hydrogen atom. There are trade-offs to be made.

However, for computation of bonding energy using my new approach, it’s OK even if a good trade-off could be reached only for the ground-state.

On this count, my recent numerical experimentation seems to suggest that using a mesh cell-side of 0.2 or 0.25 should give the most consistent results across a range of physical domain sizes (from 7.5 through 30.0 ). The atomic size extracted from the simulations also show good behaviour.

Yes, all these results are only for the hydrogen atom. But it was important that I understand the solver behaviour well enough. It’s this understanding which will come in handy while optimizing for the helium atom—which will be my next step on the simulation side.

The trends for the hydrogen atom would be used in judging the results for the the bonding energy for the helium atom.


7. The discussed “optimization” of the numerical parameters is strictly for my laptop:

Notice, if I were employed in a Western university or even at an IIT (or in an Indian government/private research lab), I would have easy access to supercomputers. In that case, much of this study wouldn’t be so relevant.

The studies regarding the atomic size determination, in particular, would still be necessary, but the results are quite stable there. And it is these results which tell me that, had I have access to powerful computational resources, I could have used larger boxes (which would minimize the edge-/size- effect due to the finite size of the box), and I could have used much, much bigger meshes, while still maintaining the all-important mesh cell-side parameter near the sweet spot of about 0.20 to 0.25. So, yes, optimization would still be required. But I would be doing it at a different level, and much faster. And, with much better accuracy levels to report for the helium atom calculations.

Inline update on 2021.02.02 19:36 IST: Addendum: I didn’t write this part very well, and a misleading statement crept in. The point is this: If my computational resources allow me to use very big meshes, and then I would also explore cell-sides that are smaller than the sweet-spot of 0.20 to 0.25. I’ve been having a hunch that the eigenvalue solver would still not show up the kind of degeneracy due to very deep PE well, provided that the physical domain size also were to be made much bigger. In short, if very big meshes are permissible, then there is a possibility that another sweet-spot at smaller cell-sizes could be reached too. There is nothing physical about the 0.20 to 0.25 range alone, that’s the point. Inline update over.

The specifics of the study mentioned in this post was largely chosen keeping in the mind the constraint of working within the limits of my laptop.

Whatever accuracy levels I do eventually end up getting for the helium atom using my laptop, I’ll be using it not just for my planned document but also for my very first arXiv-/journal- paper. The reader of the paper would, then, have to make a mental note that my machine could only support a mesh size of only 131 nodes at its highest end. For FDM computations, that still is a very crude mesh.

And, indeed, for the reasons given above, I would in fact be reporting the helium atom results for meshes in between 41 to 81 nodes per side of the cube, not even 131 nodes. All the rest of the choices of the parameters were made keeping in view this limitation.


8. “When do you plan to ship the code?”

I should be uploading the code eventually. It may not be possible to upload the “client-side” scripts for all the trials reported here (simply because once you upload some code, the responsibility to maintain it comes too!). However, exactly the same “server”- or “backend”- side code will sure be distributed, in its entirety. I will also be giving some indication of the kind of code-snippets I used in order to implement the above mentioned studies. So, all in all, it should be possible for you to conduct the same/similar trials and verify the above given trends.

I plan to clean up and revise the code for the hydrogen atom a bit further, finalize it, and upload it to my GitHub account within, say, a week’s time. The cleaned up and revised version of the helium-atom code will take much longer, may be 3–4 weeks. But notice, the helium-atom code would be giving calls to exactly the same library as that for the hydrogen atom.

All in all, you should have a fairly good amount of time to go through the code for the 3D boxes (something which I have never uploaded so far), run it, run the above kind of studies on the solid grounds of the hydrogen atom, and perhaps even spot bugs or suggest better alternatives to me. The code for the helium atom would arrive by the time you run through this gamut of activities.

So, hold on just a while, may be just a week or even less, for the first code to be put on the GitHub.


On another note, I’ve almost completed compiling a document on the various set of statements for the postulates of QM. I should be uploading it soon too.


OK, so look for an announcement here and on my Twitter thread, regarding the shipping of the basic code library and the user-script for the hydrogen atom, say, within a week’s time. (And remember, this all comes to you without any charge to you! (For that matter, I am not even in any day-job.))


A song I like:

(Hindi) दिल कहे रुक जा रे रुक जा (“dil kahe ruk jaa re ruk jaa”)
Lyrics: Sahir Ludhiyanvi
Music: Laxmikant-Pyarelal
Singer: Mohammed Rafi

[Another favourite right from my high-school days… A good quality audio is here [^]. Many would like the video too. A good quality video is here [^], but the aspect-ratio has gone awry, as usual! ]


History:
— 2020.01.30 17:09 IST: First published.
— 2021.02.02 20:04 IST: Inline updates to sections 5. and 7 completed. Also corrected a couple of typos and streamlined just a few sentences. Now leaving this post in whatever shape it is in.