## The electron always waves at you

This post has reference to Aatish Bhatia’s post on the subject matter bearing the title: “Hey There Little Electron, Why Won’t You Tell Me Where You Came From?” dated 27th September 2014 and published at the “Empirical Zeal” blog [^].

The post being referred to may immediately be perused.

“… Here’s the setup. On the table in front of me there’s a box with two thin slit-like openings at one end. We’re shooting particles into this box through these slits. I did the experiment with photons, i.e. chunks of light, but others have done it with electrons and […] For convenience, I’m going to call the objects in this experiment electrons but think of that word as a stand-in for any kind of stuff that comes in chunks, really. [bold emphases added] …”

Concerning the differences between electrons and photons, it would appear that it is not essential to look into all details of the proceedings that occurred when the present author defended his PhD thesis in mechanical engineering; however inasmuch as an inclusion of a reference increases the total number of references being cited for this post, the same [^] may perhaps be found in order.

“… And indeed, if you do this experiment with only one slit open, they behave just like baseballs, hitting the wall in a single band behind the open slit. …”

Further research is needed to develop a deeper understanding of the term: “single band.”

“… You can watch the electrons coming in one at a time in this video produced by scientists at Hitachi in 1989. …”

It is with great pleasure that the present author wishes to recall the inclusion of this video clip at the time of his conference presentation; the simulation he presented however was for photons.

“…Did the electron go through the left slit?

No! Because when you cover up the right slit, the stripey pattern disappears and you get a boring single band instead. …”

As has been mentioned above, more research is needed to develop a deeper understanding of the term: “single band.”

“…Did the electron go through the right slit?

No! For the same reason as above. When you cover up the left slit, instead of the stripey pattern you get a single band. …”

However, in the light of the further and deeper study of the reference post, it would appear that when the term “single band” is being used, the meaning being indicated is that of only one band as would be produced by a classical i.e. non-quantum mechanical particle.

“…As MIT professor Allan Adams puts it, that pretty much exhausts all the logical possibilities!…”

The quantitative logical closure contained in the salient reference which has been alluded to in the main reference post would be of great theoretical interest in general.

Therefore, it is one of the intermediate and urgent proposals of the present post to peruse this reference in an expeditious manner.

As Heisenberg and others taught us, although language fails us, it’s possible to come up with rules that correctly predict how tiny things behave. Those rules are quantum mechanics. You can learn these rules for yourself by reading Richard Feynman’s classic book QED…

It is further proposed to also include this reference in the literature review on a priority basis.

Kindly refer to Appendix A for details of the funds and employment opportunities needed. Kindly refer to Appendix B for details of justifications thereof.

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OK. I have had enough of this“research” talk (i.e. write). So let me switch back to writing in my usual blogsome way.

I think that Aatish makes a conceptual mistake here.

The electron always, say, “waves at you.” It does so even when it goes through only a single slit.

What you observe with two (or more) slits is an interference pattern. There of course is no interference pattern when only one slit is kept open. But even in this case, what you observe is not a single classical band (as for a heap of grains), but a diffraction pattern containing many fringes.

Many people, esp. science popularizers, make this error. They present QM as if a single— and by implication, classical—band is observed when one the two slits is closed.

… What could be the source of this error? Where might have it begun?

It was Feynman who highlighted the conceptual importance of the double-slit interference arrangement. He also wrote his text book in a very informal and attractive style. He has enormously influenced some two generations by now, including the pop-sci book writers. May be there was something to the way he presented this material, which misled people?

Check out the informal diagrams Feynman includes in his Lectures, esp. fig. 1.3, part (b) [^].

He draws each of the two single-slit probability curves with only single humps. Look at the curve for P2 more carefully. It gradually decreases in magnitude and becomes zero as you go up (i.e., along the positive x-axis; here the x-axis is taken vertically). However, no hint is given that this P2 curve would then once again increase in magnitude (or go to the right in this diagram) as you continue going further up, and it will thus have another, relatively much smaller peak—and then, an entire infinite series of similar (and progressively more and more faint) peaks.

In diffraction, these other peaks often are almost undetectably faint. For instance, see the two diagrams that appear just above the “Problem” section, here [^]. In fact, in terms of brightness/faintness, the first diagram just above the “Problem” section is only schematic; it depicts the other fringes with much more brightness than actually is the case. The diagram above it (i.e. the graph) is a better representation of the relative magnitudes involved. (Another point: These outer fringes in the single-slit diffraction also happen to be relatively very faint when compared to the intensity of the interference bands which appear when both the slits are kept open.)

However, in the context of the single-slit, Feynman doesn’t explicitly say anything at all about the diffraction phenomenon—either in the text or in the diagram.

People then must have over-interpreted his diagram, and wrongly thought that (i) when an electron goes through a single slit, it behaves exactly similar to how grains falling through a chute behave, and (ii) when both slits are open, the electron somehow begins to behave something like a wave, something like a superposing quantum particle.

Thus, the idea being advanced is: single-slit means classical grain nature (Bhatia uses the example of baseballs); double-slit means in part a wave nature, as in QM. This characterization is wrong.

The faulty interpretation also makes QM sound more mysterious than it actually is.

Since Feynman’s double-slit diagrams anyway were only schematic (observe that the curves for P1 and P2 are manually drawn and therefore they are not precisely symmetrical), he could have shown a bit of the fringing effect in the single-slit too, with the usual note: diagram not drawn to scale. That single feature would have saved a significant error in so many popular expositions.

Feynman himself does explicitly note the fact of appearances of fringes even in the single-slit diffraction, but only in his later QED book—and only by way of a footnote. (Sorry, can’t locate it for you. I don’t have the book ready with me right now—it is still packed in a carton when I moved from Mumbai to Pune after my job-loss in January. But I do remember that it is in the early parts of the book, very probably in chapter i.e. lecture 1.)

Since I have exhausted my ‘net bandwidth for this month, I couldn’t go through the MIT professor Allan Adam’s video that Aatish Bhatia refers to. (It’s more than 1 hour long.) Instead, I checked out his PDF course notes, to see if he too makes this common mistake. … Well, that way, I didn’t actually expect Adam to repeat this mistake, but since Bhatia makes an enthusiastic reference to it, I wanted to check out. The relevant course notes are here, L2 [(.PDF) ^]. As expected, the notes don’t actually commit the mistake, but still, they repeat the same omission (of diffraction). Adam says in his notes (p. 7–8)

Hence, determining through which slit an electron passes does away with the interference pattern.

He could add an explicit mention of the diffraction pattern.

Though both electrons and photons would show a similar behaviour, it should be easiest to demonstrate the diffraction effect using light rather than electrons. My unpublished simulation of the PhD times showed a gradual “morphing” from a full double-slit interference pattern to a full single-slit diffraction pattern, as the detection efficiency of the photon detector placed near only one of the two slits was increased. Very natural. Check out Adam’s maths on p. 6:

$A(y) = A_0 \left( e^{i\theta_1(y)} + e^{i\theta_2(y)} \right)$

Split it up keeping two different terms $A_1$ and $A_2$, even if we assume $A_1 = A_2$:

$A(y) = A_1 e^{i\theta_1(y)} + A_2 e^{i\theta_2(y)}$

Gradually take, say $A_1$, to zero. You gradually transition from interference to diffraction.

Why might Bhatia have made the error? Here is a speculation.

He mentions the experiment with photons, which he did have an opportunity to perform, as an undergraduate student.

It’s possible that the geometrical scaling of the experimental arrangement was such that the outer diffraction fringes got placed outside the detection limits of the CCD camera. It’s also possible that they ran the experiment mainly to bring out the particle nature well, and so, they effectively “expanded” the time axis a great deal (by using a very low flux rate). Now, for the single-slit version, the outer diffraction fringes are very faint, as compared to the big fringe in the middle. Therefore, in the actual experimentation, enough particles might not have been registered at the locations of these outer diffraction fringes, at least over the relatively shorter duration of the experiment.

Eleven years later, as he began sharing the joy and excitement of seeing the quantum nature in action from memory, he might have focused a bit more on the dramatic features of the experimental measurements and forgotten the correct theory that helps explain it. Possible.

Or, it is also possible that this bit simply skipped his attention. Quantum theory actually is a very big theory—it has a very large scope. As a typical university student, your goal is to master its mathematical tools. It’s easily possible that you skip over some of the rudiments rather quickly. It happens to almost everyone. The initially unintuitive nature of QM isn’t the only thing that people get used to; they also keep adding many small details in their understanding, because no one book can possibly cover all such details, applications, etc.

Anyway, he would know best.

… I have already dropped a comment at his blog, mentioning the concepts-wise serious nature of this mistake. It sets people—the newcomers, the layman—on a wrong path. It did that to me, for more than 5 years (in fact almost a decade, perhaps more, if you count the time from the very first exposure to the wave-particle duality, which was sometime in XI or XII standard in my case).

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A Song I Like:

[A very careful study of this song is advisable. There will be examination, once the song is over.]

(Marathi) “hee gulaabee hawaa, veD laavi jeevaa…”
Singer: Vaishali Samant
Lyrics: Guru Thakur

[Open Book Examination: Peruse online resources such as the material made available here [^] and here[^], and use it in order to determine whether the above-mentioned song may be deemed to be based on the North Indian Classical “raaga” “marwaa,” or otherwise. Provide detailed explanatory comments in similar English.

Extra Credit: Imagine how, in an Indian classical music concert held, for example in Goa, how Shobhaa ShiroDkar (i.e. Shobhaa GurTu) could have rendered this song. Then, using her Marathi song “maajhiyaa priyaalaa” as the propaedeutic for “ucchaaraNa bhed” and voice culture, render the above song the way GurTu would have, attempting to your fullest capacity an imitation of her voice and scale. Submit the evidence of your attempt via a CD-quality recording.

Extra Extra Credit: Repeat the aforementioned exercise (including GurTu’s voice and scale), but using another “raag” of the “maarwaa thaaT,” viz. “puriyaa kalyaaN!”]

[E&OE]

I won’t take chances. This was a comment I just made this morning at Prof. Dheeraj Sanghi’s blog, here [^]. Comments at his blog are moderated. And, I don’t know if he will allow it in. (There could perhaps even be some valid reasons for this comment not to be run there.) So, I have decided to go right ahead and note my comment here, too. (On second thoughts, as I have often said earlier on this blog, I anyway think that I should be bringing here many other comments I have made over a period of time at many other blogs, too.)

Here is the comment I have just made at Sanghi’s blog:

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Quote:

Dear Dheeraj,

You write interestingly, even engagingly. Well, at least, you write—as in contrast to mostly just excerpting from Internet links!

I don’t mean to fully defend the practice that has been adopted. I just wish to note down out a few points that seem to be contrary to the flow of your argument, a few points that passed through my mind.

When you recruit a lower-level employee, a PhD student, or a professor, you do follow the meticulous process you mentioned; it involves lengthy interviews, too. Why might someone not follow a similarly long interview while recruiting IIT directors?

I think that some at least plausible answer may be hidden right in that question.

For the starters, when it comes to the candidates for the director’s post, as against the other posts you mentioned, simply because all the candidates have already been subjected to a meticulous process, throughout their prior career, typically spanning over decades.

They have been observed and evaluated at the senior and responsible positions for at least a decade or more by multiple, disparate, parties. … Any comments they make at professional conferences, any viewpoints they offer at the industry-institute interactions, the quality of the documents they write for obtaining funding, etc. Also, the blogs they write [ ;-) ]. And, they have been continuously evaluated by various parties: h-Index (certainly), student evaluations (if these are taken seriously at IITs)—and, certainly, via the annual reviews from their seniors, which includes mandatory remarks from the viewpoint of their potential as leaders. The CRs (annual confidential reports), made over a decade+ times (through various political dispensations, under many different HoDs and Deans and Directors) do have some purpose, you know—i.e., if these are taken seriously at IITs!

They also have been short-listed by the formal selection committees. Presumably, the committee’s role does not end only with providing an unordered list of names. Presumably, the short-listing committee takes its job seriously.

IITs are not private institutes. The top decision makers here, by explicit organization structure, are the concerned ministry/ministries. Whether you like it or not, they do have their regular input channels, too—channels other than the selection committees. In India, in case you have happened to overlook it, we have more than 10 central agencies for internal intelligence gathering. When the body called the Planning Commission got dismantled, another one stepped in to fill the vacuum.

Another point: At the director’s level, IITs also typically do not go for rank outsiders. Most, if not all, of what I say would remain valid even if the candidate is an outsider.

The cumulative input from multiple sources thus is already there. It is distilled, and available, just in case not already factored in, by the time the short-list is made. And, then, there are internal reviews.

The final interview, thus, is more or less just a formality. Shocking? Why should it be, to you?

And, doesn’t this happen in the USA anyway—and I mention this point, because I know that at IITs, esp. at IITK, a top-10 US PhD is routinely valued better than a PhD that COEP graduated after a failure at a PhD program in a 50+ USA school. Thus, mentioning the US practice should be perfectly acceptable.

Would a colleague of yours in the USA—one who values your word—even bother to talk with someone you strongly recommend, i.e., with a personal touch of yours? Do they? actually? even for just five minutes? Especially if they themselves know someone trustworthy other than you, who personally knows the post-doc applicant? Do you find their practice offensive? Did you find it offensive when Manindra Agarwal’s students received offers for post-docs etc., even before submitting their PhD theses at IITK? Did you begin blogging something about the fact that there was no 30 minute interview, not even 5 minute interview for them? Do you hasten to wear your skeptical glasses if an IUCAA PhD student gets a post-doc offer at Princeton or CalTech even before submitting his thesis?

At this point, you should be a bit bemused, perhaps even a bit agitated, but you would still not be convinced. There is a bit of valid reason for it, too. I can understand and sympathize with your viewpoint.

You see, I myself have undergone a similar kind of a process—the kind that you criticize. When I applied for a professor’s position at COEP, what actually happened was that, apart from submitting my application (manually making sure that it was duly entered into the inwards register), I then dropped by a few professor’s cabins in the department, and then, also the Director’s cabin. I broached the metallurgy-to-mechanical branch-jumping issue with him, and sought his opinion about it. To cut a long story short, he bluntly told me that he has had no objection on that count (it was he who had given me an opportunity to teach an FEM course before my PhD thesis was defended), but that, as a director, what the department thinks, he said, was more important to him. And, while the department had thought differently earlier, when my PhD guide was still in it (or had just left it), now the department had begun “thinking” some “different” way.

I was duly short-listed, called for the interview, and it became evident to me within the first 1–2 minutes the nature of what to expect. (Doesn’t it, if you are past your 40?… In my case, I could tell right when I was in my 20s.) The interview did last for about 30 minutes—I stretched it, because I wanted to tell them in sufficient detail—while all along, they were just wanting to hurry it up and wrap up it all. … To cut a long story short, in the end, they selected someone whose thesis had been examined by a low-ranked NIT’s low-ranked professor, whereas every one in COEP knew that my guide had, on my informal remarks, dared contacting people from top 5 univs in the USA for examination of my thesis (including Frank Wilczek). That none of them bothered to examine it is a different story. The end result was that after almost 1.5 years, my thesis was finally picked up for examination by two senior professors from one of the five old IITs—both of whom had been HoDs and Deans, and one later on was a Director of a central lab. Now regardless of this difference, COEP showed me the door. As expected and made clear right during the interview process. (“Are you now casting aspersions that we don’t know what is good for this institute?” etc. When I say I had stretched them to 30 minutes, I mean it. After taking the decision, they did not take care to inform me of the outcome. I saw the director. He managed to sympathize with me. Though he didn’t say a thing, I knew that he knew that I knew that I should have known that I would not get selected.

Just a COEP professor’s post and an IIT director’s post, there is a difference, you say?

Well, Dheeraj, you then speak more like a typical IAS officer or a second-rate corporate MBA, than like a professor. If a director directly impacts some 500 faculty members over his entire term(s), a professor impacts some 500 students every year. And the impactees in the second case are both far more sensitive and powerless. And, with far longer period of their future at the stake.

If there were to be betting rackets for IIT Directors’ positions, the going rates would almost consistently get the selections right, regardless of change of political dispensations, and without the benefit of even a one minute interview. Why is a five minute interview so difficult to get by top ranked IIT professors, cognitively speaking.

And if you still say that the five minutes interviews still are not acceptable because the process can result in wrong/bad selections, well, you only join me, my argument—you cast doubts on the short-listing and the real reviewing processes, on the grounds that some people who could easily become second-rate directors, too, had got short-listed by the selection committee. Exactly similar to what happened to many other candidates in the COEP process. Not just short-listing, but the internal reviews before the interviews even began.

But then, who blogs about a non-JPBTI anyway—let alone for him? Who defends him? Answer: None—if his PhD guide is dead.

These are some of the things that passed by my mind, while thinking about this directors’ selection issue. I don’t pretend to know or understand the full situation. But I do know that what I said is, in many important ways, relevant.

Best,

–Ajit
[E&OE]

Unquote

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A Song I Like:
(Hindi) “aayaa hai mujhe phir yaad wo zaalim…”
Singer: Mukesh
Music: Roshan
Lyrics: Anand Bakshi

[E&OE]

I had made one comment in response to a post and a couple of replies, at the nanopolitan blog, here [^]. Since the comment was long, I had saved it. To my surprise, for some reasons not known to me, it was gone the next day.

If they were to run my comment, it would have appeared immediately after the comment by one “Sushant Rai” (on 4:36 PM, March 23, 2015).

In the next section I copy-paste my comment (which, as I said, assumes the context of the previous discussion) exactly as it originally appeared (including mistakes/typos, and the emphases in italics or bold):

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Quote:

No, Ankur, there is a basic difference between an academic institution and a corporate house. … You would know about it, but just in case you don’t, check out Dijkstra’s article on academia, here: http://www.cs.utexas.edu/users/EWD/transcriptions/EWD11xx/EWD1175.html

Now, a bit about the similarity. Mistry is the first non-Tata to head the house of Tatas. The party to hold the purse-strings does make the key difference. Whether it’s the majority or the critical shareholders, or, the controlling politicians and/or bureaucrats.

If TIFR were to be a private company (e.g. a coaching classes company), then what you say would have been applicable. It emphatically is not.

Government interference in economy is always bad; the public sector science and academia is no exception. The academia would survive (cf Dijkstra) even on public money, but don’t count on keeping quality. The broader context itself is wrong.

If you ask me: If anything, be thankful to your luck/stars/etc. that you all at IIXs etc. still get to exercise at least as much freedom as you do. The broad systemic nature doesn’t actually allow it. … Some memories of some decent traditions of the yesteryears’ private universities abroad, and some memories of some decent simulation thereof here, is the reason why you still get as much decent a treatment as you do. Visit a “private” engineering college and ask around.

As to a director’s post, I do think that these, too, should be publicly advertised. And, the names of all the people involved in the decision-making process should also be publicly declared as well. One shouldn’t have to file an RTI application for that.

After all, the government/public sector also is far more easily susceptible to the old boys network sort of a thing, as compared to the corporate sector. (There already have been articles in the media about how even some retired judges have landed plush jobs immediately after their retirement, and how courting for favours (!) might have gone before their retirement from career 1.0.)

As a long-time sufferer at the hands of those who have peopled the premier institutes in the Indian education system (“What? Metallurgy? Why did you come here? Don’t you know this is the Mechanical department?” and “What, only GATE? No JEE? You are worse than a dog then!” (Ok, this second bit is a bit of an exaggeration)), I do like it when it does receive a dose of its own medicine once in a while.

And I am sure, Sandip (Trivedi) won’t go jobless in the meanwhile—he cleared the JEE, did PhD under Preskill, and co-authored with Frank Wilzcek. He will get to continue at the scenic Colaba campus in the meanwhile, too. That’s the bottom-line. (Or at least the one I draw.)

Best,

–Ajit
[E&OE]

Unquote
[The time of my comment, soon after posting it, was shown at the nanopolitan blog as “5:58 PM, March 23, 2015”]

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Oh, BTW, I of course welcome the recent Supreme Court judgment scrapping the section 66A of the IT act.

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A Song I Like:
[How I wish the recording technique were better here!]
Music: Ilaiyaraaja
Singers: S. P. Balasubrahmanyam and S. Janaki
Lyrics: R. N. Jayagopal

[E&OE]

## Certain features of Dirac’s notation and a physical analog

Important Update on 2015.03.20:

tl;dr version: Don’t bother with this post. It’s in error.

Long version:

On second [and third…] thoughts, I think that this post has turned out to be just bad. (I am being serious here.) Regardless of whatever seeds of some good or promising ideas there may be in it (and I do think there are some), there also are far too many errors or wrong ideas in it, and the errors make the overall description just plain wrong.

If you are interested in knowing which ones I now think are bad or very bad, drop me a line. That is, should you decide to read this post at all, in the first place—something I won’t recommend. The only reason I am keeping this post is to keep a record of how crazy QM can sometimes get to get, especially to me. [Yes, even if I have published a paper on some aspects of the foundations of QM.]

Yet, if you still choose to go through this post, then I would say: OK, go through it, finish reading it, and then come back to this point once again, and think about points like these: (i) Why two chambers? Ideally, there should be only one chamber. (ii) Does the system really model a complex-valued vector and its conjugate correctly? Answer: no. (iii) Does the system model the vector-matrix-vector multiplication right? Answer: no. (iv) Does it even model the multiplication? Answer: no, not really. (v) There also are other inconsistencies.

Of course it’s a fact that as far as QM is concerned, I don’t get to discuss ideas with any one—there is absolutely no informal tossing of ideas back and forth with any one—no fleshing out (or thrashing out) of ideas at the blackboard, gaining clarity as you go on explaining them to someone else (say to a student), nothing. … So, things do get a bit crazy. … Yesterday, I met an engineer friend, and thus had my very first chance to speak with anyone else about the ideas of this post. I could not discuss the QM aspects of it because he hasn’t studied it, but I could at least discuss phasors and conjugates, vectors and matrices, Fourier transforms and waves, etc. I told him the kind of error I thought I was making, and asked him to confirm it. Frankly speaking, he was not sure. He could give me a benefit of doubt because of symmetries, though, being an informal discussion (over a small drink), we let it go at that. But whatever he happened to mention also brought phasors into full focus for me. That was enough to confirm my suspicions. … Finally, today, I decided to put on record the bad points, too.

No, I will not give up attempting to model the Dirac notation via some easily understandable physical analogs. And if I get to something right, I will sure post about it.

That way, these days, I hardly even look at QM (except for browsing of others’ blogs now and then). I am mostly thinking or reading or working something about my other researches—water conservation, CFD, FEM, etc. So, it will be a long while before I could possibly take out some time to get down to thinking about the Dirac notation and all, as my primary thinking goal. And, it can only be after that, that if I at all get something about it consistently right, I could post something about it.

All that I am saying, in the meanwhile, is that no matter how many seeds of some workable ideas this post might otherwise have, the system description in this post is in error. It is bad—bad, even as an analogy. Treat it that way.

Let me not bother with this post any further.

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[Note: I have added a significant update (more like an extension) on 2015.03.19]

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This post follows my browsing of Piotr Migdal’s guest post on John Baez’ blog, here [^], yesterday. Migdal’s aim is make QM simple to understand. He somehow begins with Dirac’s notation, and rapidly comes to stating this formalism:

$E = \langle \psi | H | \psi \rangle$

I read through about half of this post, and then rapidly browsed through the remaining part, before returning to this formalism and begin thinking a bit about it. … After all, he was doing something about presenting the QM ideas as simply as possible, you know…

Then, an analogy struck me. It’s based on my ideas of QM, of course—remember those pollen grains and the bumping particles and all that stuff which I had written a couple of months ago or so? (On second thoughts, here it is: [^].)

Anyway, let me share with you the analogy that struck me today. If you find something objectionable with it, sure feel free to drop me a line.

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A physical system with a gas-filled cylinder:

Consider a cylinder with two pistons, one at each end, and a rigid, impermeable but movable partition in the middle. Assume that the system is frictionless.

Suppose that both the chambers of the cylinder are of the same length and that both are filled with the ideal gas to the same pressure—some sufficiently low pressure.

Now suppose that the piston on the right hand side (RHS for short) is moved to and fro at a constant angular frequency $\nu$, a certain maximum displacement $A$, and a certain initial phase $\theta_0$. This motion can be specified using a phasor, i.e. a complex number; the phasor rotates in the CCW sense in the abstract phasor plane.

The RHS piston imparts momentum to the gas molecules in the right chamber. The generated sound waves hit the central partition, impart it the momentum, and thus tend to make it move back and forth as well.

But suppose we wish to ensure that the partition in the middle remained stationary. How could we accomplish this goal?

If you were allowed to move the piston on the left, in precisely what way would you move it so that the central partition remained motionless at all times?

Obviously, you would have to move the LHS piston in such a way that its frequency and maximum amplitude are the same as for the RHS piston, viz., the same values as $\nu$ and $A$. However, the initial phase of the phasor for the LHS piston must be made  $-\theta_0$ (opposite to that of the RHS piston), and the sense of rotation of the phasor for the LHS piston must be made CW (whereas that for the RHS piston had the CCW sense).

If the pistons were to be linked to the central partition via ideal continuous springs, then the central partition would always remain perfectly standstill.

However, if instead of springs, a gas is used for filling the chambers, then since a gas is made of only a finite number of discrete molecules, the transmission of momentum to the central partition acquires a discrete character. Further, if the molecules are randomly distributed (in terms of either positions, momenta, or both), then the momentum transmission acquires a stochastic character.

As a result, the partition does not remain perfectly standstill at all times, but undergoes a small, random, vibratory motion.

In the terminology deployed by QM, the position of the partition is said to be, you know, uncertain.

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Update the next day (on 2015.03.19)

Let me rapidly note down a few additional points (some of which should be very obvious to many):

(i) Irregular pulses instead of a regular (single) sine wave:

The motion of the RHS piston doesn’t have to be perfectly sinusoidal. Even if the motion is a rather irregular wave (as is the case when one side of a drum is banged), such a motion can always be analysed via the Fourier transform. In other words, $|\psi\rangle$ now has several basis components of different frequencies. Doesn’t matter; just make sure that for each frequency component, the LHS piston too perfectly opposes the motion.

(ii) A system with parallel grooves:

For illustration via a working physical model (or for implementation in a C++ program), I think it could be better to think of the following situation.

Suppose there are ten (/hundred/thousand) straight-line grooves smoothly carved into a horizontal platform. All the grooves are of same width and lie parallel to each other. Suppose, there are several ball-bearing balls placed in each groove (the number per groove may or may not be constant). At the initial time, the balls are placed at randomly different distances. Instead of the RHS piston, we now have a rigid plunger normal to the grooves; it simultaneously moves through the same distance over all the grooves—something like a comb going over some parallel scratches. The middle partition and the LHS piston, too, of course are something like this “comb.” The balls represent the gas molecules. This mechanism makes the one-dimensionality of motion (positions and momenta) inescapable. You can figure out the rest. (For instance, ask yourself what role does the initial speed of a ball has? Does it imply anything towards an independent frequency component, energy, basis vector? Can all balls in a given groove have random initial positions but the same initial speeds, with balls from different grooves differing in speeds? Etc.). You can more easily implement a software program than a build a physical model, to study the behaviour.

(iii) Trying something for the quantum discreteness:

If you wish to go even further, think of having side-walls parallel to and outside of the extreme grooves, and suppose that these walls carry some serrations. Suppose also that the middle partitioning “comb” carries a small ball and a spring (lying in the plane of the comb) in such a way that the comb successively halts only in the valleys of the serrations, The middle partition thus snaps in at discrete positions, say, $0$, $\pm \Delta x$, $\pm 2 \Delta x$, $\cdots$, etc., thereby imparting the motion of the partition something like a discrete character.

Finally, if you must have something to stand in for that $H$ symbol, think of a system with two symmetrically placed middle partitions instead of just one—say, one each at $\pm x$. This gives rise to a system of three chambers. For a system with the ideal gas, insert a sensitive thermometer in the central chamber. It will measure the level of the kinetic energy contained within the central chamber. …

Honestly, though, at least to me, this idea looks like an overkill. After all, the entire system still remains only classical. It merely serves to highlight some of the features of QM—not all.

(iv) What all these systems are good for:

Realize, all the above models are purely classical. None is fully quantum. They do, however, help simplify and bring out certain features of QM.

As far as I am concerned, even a simple C++ program with just two chambers (or parallel grooves with just one partition) might be enough—it will still bring out the the discrete and stochastic momentum-transmission events, and the 1D random walk undergone by the middle partition.

And even this simple a system should bring out many more features of the quantum formalism pretty well… Features like: the necessity of complex numbers in the Dirac notation, the necessity to define the row vectors with complex conjugates, the idea of basis vectors for the column and row vectors, etc.

This is good enough. It is much better than letting your ideas float in an abstract Dirac sea the thin air—thereby making you susceptible for recruitment by many quantum interpretations [^]. The chance that irrational ideas have to grab or overpower your mind is inversely proportional to the clarity which you derive about even simple-looking, basic, concepts. Even a partial clarity can be sometimes good enough. I mean not some half-baked knowledge, but a full clarity on some aspects of a very complex phenomenon. You can always build on it, later.

Bye for now. In the next post, I will return to some notes from my studies of the micro-level water resources engineering.

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A Song I Like:
Music: Hridaynath Mangeshkar
Lyrics: N. D. Mahanor
Singer: Asha Bhosale

[E&OE]

## Micro-level water-resources engineering—2

As mentioned in my last blog post, I have been browsing material on the title subject.

In this post, let me note down a few informative links that I have (only) browsed (but not completely read through) thus far. I will come back to my own notes and observations (based on them) in the subsequent posts. BTW, I intend to keep this post as a catch-all thing: whenever I find a new interesting link, I will come back and note it here, without separately mentioning update dates and all. (I think I will also consider converting this post into a separate page of this blog or my personal Web site.)

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Portals:

India Water Portal [^] (Web sites like these are, IMO, better than novels in English :) )
Rainwater Harvesting [^]

Government and Public Sector Portals/Sites:

IMD: [^]

IITM: [^]

Central Groundwater Board:  [^]. This is a big site/portal. So, let me note down the links to some specifically relevant parts of it:  Downloads [^], Watershed [^], Aquifer Systems of India (and a few states) [^], Groundwater Yearbooks [^], Groundwater Scenario in India [^].

The World Bank funded projects in India, phase I and II: [^][^]

A popularization kind of a book on Rajasthan’s water culture: [^]. Incidentally, it is through this book that I came to place Rajendra Singh’s work in a better context. The last time I had wondered why Singh didn’t go 400 km West. This book clarified the matter to me.

US Dept of Agriculture Report: Technical Guide to Managing Ground Water Resources [(.PDF) ^]

Groundwater Manual [^]

A book at the US GS site: [^]

A research group at CTARA, IIT Bombay: [^]. Reports and Course Materials at [^],  [^],  and [^]

A research group at IISc Bangalore: [^]. An example of a project they are carrying out: [^]

A private research cum consulting group from Pune (with many academic projects conducted with the Geology Dept. of S. P. University of Pune, too): [^]

CP Kumar’s links on hydrology [^] and on hydrology resources [^]. He works at the National Institute of Hydrology: [^]. There is a learning package for hydrology for the beginners, too: [^]

Indian Association of Hydrologists [^]

Software:

Hydrology Software:

Lists of software maintained at the USGS site, in general [^], and for groundwater in particular [^].

A proprietory software developed for use by the government agencies in India [^]:

Open-source GIS software:

Wiki list [^].

The following two seem to be more general purpose and/or leading; they also are multi-platform: QGIS (I think IIT Bombay people use it) [^], and GRASS [^].

An open-source GIS software on Windows (.NET) platform: [^]. US EPA uses it: [^]. I installed and tried it, but the documentation seems to be lagging behind the software.

ParFlow: [^]

List at the GIS Lounge: [^]

Rainfall and Its Measurement:

Annual rainfall animation [^]. Check out the animated GIF [^]. A surprise: check out the low rainfall area which the animation shows for the Konkan region. That is because while creating the animation, they coarse-grained the data. There are unexpectedly low-rainfall region even in Konkan, but these are rather isolated. Once again highlights the importance of the local data. But, it’s entertaining anyway.

Another royal entertainment (reduce your computer’s volume before hitting the link): [^]. Then, to see the actual action, hit the “Play the whole sequence” button. (This is one of the rare times that you would wish you had an Intel 386.)

Just in case you want to keep a record of the rainfall in your area, in India, we follow these specs  [(.PDF) ^].

In case you didn’t know, 1 mm of rainfall at a point means “A 0.001 m3, or 1 litre of water to each square metre of the field” [^]. … 1 cm of rainfall is ten times that number.

Exercise:

On the Internet, look up the area of a state, district, taluka, or city; look up its average annual rainfall; then find the total quantity of water (in litres) it receives via rainfall in a typical year.

Then, also do searches and find out data about its total water demand. Also, find out its current water availability, and the short-fall in the supply.

Trivia:

The average annual rainfall for India is about 70 cm in monsoon alone, and about 110 cm for the entire year (including the non-monsoon rains, snow-fall, etc.) (Source: [^]. Also see: [^]).

Floods and droughts still visit India every year.

The average annual rainfall in Jaisalmer is just 16.4 cm (less than one-fifth of that at Delhi), and all of it is received over only 10 days. (Yes, statistically speaking, as many as 355 days in a year go completely dry there.) The water-table depth there is really bad; it ranges between about 40 to 80 m (i.e., about 125 to 250 feet) [^].

Jaisalmer nevertheless has a huge lake that would supply water to the city [^] all through the year—the lake would not go dry even in summer! This lake: Image [^], video [^].

No, that lake doesn’t get its water supply from a river or groundwater sources; there is in fact no mountainous or hilly region around it. The only source of water for this lake is: an ingenious scheme for rainwater harvesting. A scheme that is almost 7 centuries old.

Now, go, figure how wasteful—and flood-hit—and water-scarce—the rest of us manage to remain even today.

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A Song I Like:
(Marathi) “ye re ghanaa ye re ghanaa…”
Music: Hridaynath Mangeshkar
Lyrics: Aarati Prabhu
Singer: Asha Bhosale

[E&OE]