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

Links, and my comments:


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

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

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

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

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

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

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

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

There are many neat passages like this in the text.


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

Then I ran into

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

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

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

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

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

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

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

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


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

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

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

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

Err… Why “almost,” Lubos? 🙂

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

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


Finally, a post at the SciAm blog:

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

The authors say:

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

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

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

Roger Schlafly’s blog [^],

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


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

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


A general update:

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

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

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

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

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

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

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

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

So, there. Bye for now.


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


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

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

 

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Yes I know it!—Part 2

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


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

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

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

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

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

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

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

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

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

Yes I know it!

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


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

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


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

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

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

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

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


Just wanted to remind you, that’s all!


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

I have moved the update to a new post.

 


A Song I Like:

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

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

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

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


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

 

 

Busy, busy, busy… And will be. (Aka: Random Notings in the Passing)

Have been very busy. [What’s new about that? Read on…]


First, there is that [usual] “busy-ness” on the day job.


Then, Mary Hesse (cf. my last post) does not cover tensor fields.

A tensor is a very neat mathematical structure. Essentially, you get it by taking a Cartesian product of the basis vectors of (a) space(s). A tensor field is a tensor-valued function of, say, the physical (“ambient”) space, itself a vector space and also a vector field.

Yes, that reads like the beginning paragraph of a Wiki article on a mathematical topic. Yes, you got into circles. Mathematicians always do that—esp. to you. … Well, they also try doing that, on me. But, usually, they don’t succeed. … But, yes, it does keep me busy. [Now you know why I’ve been so busy.]


Now, a few other, mostly random, notings in the passing…


As every year, the noise pollution of the Ganapati festival this year, too, has been nothing short of maddening. But this year, it has not been completely maddening. Not at least to me. The reason is, I am out of Pune. [And what a relief it is!]


OK, time to take some cognizance of the usual noises on the QM front. The only way to do that is to pick up the very best among them. … I will do that for you.

The reference is to Roger Schlafly’s latest post: “Looking for new quantum axioms”, here [^]. He in turn makes a reference to a Quanta Mag article [^] by Philip Ball, who in turn makes a reference to the usual kind of QM noises. For the last, I shall not provide you with references. … Then, in his above-cited post, Schlafly also makes a reference to the Czech physicist Lubos Motl’s blog post, here [^].

Schlafly notes that Motl “…adequately trashes it as an anti-quantum crackpot article,” and that he “will not attempt to outdo his [i.e. Motl’s] rant.” Schlafly even states that he agrees with him Motl.

Trailer: I don’t; not completely anyway.

Immediately later, however, Schlafly says quite a remarkable thing, something that is interesting in its own regard:

Instead, I focus on one fallacy at the heart of modern theoretical physics. Under this fallacy, [1] the ideal theory is one that is logically derived from postulates, and [2] where one can have a metaphysical belief in those postulates independent of messy experiments.” [Numbering of the clauses is mine.]

Hmmm…

Yes, [1] is right on, but not [2]. Both the postulates and the belief in them here are of physics; experiments—i.e. [controlled] observations of physical reality—play not just a crucial part; they play the “game-starting” part. Wish Schlafly had noted the distinction between the two clauses.

All in all, I think that, on this issue of Foundations of QM, we all seem to be not talking to each other—we seem to be just looking past each other, so to say. That’s the major reason why the field has been flourishing so damn well. Yet, all in all, I think, Schlafly and Motl are more right about it all than are Ball or the folks he quotes.

But apart from it all, let me say that Schlafly and Motl have been advocating the view that Dirac–von Neumann axioms [^] provide the best possible theoretical organization for the theory of the quantum mechanical phenomena.

I disagree.

My position is that the Dirac-von Neumann axioms have not been done with due care to the scope (and applicability) of all the individual concepts subsuming the different aspects of the quantum physical phenomena. Like all QM physicists of the past century (and continuing with those in this century as well, except for, as far as I know, me!), they confuse on one crucial issue. And that issue is at the heart and the base of the measurement/collapse postulate. Understand that one critical issue well, and the measurement/collapse postulate itself collapses in no time. I can name it—that one critical issue. In fact, it’s just one concept. Just one concept that is already well-known to science, but none thinks of it in the context of Foundations of QM. Not in the right way, anyway. [Meet me in person to learn what it is.]


OK, another thing.

I haven’t yet finished Hesse’s book. [Did you honestly expect me to do that so fast?] That, plus the fact that in my day-job, we would be working even harder, working extra hours (plus may be work on week-ends, as well).

In fact, I have already frozen all my research schedule and put it in the deep freeze section. (Not even on the back-burner, I mean.)

So, allow me to go off the blog once again for yet another 3–4 weeks or so. [And I will do that anyway, even if you don’t allow.]


A Song I Like:

[The value of this song to me is mostly nostalgic; it has some very fond memories of my childhood associated with it. As an added bonus, Shammi Kapoor looks slim(mer than his usual self) in this video, the so-called Part 2 of the song, here [^]—and thereby causes a relatively lesser irritation to the eye. [Yes, sometimes, I do refer to videos too, even in this section.]]

(Hindi) “madahosh hawaa matawaali fizaa”
Lyrics: Farooq Qaisar
Singer: Mohammed Rafi
Music: Shankar-Jaikishan

[BTW, did you guess the RD+Gulzar+Lata song I had alluded to, last time? … May be I will write a post just to note that song. Guess it might make for a  good “blog-filler” sometime during the upcoming several weeks, when I will once again be generally off the blog. … OK, take care, and bye for now….]

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

What am I thinking about?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


What should I think about?

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

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


If not…

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

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

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

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

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

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

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

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


OK then, bye for now.


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


[E&OE]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Anyway, interesting reading.


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

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

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

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


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


[E&OE]

In QM, local action does make sense

We have been too busy with our accreditation-related work, but I still had to squeeze in a comment at Scott Aaronson’s blog.

In case you don’t know, Scott Aarsonson is a tenured Associate Professor in CS at MIT (I mean the one in Cambridge, MA, USA). Scott’s opinions count—at least, they are very widely read (and often, also very extensively commented on and discussed).

This year, Scott was invited to respond to the Edge’s annual question [^]. In the latest post on his blog covering his and others’ responses to the question [^], Scott singled out three answers by others (at the Edge forum) which he thought were heading in the wrong direction. In Scott’s own words:

Then there were three answers for which the “progress” being celebrated, seemed to me to be progress racing faster into WrongVille

In particular, the following residents of the so-called “WrongVille” were of immediate interest to me; let me continue quoting Scott’s words:

Ross Anderson on an exciting conference whose participants aim to replace quantum mechanics with local realistic theories.  (Anderson, in particular, is totally wrong that you can get Bell inequality violation from “a combination of local action and global correlation,” unless the global correlation goes as far as a ‘t-Hooft-like superdeterministic conspiracy.) [Emphasis in bold is mine.]

The minimum implications here are these two: (i) quantum mechanics—not this interpretation or that interpretation of its existing mathematics, but the entire mechanics of the quanta itself—cannot ever be local, and (ii) therefore, any attempts to build a local theory to explain the quantum phenomena must be seen as a replacement for QM [a lock, stock and barrel replacement, I suppose].

One further implicit idea here seems to be that any local theory, if it yields the necessary global correlation, must also imply superdeterminism. In case you don’t know, “superdeterminism” here is primarily a technical term, not philosophical; it is about a certain idea put forth by the Nobel laureate ‘t Hooft.

As you know, my theorization has been, and will always remain, local in nature. Naturally, I had to intervene! As fast as I could!!

So I wrote a comment at Scott’s blog, right on the fly. (Literally. By the time I finished typing it and hit the Submit Comment button, I was already in the middle of some informal discussions in my cabin with my colleagues, regarding arrangements to be made for the accreditation-related work.)

Naturally, my comment isn’t as clear as it should be.

It so happens that our accreditation-related activities would be over on the upcoming Sunday, and so, I should be able to find the time to come back and post an expanded and edited version early next week. Until then, please make do with my original reply at Scott’s blog [^]; I am copy-pasting the relevant portion “as is” below:

Anderson’s (or others’) particular theory (or theories) might not be right, but the very idea that there can be this combination of a local action + a global correlation, isn’t. It is in fact easy to show how:

The system evolution in QM is governed by the TDSE, and it involves a first derivative in time and a second in space. TDSE thus has a remarkable formal similarity to the (linear) diffusion equation (DE for short).

It is easy to show that a local solution to the DE can be constructed. Indeed, any random walks-based solution involves only a local action. More broadly, starting with any sub-domain method and using a limiting argument, a deterministic solution that is local, can always be constructed.

Of course, there *are* differences between DE and TDSE. TDSE has the imaginary $i$ multiplying the time derivative term (I here assume TDSE in exactly that form as given on the first page of Griffith’s text), an imaginary “diffusion coefficient,” and a complex-valued \Psi. The last two differences are relatively insignificant; they only make the equation consistent with the requirement that the measurements-related eigenvalues be real. The “real” difference arises due to the first factor, i.e. the existence of the i multiplying the $\partial \Psi/\partial t$ term. Its presence makes the solution oscillatory in time (in TDSE) rather than exponentially decaying (as in DE).

However, notice, in the classical DE too, a similar situation exists. “Waves” do exist in the space part of the solution to DE; they arise due to the separation hypothesis and the nature of the Fourier method. OTOH, a sub domain-based or random walks-based solution (see Einstein’s 1905 derivation of the diffusion equation) remains local even if eigenwaves exist in the Fourier modeling of the problem.

Therefore, as far as the local vs. global debate is concerned, the oscillatory nature of the time-dependence in TDSE is of no fundamental relevance.

The Fourier-theoretical solution isn’t unique in DE; hence local solutions to TDSE are possible. Local and propagating processes can “derive” diffusion, and therefore, must be capable of producing the TDSE.

Note, my point is very broad. Here, I am not endorsing any particular local-action + global-correlation theory. In fact, I don’t have to.

All that I am saying is (and it is enough to say only this much) that (i) the mathematics involved is such that it allows building of a local theory (primarily because Fourier theoretical solutions can be shown not to be unique), and (ii) the best experiments done so far are still so “gross” that existence of such fine differences in the time-evolution cannot be ruled out.

One final point. I don’t know how the attendees of that conference think like, but at least as far as I am concerned, I am (also) informally convinced that it will be impossible to give a thoroughly classical mechanics-based mechanism for the quantum phenomena. The QM is supposed to give rise to CM (Classical Mechanics) in the “grossing out” limit, not the other way around. Here, by CM, I mean: Newton’s postulates (and subsequent reformulations of his mechanics by Lagrange and Hamilton). If there are folks who think that they could preserve all the laws of Newton’s, and still work out a QM as an end product, I think, they are likely to fail. (I use “likely” simply because I cannot prove it. However, I *have* thought about building a local theory for QM, and also do have some definite ideas for a local theory of QM. One aspect of this theory is that it can’t preserve a certain aspect of Newton’s postulates, even if my theorization remains local and propagational in nature (with a compact support throughout).)

OK. So think about it in the meanwhile, and bye for now.

[BTW, though I believe that QM theory must be local, I don’t agree that something such as superdeterminism is really necessary.]


A Song I Like:

(Hindi) “aaj un se pehli mulaaqaat hogi…”
Singer: Kishore Kumar
Music: R. D. Burman
Lyrics: Anand Bakshi

[E&OE]

 

 

 

For your holiday reading…

So… it’s the holiday season [for you], and so, you are looking for some light reading on quantum mechanics, right?

Well, I won’t disappoint you.

Check out this document [^].

Happy reading. Happy Christmas. Happy New Year! See you at this blog the next year!

[I am running the initial sanity-check on my new QM ideas. [I am, like, 99.99% certain to declare in the new year that my approach as stated in my published papers is wrong, and that it needs to be replaced by some new ideas such as what I now have [though I will not be retracting my papers just as yet]], and thus am likely to share my new thoughts on QM the next year. In the meanwhile, once again, happy: reading, Christmas, and New Year! Oh, BTW, writing on my new QM ideas (and explaining why my old ones are wrong, and how they are not all that completely wrong), is my NYR for 2016.]

 


 

A Song I Like:

(Western, Instrumental) “The Girl from Ipanema”

[I mean the instrumentals version of this song. Not (any of) the sung version(s). And certainly not the one where the singer has a double-decker of a hair-style.

As far as versions are concerned, Wiki tells [^] me that this song “is believed to be the second most recorded pop song in history, after `Yesterday’ by The Beatles” [and the needed citation is supplied, too; it refers to an article in WSJ]. Thus, it’s useless trying to be knowledgeable about this song—all its different versions.

I will therefore come straight to the version which I listened to for the first time in life, and which is one of the versions I still like: it’s the recording by the 101 Strings Orchestra. [Apparently, there are some good things in life that go by the name “101”, too.]

In my book, The 101 Strings Orchestra has a tie, actually, with the version by James Last and his orchestra. And, with just one page of one google search today, I now find that I like this version [^] equally well, too! In all the three cases, the instrumentals beat the human voice-sung versions by a galactic margin. … There is something about the instrumentals that make them appear a bit more restrained (and therefore more deep or even profound!), and yet, at the same time, also more suggestive. May be it’s because of the fact that they are more abstract—I don’t know. Anyway, enjoy the music, best wishes, and bye for now!]

[E&OE]

Why do physicists use infinity?

This post continues from my last post. In this post, I deal with the question: Even if infinity does not physically exist, why do physicists use it?


The Thermometer Effect

Consider the task of measuring the temperature of a hot body. How do you do that?

The simplest instrument to use would be: a “mercury” thermometer (i.e. of the kind you use when you come down with fever). Heat flows from the hot object to the “mercury” in the thermometer, and so, the temperature of the “mercury” rises. When its temperature increases, the “mercury” expands and so, its level in the tube rises. This rise in the level of the “mercury” can be calibrated to read out the temperature—of the mercury. After a while, the temperatures of the hot object (say the human body) and the thermometer become equal (practically speaking), and so, the calibrated tube gives you a reading of the temperature of the hot object. … We all learnt this in high-school (and many of us understood it (or at least had it explained to us) before the topic was taught to us in the school). What’s new about it?

It’s this (though even this point wouldn’t be new to you; you would have read it in some popular account of quantum mechanics): In the process of measurement, the thermometer takes away some heat from the hot object, and in the process makes the latter’s temperature fall. Therefore, the temperature that is read out refers to the temperature of the {hot object + thermometer} system, not to the initial temperature of the hot object itself. However, we were interested in measuring the temperature only of the hot object, not of the {hot object + thermometer} system.

QM folks, and QM popularizers, habitually go on a trip from this point on. We let them. Our objective is something different, something (hopefully) new. Our objective is the objectivity of measurements in an objectively understandable universe, not a subjective trip into an essentially subjective/unknowable universe. How do we ensure that? How do we ensure the objectivity of the temperature readings?

Enter infinity. Yes. Let me show you how.

We realize that the bigger the thermometer, the bigger is the heat leaked out from the hot object to the thermometer, and therefore, the bigger is the error in the measurement. So, we try something practical and workable. We try a smaller thermometer. Let’s be concrete.

Suppose that the hot object  remains the same for each experiment in this series. Suppose that thermometer T1 holds 10 ml of the sensing liquid, and suppose that at the end of the measurement process, it registers a temperature of 99 C. To decrease the amount of heat leaking into the thermometer, we get a second thermometer, T2; it holds only 1 ml of the sensing liquid. Suppose it registers a temperature of 99.9 C. We know that as the thermometer becomes smaller and still smaller, the reading read off from it will grow ever more accurate. For instance, with development in technology, yet another thermometer T3 may be built. It contains only 0.1 ml of the sensing liquid! It is found to register a temperature of 99.99 C. Etc. Yet, a thermometer of 0 ml liquid is never going to work, in the first place!

So, we do something new. We decide not to remain artificially constrained by the limits of the available technology, because we realize that here we don’t actually have to. We realize that we can take an inductive leap into the abstract. We plot a graph of the measured temperature against the size of the sensing liquid, and find that this graph has begun to “plateau out”. Given the way it is curving, it is obvious that it is coming to a definite limit.

We translate the graph into algebraic terms, i.e., as an infinite sequence, formulated via a general `n’th term. Using calculus, we realize that as `n’ approaches infinity, the amount of the sensing liquid approaches zero, and the temperature T_n registered by the thermometer approaches 100 C.

We thus conclude that in the limit of vanishing quantity of the sensing liquid, the measured temperature would be the true i.e. the unfallen or the initial temperature—viz. 100 C.

Carefully go through the above example. It shows the essence of how physicists use infinity. Indeed, if they were not to do so, they could not tie the purely imagined notion of a true temperature with any actual measurement done with any actual thermometer. Their theories would be either lacking in any generalization concerning temperature measurement, or it would be severed from the physical reality. However, one sure way that they can reach reality-based abstractions is via the idea of infinity. Before the 20th century, they did.


The sound-meter effect

Suppose you have a stereo system installed in your living room. For the sake of argument, just one speaker is enough. The logic here is simply additive: it applies even if you have a stereo system (two speakers) or a surround sound system (5 speakers); just find the effect that a single speaker produces one at a time, and add them all together, that’s all! That’s why, from now on, we consider only a single speaker.

Suppose you play a certain test sound, something like a pure C tone (say, as emitted by a flute), at a certain sound level—i.e. the volume knob on your music system is turned to some specific position, and thus, the energy input to the speaker is some definite fixed quantity.

You want to find out what the intensity of the sound actually emitted by a speaker is, when the volume knob on the music system is kept fixed at a certain fixed position.

Now, you know what the phenomenon of sound is like. It exists as a kind of a field. The sound reaches everywhere in the room. Once it reaches the walls of the room, many things happen. Here is a simplified version: a part of the sound reaching the wall gets reflected back into the room, a part of it gets transmitted beyond the wall (that’s the physical principle on the basis which your neighbors always harass you, but you never ever disturb anyone else’s sleep because that’s what your “dharma,” of course, teaches you), and a part of it gets absorbed by the wall (this is the part that ultimately gets converted into heat, and thereby loses relevance to the phenomenon of sound as such).

Now, if you keep a decibel-meter at some fixed position in the room, then the sound-level registered by it depends on both these factors: the level of the sound directly received from the speaker, as well as the level of the reflected sound. (Yep, we are getting closer to the thermometer logic once again).

Our task here is to find the intensity of the sound emitted by the speaker. To do that, all we have is only the decibel-meter. But the decibel meter is sensitive to two things: the directly received sound, as well as the indirectly received sound, i.e., the reflected sound. What the decibel meter registers thus also includes the effect of the walls of the room.

For instance, if the walls are draped with large, thick curtains, then the effective absorptivity of the walls increases, and so, the intensity of the reflected sound decreases. Or, if the walls are thinner (think the walls of a tent), then the amount leaked out to the environment increases, and therefore, the amount reflected back to the decibel-meter decreases.

The trouble is: we don’t know in advance what kind of a wall it is. We don’t know in advance the laws that apportion the incident sound energy into the reflecting, absorbing and transmitting parts. And therefore, we cannot use the decibel meter to calculate the true intensity of the sound emitted by the speaker itself. Or so it seems.

But here is the way out.

Since we don’t precisely know the laws operative at the wall, altering the material or thickness of the wall is of no use. But what we can do is: we can change the size of the room. This part is in our hands, and we can use it—intelligently.

So, once again, we conduct a series of experiments. We keep everything else the same: the speaker, the position of the decibel-meter relative to the speaker, the position of the volume-level knob, the MP3 file playing the C-note, etc. They all remain the same. The only thing that changes is: the size of the room.

So, suppose we first conduct the experiment in our own living room, and register a decibel meter reading of, say, 110 (in some arbitrary units). Then, we go to a friend’s house; it has bigger rooms. We conduct our experiment there. Suppose the reading is: 108 units. Then, we take the permission of the college lab in-charge, and conduct our experiment in the big laboratory hall: 106 units. We go into an in-door stadium in our town: 102 units. We go into an in-door stadium in another town: 102.5 units. We go out in a big open field out of town, and conduct the experiment at late night: 101.5 units. We go to that open salt field in the rann of the Kutch: 101.1 units. We take the measurement at a high level in the rann: 100.91 units.

Clearly, the nature of the wall has always been effecting our measurements. Clearly, the wall has always been different in different places—and we didn’t have any control over the kind of a wall there may be, neither do we know the kind of laws it follows. And yet, the size of the room has clearly emerged as the trend-setter here.

If we plot the intensity of the sound vs. the size of the room, the trend is not as simple (or monotonic) as in the thermometer case. There are slight ups and downs: even for a room of the same size, different readings do result. Yet, the overall trend is very, very clear. As the size of the room increases, the measurements go closer and closer to: 100 units.

Why? It’s because, choosing a bigger room leads to one definite effect: the effect of the wall on the measured sound level goes on dropping. The drop may be different for different kinds of walls. Yet, as the size of the room becomes really large, whatever be the nature of the wall and whatever be the laws operating at that remote location, they begin to exert smaller and ever so smaller effect on our measurements.

In the limit that the size of the room approaches infinity, the measurement procedure tends to yield an unchanging datum for the intensity of the measured sound. Indeed, in this limit, it would be co-varying with the intensity of the emitted sound, in a most simple, direct, manner. We have, once again, arrived at a stable, orderly datum—even if there were so many things affecting the outcome. We have, once again, managed to reach a universal principle—even if our measurement procedures were constrained by all kinds of limits; all kinds of superfluous influences of the walls.


For the advanced student of science/engineering:

In case you know differential equations (esp. computational modeling), the use of infinity makes the influence of the boundary conditions superfluous.

For instance, take a domain, take the Poisson equation, and use various boundary conditions—absorbing, partially absorbing, periodic, whatever—to find the field strengths at a point within the domain (as controlled by the various boundary conditions). Now, enlarge the domain, and once again try out the same boundary conditions. Go on increasing the domain size. Observe the logic. In the limit that the domain size approaches infinity, the value of the field variable approaches a certain limit—and this limit is given, for the Poisson problem, by the simple inverse-square law!


The Infinity, and Philosophy of Physics:

Increasing the size of the domain to the infinitely large serves the same purpose as does decreasing the amount of the sensing liquid to the infinitely small. The infinitely large or the infinitely small does not exist—the notion has no physical identity. But the physical outcomes in definitely arranged sequences do, and, in fact, even an only imaginary infinite sequence of these does help establish the physical identity of the phenomenon under discussion.

In both cases, infinity allows physicists the formulation of universal laws even if all the preceding empirical measurements are made in reference only to finite systems.

That incidentally is the answer to the question with which we began this post: Even if infinity does not physically exist, why do physicists use it?

It’s because, the idea allows them to objectively isolate the universal phenomena from the local physical experiences.


Homework for you:

In the meanwhile, here are a few questions for you to think about, loosely grouped around two (not unrelated) themes:

Group I: The Argument from the Arbitrary:

Is the question: “Is the physical universe infinite?” invalid? Can it be answered in the yes/no (or true/false) terms alone? Does it involve any arbitrary idea (in Ayn Rand’s sense of the term)? Is the very idea of the arbitrary valid?

Is there any sense to the idea of a finite physical universe?

Is there any sense to the supposition that you could reach the end of the universe?

Group II: The Argument from the Unseen Universe:

Think whether you would refute the following argument, and if yes, how: We cannot rely on physics, because the entirety of physics has been derived only in reference to a finite portion of the universe. Therefore, physics does not represent a truly universal knowledge. Our knowledge, as illustrated by its most famous example viz. physics, has no significance beyond being of a severely limited practical art. Knowledge-wise, it’s not a true form of knowledge; it’s only nominal. Some day it is bound to all break down, as influences from the unseen portion of the universe finally reach us.


Additional homework for the student of quantum mechanics:

Find out the relevance of this post to your course in quantum mechanics, as is covered usually in the universities, (e.g. Griffith’s text).

A Hint: No, this is not a “philosophy” related homework.

Spoiler Alert: Jump to the next section (on songs) if you don’t want to read a further hint, a very loud hint, about this homework.

A Very Loud Hint: Copy paste the following text into a plain text editor (such as the Notepad):

The Sommerfeld radiation condition

The Answer: In the next post, of course!


A Couple of Songs I Don’t Particularly Like:

Both are merely passable.

[This song is calculus-based. Really. In the reel life, it makes a monkey of the dashing young hero (and also of his dog), just the way the calculus does of most any one, in the real life.]

(Hindi) “samundar, samundar, yahaan se wahaa tak…”
Music: S. D. Burman
Singer: Lata Mangeshkar
Lyrics: Anand Bakshi

[This song used to be loved by the Americans (and many others, including Indians) when I was at UAB—and also for some time thereafter. It, or the quotable phrase that its opening line had become, doesn’t find too much of a mention anywhere. … Just the way neither does the phrase: the brave new world!]

(English) “A whole new world…”
Singers: Brad Kane and Lea Salonga
Music: Alan Menken
Lyrics: Tim Rice


[I intended to finish this thread off right in this post, but it grew too big. Further, I will be preoccupied in teaching activities (the beginning time is always the more difficult time), and so, there may be some time before I come back for the next post—may be the next weekend, or possibly even later—even though my attempt always would be to try to wrap this thing off as soon as possible anyway.]
[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:
(Marathi) “jaambhuL pikalyaa jhaaDaakhaali…”
Music: Hridaynath Mangeshkar
Lyrics: N. D. Mahanor
Singer: Asha Bhosale

[E&OE]