# The goals are clear, now

This one blog post is actually a combo-pack of some 3 different posts, addressed to three different audiences: (i) to my general readers, (ii) to the engineering academics esp. in India, and (iii) to the QM experts. Let me cover it all in that order.

(I) To the general reader of this blog:

I have a couple of neat developments to report about.

I.1. First, and of immediate importance: I have received, and accepted, a job offer. Of course, the college is from a different university, not SPPU (Savitribai Phule Pune University). Just before attending this interview (in which I accepted the offer), I had also had discussions with the top management of another college, from yet another university (in another city). They too have, since then, confirmed that they are going to invite me once the dates for the upcoming UGC interviews at their college are finalized. I guess I will attend this second interview only if my approvals (the university and the AICTE approvals) for the job I have already accepted and will be joining soon, don’t go through, for whatever reason.

If you ask me, my own gut feel is that the approvals at both these universities should go through. Historically, neither of these two universities have ever had any issue with a mixed metallurgy-and-mechanical background, and especially after the new (mid-2014) GR by the Maharashtra State government (by now 2.5+ years old), the approval at these universities should be more or less only a formality, not a cause for excessive worry as such.

I told you, SPPU is the worst university in Maharashtra. And, Pune has become a real filthy, obnoxious place, speaking of its academic-intellectual atmosphere. I don’t know why the outside world still insists on calling both (the university and the city) great. I can only guess. And my guess is that brand values of institutions tend to have a long shelf life—and it would be an unrealistically longer shelf life, when the economy is mixed, not completely free. That is the broad reason. There is another, more immediate and practical reason to it, too—I mean, regarding how it all actually has come to work.

Most every engineer who graduates from SPPU these days goes into the IT field. They have been doing so for almost two decades by now. Now, in the IT field, the engineering knowledge as acquired at the college/university is hardly of any direct relevance. Hence, none cares for what academically goes on during those four years of the UG engineering—not in India, I mean—not even in IITs, speaking in comparison to what used to be the case some 3 decades ago. (For PG engineering, in most cases, the best of them go abroad or to IITs anyway.) By “none” I mean: first and foremost, the parents of the students; then the students themselves; and then, also the recruiting companies (by which, I mostly mean those from the IT field).

Now, once in the IT industry and thus making a lot of money, these people of course find it necessary to keep the brand value of “Pune University” intact. … Notice that the graduates of IITs and of COEP/VJTI etc. specifically mention their college on their LinkedIn profiles. But none from the other colleges in SPPU do. They always mention only “University of Pune”. The reason is, their colleges didn’t have as much of a brand value as did the university, when all this IT industry trend began. Now, if these SPPU-graduated engineers themselves begin to say that the university they attended was in fact bad (or had gone bad at least when they attended it), it will affect their own career growth, salaries and promotions. So, they never find it convenient to spell out the truth—who would do that? Now, the Pune education barons (not to mention the SPPU authorities) certainly are smart enough to simply latch on to this artificially inflated brand-value. The system works, even though the quality of engineering education as such has very definitely gone down. (In some respects, due to expansion of the engineering education market, the quality has actually gone up—even though my IIT/COEP classmates often find this part difficult to believe. But yes, there have been improvements too. The improvements pertain to such things as syllabii and systems (in the “ISO” sense of the term). But not to the actual delivery—not to the actually imparted education. And that‘s my point.)

When parents and recruiting companies themselves don’t care for the quality of education imparted within the four years of UG engineering, it is futile to expect that mere academicians, as a group, would do much to help the matters.

That’s why, though SPPU has become so bad, it still manages to keep its high reputation of the past—and all its current whimsies (e.g. such stupid issues as the Metallurgy-vs-Mechanical branch jumping, etc.)—completely intact.

Anyway, I am too small to fight the entire system. In any case, I was beyond the end of all my resources.

All in all, yes, I have accepted the job offer.

But despite the complaining/irritating tone that has slipped in the above write-up, I would be lying to you if I said that I was not enthusiastic about my new job. I am.

I.2. Second, and from the long-term viewpoint, the much more important development I have to report (to my general readers) is this.

I now realize that I have come to develop a conceptually consistent physical viewpoint for the maths of quantum mechanics.

(I won’t call it an “interpretation,” let alone a “philosophical interpretation.” I would call it a physics theory or a physical viewpoint.)

This work was in progress for almost a year and a half or more—since October 2015, if I go by my scribblings in the margins of my copy of Griffiths’ text-book. I still have to look-up the scribblings I made in the small pocket notebooks I maintain (more than 10 of them, I’ve finished already for QM alone). I also have yet to systematically gather and order all those other scribblings on the paper napkins I made in the restaurants. Yes, in may case, notings on the napkins is not just a metaphor; I have often actually done such notings, simply because sometimes I do forget to carry my pocket notebooks. At such times, these napkins (or those rough papers from the waiter’s order-pad), do come in handy. I have been storing them in a plastic bag, and a drawer. Once I look up all such notings systematically, I will be able to sequence the progression of my thoughts better. But yes, as a rough and ready estimate, thinking along this new line has been going on for some 1.5 years or more by now.

But it’s only recently, in December 2016 or January 2017, that I slowly grew fully confident that my new viewpoint is correct. I took about a month to verify the same, checking it from different angles, and this process still continues. … But, what the heck, let me be candid about it: the more I think about it, all that it does is to add more conceptual integrations to it. But the basic conceptual scheme, or framework, or the basic line of thought, stays the same. So, it’s it and that’s that.

Of course, detailed write-ups, (at least rough) calculations, and some (rough) simulations still have to be worked out, but I am working on them.

I have already written more than 30 pages in the main article (which I should now be converting into a multi-chapter book), and more than 50 pages in the auxiliary material (which I plan to insert in the main text, eventually).

Yes, I have implemented a source control system (SVN), and have been taking regular backups too, though I need to now implement a system of backups to two different external hard-disks.

But all this on-going process of writing will now get interrupted due to my move to the new job, in another city. My blogging too would get interrupted. So, please stay away from this blog for a while. I will try to resume both ASAP, but as of today, can’t tell when—may be a month or so.

I have changed my stance regarding publications. All along thus far, I had maintained that I will not publish anything in one of those “new” journals in which most every Indian engineering professor publishes these days.

However, I now realize that one of the points in the approvals (by universities, AICTE, UGC, NAAC, NBA, etc.) concerns journal papers. I have only one journal paper on my CV. Keeping the potential IPR issues in mind, all my other papers were written in only schematic way (the only exception is the diffusion paper), and for that reason, they were published only in the conference proceedings. (I had explicitly discussed this matter not just with my guide, but also with my entire PhD committee.) Of course, I made sure that all these were international conferences, pretty reputed ones, of pretty low acceptance rates (though these days the acceptance rates at these same conferences have gone significantly up (which, incidentally, should be a “good” piece of news to my new students)). But still, as a result, all but one of my papers have been only conference papers, not journal papers.

After suffering through UGC panel interviews at three different colleges (all in SPPU) I now realize that it’s futile to plead your case in front of them. They are insufferable in every sense; they stick to their guns. You can’t beat their sense of “quality,” as it were.

So, I have decided to follow their (I mean my UGC panel interviewers’) lead, and thus have now decided to publish at least three papers in such journals, right over the upcoming couple of months or so.

Forgive me if I report the same old things (which I had reported in those international conferences about a decade ago). I have been assured that conference papers are worthless and that no one reads them. Reporting the same things in journal papers should enhance, I guess, their readability. So, the investigations I report on will be the same, but now they will appear in the Microsoft Word format, and in international journals.

That’s another reason why my blogging will be sparser in the upcoming months.

That way, in the world of science and research, it has always been quite a generally accepted practice, all over the world, to first report your findings in conferences, seek your peers’ opinions on your work or your ideas, and then expand on (or correct on) the material/study, and then send it to journals. There is nothing wrong in it. Even the topmost physicists have followed precisely the same policy. … Why, come to think of it, the very first paper that ushered humanity into the quantum era itself was only a conference talk. In fact it was just a local conference, albeit in an advanced country. I mean Planck’s very first announcement regarding quantization. … So, it’s a perfectly acceptable practice.

The difference this time (I mean, in my, present, case) will be: I will contract on (and hopefully also dumb down) the contents of my conference papers, so as to match the level of the journals in which my UGC panel interviewers themselves publish.

No, the above was not a piece of sarcasm—at least I didn’t mean it, when I wrote it. I merely meant to highlight an objective fact. Given the typical length, font size, gaps in sections, and the overall treatment of the contents of these journals, I will have to both contract on and dumb down on my write-ups. … I will of course also add some new sentences here and there to escape the no-previous-publication clause, but believe me, in my case, that is a very minor worry. The important thing would be to match the level of the treatment, to use the Microsoft Word’s equation editor, and to cut down on the length. Those are my worries.

Another one of my worries is how to publish two journal papers—one good, and one bad—based on the same idea. I mean, suppose I want to publish something on the nature of the $\delta$ of the calculus of variations, in one of these journals. … Incidentally, I do think that what I wrote on this idea right here on this blog a little ago, is worth publishing even in a good journal, say in Am. J. Phys., or at least in the Indian journal “Resonance.” So, I would like to eventually publish it one of these two journals, too. But for immediately enhancing the number of journal papers on my CV, I should immediately publish a shorter version of the same in one of these new international journals too, on an urgent basis. Now the question is: what all aspects I should withhold for now. That is my worry. That’s why, the way my current thinking goes, instead of publishing any new material (say on the $\delta$ of CoV), I should instead simply recycle the already conference-published material.

One final point. Actually, I never did think that it was immoral to publish in such journals (I mean the ones in which my interviewers from SPPU publish). These journals do have ISSN, and they always are indexed in the Google Scholar (which is an acceptable indexing service even to NBA), and sometimes even in Scopus/PubMed etc. Personally, I had refrained from publishing in them not because I thought that it was immoral to do so, but rather because I thought it was plain stupid. I have been treating the invitations from such journals with a sense of humour all along.

But then, the way our system works, it does have the ways and the means to dumb down one and all. Including me. When my very career is at the stake, I will very easily and smoothly go along, toss away my sense of quality and propriety, and join the crowd. (But at least I will be open and forth-right about it—admitting it publicly, the way I have already done, here.)

So, that’s another reason why my blogging would be sparser over the upcoming few months, esp. this month and the next. I will be publishing in (those) journals, on a high priority.

(III) To the QM experts:

Now, a bit to QM experts. By “experts,” I mean those who have studied QM through university courses (or text-books, as in my case) to PG or PhD level. I mean, the QM as it is taught at the UG level, i.e., the non-relativistic version of it.

If you are curious about the exact nature of my ideas, well, you will have to be patient. Months, perhaps even a year, it’s going to take, before I come to write about it on my blog(s). It will take time. I have been engaged in writing about it for about a month by now, and I speak from this experience. And further, the matter of having to immediately publish journal papers in engineering will also interfere the task of writing.

However, if you are an academic in India (say a professor or even just a serious and curious PhD student of physics/chemistry/engg physics program, say at an IIT/IISc/IISER/similar) and are curious to know about my ideas… Well, just give me a call and let’s decide on a mutually convenient time to meet in person. Ditto, for academics/serious students of physics from abroad visiting India.

No, I don’t at all expect any academics in (or visiting) India to be that curious about my work. But still, theoretically speaking, assuming that someone is interested: just send me an email or call me to fix an appointment, and we will discuss my ideas, in person. We will work out at the black-board (better than working on paper, in my experience).

I am not at all hung up about maintaining secrecy until publication. It’s just that writing takes time.

One part of it is that when you write, people also expect a higher level of precision from you, and ensuring that takes time. Making general but precise statements or claims, on a most fundamental topic of physics—it’s QM itself—is difficult, very difficult. Talking to experts is, in contrast, easy—provided you know what you are talking about.

In a direct personal talk, there is a lot of room for going back and forth, jumping around the topics, hand-waving, which is not available in the mode of writing-by-one-then-reading-by-another. And, talking with experts would be easier for me because they already know the context. That’s why I specified PhD physicists/professors at this stage, and not, say, students of engineering or CS folks merely enthusiastic about QM. (Coming to humanity folks, say philosophers, I think that via this work, I have nothing—or next to nothing—to offer to their specialty.)

Personally, I am not comfortable with video-conferencing, though if the person in question is a serious academic or a reputed corporate/national lab researcher, I would sure give it a thought to it. For instance, if some professor from US/UK that I had already interacted with (say at iMechanica, or at his blog, or via emails) wants to now know about my new ideas and wants a discussion via Skype, I could perhaps go in for it—even though I would not be quite comfortable with the video-conferencing mode as such. The direct, in person talk, working together at the black-board, works best for me. I don’t find Skype comfortable enough even with my own class-mates or close personal relations. It just doesn’t work by me. So, try to keep it out.

For the same reason—the planning and the precision required in writing—I would mostly not be able to even blog about my new ideas. Interactions on blogs tends to introduce too many bifurcations in the discussion, and therefore, even though the different PoV could be valuable, such interactions should be introduced only after the first cut in the writing is already over. That’s why, the most I would be able to manage on this blog would be some isolated aspects—granted that some inconsistencies or contradictions could still easily slip in. I am not sure, but I will try to cover at least some isolated aspects from time to time.

Here’s an instance. (Let me remind you: I am addressing this part to those who have already studied QM through text-books, esp. to PhD physicists. I am not only referring to equations, but more importantly, I am assuming the context of a direct knowledge of how topics like the one below are generally treated in various books and references.)

Did you ever notice just how radical was de Broglie’s idea? I mean, for the electron, the equations de Broglie used were:

$E = \hbar \nu$ and $p = \hbar k$.

Routine stuff, do you say? But notice, in the special relativity, i.e. in the classical electrodynamics, the equation for the energy of a massive particle is:
$E^2 = (pc)^2 + (m_0 c^2)^2$

In arriving at the relation $p = \hbar k$, Einstein had dropped the second term ($m_0^2 c^4$) from the expression for energy because radiation has no mass, and so, his hypothetical particles also would carry no mass.

When de Broglie assumed that this same expression holds also for the electron—its matter waves—what he basically was doing was: to take an expression derived for a massless particle (Einstein’s quantum of light) as is, and to assume that it would apply also for the massive particle (i.e. the electron).

In effect, what de Broglie had ended up asserting was that the matter-waves of the electron had a massless nature.

Got it? See how radical—and how subtly (indirectly, implicitly) slipped in—is that suggestion? Have you seen this aspect highlighted or discussed this way in a good university course or a text-book on modern physics or QM? …

…QM is subtle, very subtle. That’s why working out a conceptually consistent scheme for it is (and has been) such a fun.

The above observation was one of my clues in working out my new scheme. The other was the presence of the classical features in QM. Not only the pop-science books but also text-books on modern physics (and QM) had led me to believe that what the QM represented was completely radical break from the classical physic. Uh-oh. Not quite.

QM, actually, is hybrid. It does have a lot of classical elements built into it, right in its postulates. I had come to notice this part and was uncomfortable with it—I didn’t have the confidence in my own observation; I used to think that when I study more of QM, I would be shown how these classical features fall away. That part never happened, not even as my further studies of QM progressed, and so, I slowly became more confident about it. QM is hybrid, full stop. It does have classical features built right in its postulates, even in its maths. It does not represent a complete break from the classical physics—not as complete a break as physicists lead you to believe. That was my major clue.

Other clues came as my grasp of the nature of the QM maths became better and firmer, which occurred over a period of time. I mean the nature of the maths of: the Fourier theory, the variational calculus, the operator theory, and the higher-dimensional spaces.

I had come to understand the Fourier theory via my research on diffusion, and the variational calculus, via my studies (and teaching!) of FEM. The operator theory, I had come to suspect (simply comparing the way people used to write in the early days of QM, and the way they now write) was not essential to the physics of the QM theory. So I had begun mentally substituting the operators acting on the wavefunction by just a modified wavefunction itself. … Hell, do you express a classical problem—say a Poisson equation problem or a Navier-Stokes problem—via operators? These days people do, but, thankfully, the trend has not yet made it to the UG text-books to a significant extent. The whole idea of the operator theory is irrelevant to physics—its only use and relevance is in maths. … Soon enough, I then realized that the wavefunction itself is a curious construct. It’s pointless debating whether the wavefunction is ontic or epistemic, primarily because the damn thing is dimensionless. Physicists always take care to highlight the fact that its evolution is unitary, but what they never tell you, never ever highlight, is the fact that the damn thing has no dimensions. Qua a dimensionless quantity, it is merely a way of organizing some other quantities that do have a physical existence. As to its unitary evolution, well, all that this idea says is that it is merely a weighting function, so to speak. But it was while teaching thermodynamics (in Mumbai in 2014 and in Pune in 2015) that I finally connected the variational principles with the operator theory, the thermodynamic system with the quantum system, and this way then got my breakthroughs (or at least my clues).

Yet another clue was the appreciation of the fact that the world is remarkably stable. When you throw a ball, it goes through the space as a single object. The part of the huge Hilbert space of the entire universe which represents the ball—all the quantum particles in it—somehow does not come to occupy a bigger part of that space. Their relations to each other somehow stay stable. That was another clue.

As to the higher-dimensional function spaces, again, my climb was slow but steady. I had begun writing my series of posts on the idea of space. It helped. Then I worked through higher-dimensional space. A rough-and-ready version of my understanding was done right on this blog. It was then that my inchoate suspicions about the nature of the Hilbert space finally began to fall in place. There is an entrenched view, viz., that the wavefunction is a “vector” that “lives” only in a higher-dimensional abstract space, and that the existence of the tensor product over the higher-dimensional space makes it in principle impossible to visualize the wavefunction for a multi-particle quantum system, which means, any quantum system which is more complex than the hydrogen atom (i.e. a single electron). Schrodinger didn’t introduce this idea, but when Lorentz pointed out that a higher-dimensional space was implied by Schrodinger’s procedure, Schrodinger first felt frustrated, and later on, in any case, he was unable to overcome this objection. And so, this part got entrenched—and became a part of the mathematicians’ myths of QM. As my own grasp of this part of the maths became better (and it was engineers’ writings on linear algebra that helped me improve my grasp, not physicists’ or mathematicians’ (which I did attempt valiantly, and which didn’t help at all)) I got my further clues. For a clue, see my post on the CoV; I do mention, first, the Cartesian product, and then, a tensor product, in it.

Another clue was a better understanding of the nonlinear vs. linear distinction in maths. It too happened slowly.

As to others’ writings, the most helpful clue came from the “anti-photon” paper by (the Nobel laureate) W. E. Lamb. Among the bloggers, I found some of the write-ups by Lubos Motl to be really helpful; also a few by Schlafly. Discussions on Scott Aaronson’s blog were useful to check out the different perspectives on the quantum problems.

The most stubborn problem for me perhaps was the measurement problem, i.e. the collapse postulate. But to say anything more about it right away would be premature—it would too premature, in fact. I want to do it right—even though I will surely follow the adage that a completed document is better than a perfect document. Perfection may get achieved only on collapse, but I happily don’t carry the notion that a good treatment on the collapse postulate has to be preceded by a collapse.

Though the conceptual framework I now have in mind is new, once it is published, it would not be found, I think, to be very radically new—not by the working physicists or the QM experts themselves anyway. …

.. I mean, personally speaking, when I for the first time thought of this new way of thinking about the QM maths, it was radically new (and radically clarifying) to me. (As I said, it happened slowly, over a period of time, starting, may be, from second half of 2015 or so if not earlier).

But since then, through my regular searches on the ‘net, I have found that other people have been suggesting somewhat similar ideas for quite some time, though they have been, IMO, not as fully consistent as they could have been. For example, see Philip Wallace[^]’s work (which I came across only recently, right this month). Or, see Martin Ligare[^]’s papers (which I ran into just last month, on the evening of 25th January, to be precise). … Very close to my ideas, but not quite the same. And, not as conceptually comprehensive, if that’s the right word to use for it.

My tentative plan as of now is to first finish writing the document (already 30+ pages, as I mentioned above in the first section). This document is in the nature of a conceptual road-map, or a position/research-program paper. Call it a white-paper sort of a document, say. I want to finish it first. Simultaneously, I will also try to do some simulations or so, and only then go for writing papers for (good) journals. … Sharing of ideas on this blog wouldn’t have to wait until the papers though; it could begin much earlier than that, in fact as soon as the position paper is done, which should be after a few months—say by June-July at the earliest. I will try to keep this position paper as brief as possible, say under 100 pages.

Let’s see how it all goes. I will keep you updated. But yes, the goals are clear now.

I wrote this lengthy a post (almost 5000 words) because I did want to get all these things from my mind and on to the blog. But since in the immediate future I would be busy in organizing for the move (right from hunting for a house/flat to rent, to deciding on what all stuff to leave in Pune for the time being and what all to take with me), to the actual move (the actual packing, moving, and unpacking etc.), I wouldn’t get the time to blog over the next 2–3 weeks, may be until it’s March already. Realizing it, I decided to just gather all this material, which is worth 3 posts, and to dump it all together in this single post. So, there.

Bye for now.

[As usual, a minor revision or two may be done later.]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

If not…

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

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

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

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

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

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

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

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

OK then, bye for now.

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

[E&OE]

# At the turkey table…

Here is a poem (and some comments) I have for now!

At the turkey table

America discovers

Wonderful husbands

And wonderful wives!

And family!!

And family values!!!

Wooooo  Wooooooooo Wooooooooooo!!!!

…Oh, how nice! Oh, how wonderful!! Oh, how beautiful!!!

… Etc.

[This is supposed to be a song—even music, for that matter.]

Yes, I am intelligent.

PS: It had nothing to do with the country known as the USA. But, our opinions may differ.

[E&OE]

/

# Miscellaneous: books to read, a new QM journal, the imposter syndrome, the US presidential elections

While my mood of not wanting to do anything in particular still continues (and also, there is no word yet on the job-related matters, including on whether I might qualify as a Professor of Mechanical Engineering in SPPU or not), there are a few quick things that I may as well note.

Updates on 17th, 18th and 22nd Nov. 2016: See my English translation[s] of the song, at the end of the post.

First, the books to read. Here are a few books on my to-read list:

1. Sean Carroll, “The Big Picture” [^]. I have been browsing through Sean’s blog-posts since before the time the book was published, and so have grown curious. I don’t have the money to buy it, right now, but once I get the next job, I sure plan to buy it. Here is the review in NY Times [^]. And, here is a latest review, written by a software engineer (whose link appeared in Sean’s twitter feed (I don’t myself use my Twitter account, but sometimes do check out the feeds of others via browser))[^]. Judging from his posts, I do know that Sean writes really well, and I would certainly want to check out this book, eventually.
2. Roger Penrose, “Fashion, Faith, and Fantasy in the New Physics of the Universe,” [^]. This is the latest offering by Penrose. Sometimes I simply type “quantum physics” in Google, and then, in the search results, I switch the tab over to “news.” I came to know of this book via this route, last week, when I ran into this review [^].
3. Roger Schlafly, “How Einstein Ruined Physics: Motion, Symmetry and Revolution in Science,” [^]. Here is a review [^], though my curiosity about the book rests not on the review but on two things: (i) what I had thought of Einstein myself, as far back as in early 1990s, while at UAB (hint: Schlafly’s thesis wouldn’t be out of bounds for me), and (ii) my reading the available portions of the book at Google Books. …This book has been on my “to-read” list for quite some time, but somehow it keeps slipping off. … Anyway, to be read, soon after I land a job…

A New QM Journal:

A new journal has arrived on the QM scene: [^]. Once again, I got to know of it through the “news” tab in a Google search on “quantum physics”, when I took this link [^].

It’s an arXiv-overlay journal. What it means is that first you submit your paper to arXiv. … As you know, getting something published at arXiv carries a pretty low bar (though it is not zero, and there have been some inconsistencies rarely reported about improper rejections even at arXiv). It’s good to bring your work to the notice of your peers, but it carries no value in your academic/research publications record, because arXiv is not a proper journal as such. … Now, if your work is good, you want to keep it open-access, but you don’t want to pay for keeping it open-access, and, at the same time, you also want to have the credentials of a proper journal publication to your credit, you have a solution, in the form of this arXiv-overlay journal. You send the link to your arXiv-published paper to them. If their editorial board finds it fitting the standards and purpose of their journal, they will include it.

The concept originated, I guess, with Timothy Gowers [^] and others’ efforts, when they started a maths journal called “Discrete Analysis.” At least I do remember reading about it last year [^]. Here is Gowers’ recent blog post reflecting on the success of this arXiv-overlay journal [^]. Here is what Nature had to report about the movement a few months ago [^].

How I wish there were an arXiv for engineering sciences too.

Especially in India, there has been a proliferation of bad journals: very poor quality, but they carry an ISSN, and they are accepted as journals in the Indian academia. I don’t have to take names; just check out the record of most any engineering professor from outside the IISc/IIT system, and you will immediately come to know what I mean.

At the same time, for graduate students, especially for the good PhD students who happen to lie outside the IIT system (there are quite a few such people), and for that matter even for MTech students in IITs, finding a good publication venue sometimes is difficult. Journal publications take time—1 or 2 years is common. Despite its size, population, or GDP, India hardly has any good journals being published from here. At the same time, India has a very large, sophisticated, IT industry.

Could this idea—arXiv-overlay journal—be carried into engineering space and in India? Could the Indian IT industry help in some ways—not just technical assistance in creating and maintaining the infrastructure, but also by way of financial assistance to do that?

We know the answer already in advance. But what the hell! What is the harm in at least mentioning it on a blog?

Just an Aside (re. QM): I spent some time noting down, on my mental scratch-pad, how QM should be presented, and in doing so, ended up with some rough outlines of  a new way to do so. I will write about it once I regain enough levels of enthusiasm.

The Imposter Syndrome:

It seems to have become fashionable to talk of the imposter syndrome [^]. The first time I read the term was while going through Prof. Abinandanan’s “nanopolitan” blog [^]. Turns out that it’s a pretty widely discussed topic [^], with one write-up even offering the great insight that “true imposters don’t suffer imposter syndrome” [^]. … I had smelled, albeit mildly, something like a leftist variety of a dead rat here… Anyway, at least writing about the phenomenon does seem to be prevalent among science-writers; here is a latest (H/T Sean Carroll’s feed) [^]…

Anyway, for the record: No, I have not ever suffered from the imposter syndrome, not even once in my life, nor do I expect to do so in future.

I don’t think the matter is big enough for me to spend any significant time analyzing it, but if you must (or if you somehow do end up analyzing it, for whatever reasons), here is a hint: In your work, include the concept of “standards,” and ask yourself just one question: does the author rest his standards in reason and reality, or does he do so in some people—which, in case of the imposter syndrome, would be: the other people.

Exercise: What (all) would stand opposite in meaning to the imposter syndrome? Do you agree with the suggestion here [^]?

The US Presidential Elections: Why are they so “big”? should they be?

Recently, I made a comment at Prof. Scott Aaronson’s blog, and at that time, I had thought that I would move it here as a separate post in its own right. However, I don’t think I have the energy right now, and once it returns, I am not sure if it will not get lost in the big stack of things to do. Anyway, here is the link [^]. … As I said, I am not interested much—if at all—in the US politics, but the question I dealt with was definitely a general one.

Overall, though, my mood of boredom continues… Yaawwwnnnn….

A Song I Like:

(Hindi) “seene mein jalan…”
Lyrics: Shahryar
Music: Jaidev

[Pune today is comparable to the Bombay of 1979 1978—but manages to stay less magnificent.]

Update on 2016.11.17: English translation of the song:

For my English blog-readers: A pretty good translation of the lyrics is available at Atul’s site; it is done by one Sudhir; see here [^]. This translation is much better than the English sub-titles appearing in this YouTube video [^] which comes as the first result when you google for this song. …

I am not completely happy with Sudhir’s translation (on Atul’s site) either, though it is pretty good. At a couple of places or so, it gives a slightly different shade of meaning than what the original Urdu words convey.

For instance, in the first stanza, instead of

“Just for that there is a heart inside,
one searches a pretext to be alive,”

it should be something like:

“just because there is a heart,
someone searches (i.e., people search) for an excuse which can justify its beating”

Similarly, in the second stanza,  instead of:

“what is this new intensity of loneliness, my friend?”,

a more accurate translation would be:

“what kind of a station in the journey of loneliness is this, my friends?”.

The Urdu word “manzil” means: parts of the Koran, and then, it has also come to mean: a stage in a journey, a station, a destination, or even a floor in a multi-storied building. But in no case does it mean intensity, as such. The underlying thought here is something like this: “loneliness is OK, but look, what kind of a lonely place it is that I have ended up in, my friends!” And the word for “friend” appears in the plural, not the singular. The song is one of a silent/quiet reflection; it is addressed to everyone in general and none in particular.

… Just a few things like that, but yes, speaking overall, Sudhir’s translation certainly is pretty good. Much better than what I could have done purely on my own, and in any case, it is strongly recommended. … The lyrics are an indispensable part of the soul of this song—in fact, the song is so damn well-integrated, all its elements are! So, do make sure to see Sudhir’s translation, too.

Update on 2016.11.18: My own English translation:

I have managed to complete my English translation of the above song. Let me share it with you. I benefitted a great deal from Sudhir’s translation and notes about the meanings of the words, mentioned in the note above, as well as further from “ek fankaar” [^]. My translation tries to closely follow not only the original words but also their sequence. To maintain continuity, the translation is given for the entire song as a piece.

First, the original Hindi/Urdu words:

seene mein jalan aankhon mein toofaan sa kyun hai
is shehar mein har shakhs pareshaan saa kyun hai

dil hai to dhadakne ka bahaanaa koi dhoondhe
patthar ki tarah behis-o-bejaan sa kyun hai

tanahaai ki ye kaun si manzil hai rafeeqon
ta-hadd-e-nazar ek bayaabaan saa kyon hai

kyaa koi nai baat nazar aati hai ham mein
aainaa hamen dekh ke hairaan sa kyon hai

Now, my English translation, with some punctuation added by me [and with further additions in the square brackets indicating either alternative words or my own interpolations]:

Why is there jealousy in the bosom; a tempest, as it were, in the eyes?
In this city, every person—why does it seem as if he were deeply troubled [or harassed]?

[It’s as if] Someone has a heart, so he might go on looking for an alibi [or a pretext] to justify [keeping it] beating
[But] A stone, as if it were that, why is it so numb and lifeless [in the first place]?

What kind of a station in the journey of the solitude is this, [my noble] friends?
Right to the end of the sight, why is there [nothing but] a sort of a total desolation?

Is there something new that has become visible about me?
The mirror, looking at me, why does it seem so bewildered [or perplexed]?

Update on 22nd Nov. 2016: OK, just one two more iterations I must have; just a slight change in the second [and the first [, and the third]] couplet[s]. (Even if further improvements would may be possible, I am now going to stop my iterations right here.):

Why is there jealousy in the bosom; a tempest, as it were, in the eyes?
In this city, every silhouette [of a person]—why does it seem as if he were deeply troubled [or harassed]?

[It’s as if] A heart, one does have, and so, someone might go on looking for an alibi [or a pretext] to justify [keeping it] beating
[But] A stone, as if it were that, why is it so numb and lifeless [in the first place]?

What kind of a station in the journey of the solitude is this, [my noble] friends?
[That] Right to the end of the sight, why is there [nothing but] a sort of a total desolation?

Is there something new that has become visible about me?
The mirror, looking at me, why does it seem so bewildered [or perplexed]?

[E&OE]