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]