# Three neat resources on QM. Deduction-based vs. historically oriented approaches in teaching physics. A new “ambition”, + miscellaneous.

Okaay… So… How are you?

… On my side, I’ve been having quite some fun studying QM. I’ve reached a certain point in my studies, and it seems like this is a right time to take a little break, and write down an update, and thereby keep some momentum going at this blog.

Today I am going to write a little bit about three neat resources on QM, and also share some random thoughts, which occurred to me as a result of my wondering as to why I find these resources useful. In the process, I am going to touch a bit on the various approaches for systematically presenting a difficult topic like QM. The approaches I have in mind are: deduction-oriented, historically sequenced, and some combination of the two. Finally, I will also write a bit about a new ambition that has arisen in my mind. … OK, so let’s get going…

0. Preliminaries:

Lecture notes (and even full text-books) on QM tend to be organized in a highly deductive manner, especially when it comes to the topic of the QM spin. OTOH, many of you probably are very well aware that I tend to dislike any deduction-heavy treatments, even if these come from Nobel laureates and are highly praised, e.g., The Feynman Lectures vol. 2 and 3. … If fully or almost fully deductive, then sometimes, I even hate them—I mean such notes and  books, the prime example here being: Dirac’s book on QM! … So… Where do we begin?

Ummm… First, a word about how the element of deduction can be put to some good use too!…

1. Some textbooks / lecture-notes / course-videos lean towards a deductive approach but still are good:

One good point with a somewhat deductively organized presentation is that it can achieve a greater efficiency in teaching—assuming that a class-room teacher (not to mention TAs for conducting the recitation sessions) are available to the student for clarifying his doubts and difficulties right then and there in an informal, personal, settings. A somewhat deduction-oriented pedagogy has its uses, given the time constraints of the typical university schedules—provided that it’s done right—and provided that resources like systematic recitation (or tutorial) sessions are available.

However, in general, if the organization of topics is more heavily slanted towards deduction, then, even if TAs are available, the main teacher himself has to be very careful. He has to exercise special care, especially on these two counts:

• he has to keep on giving at least some “intuitive” feel for the logical starting point of each topic separately, again and again, and
• the scope of the logical “fanning out” to implications, starting from the major premise(s) selected for deduction, also has to be kept judiciously delimited.

Deduction is powerful—too powerful, in fact! Just like a sharp, double-edged sword. If handled right, it can work wonders. In teaching-learning, it means: Deduction is useful to the student provided he already knows the general outline and meaning of a topic and its scope. But if you are completely new to a topic, then a deduction-heavy treatment is more likely to induce in you a large number of small but enduring misconceptions. Reason: Objectively speaking, induction has primacy. Deny it its rightful role by suppressing it, and it’s going to try and figure out some way out of the suppression. Splintering of knowledge, therefore, is a very easy possibility—if not splintering of the mind as well!

However, if due care is exercised on the aforementioned two counts, then a semi-deductive treatment can come in quite handy. Let me give you an example from engineering, just to illustrate what I have in mind here.

Consider the very first course in engineering mechanics, viz., the vector mechanics of Newtonian particles and rigid bodies.

For more than two centuries, physicists actually theorized certain natural phenomena, and solved problems related to these, without ever using the idea of vectors. They explicitly worked with systems of equations: three coupled scalar equations, one each formulated along a principal coordinate axis.

So, a pedagogical approach that sticks to a purely historical order would have to teach all the topics in applied mechanics—not just acceleration in $3D$, and angular momentum, and the Coriolis forces, and everything else, but eventually, also the entirety of Maxwell-Lorentz EM—using only systems of coupled scalar equations! The task of teaching would become unwieldy in practice (and require black-boards on all sides of a class-room, with student-chairs that can rotate through $360$ degrees). The task of learning would become even harder, and therefore, knowledge would become accessible to relatively very few students.

However, experience shows that a combination of the historical and the deductive approaches does work great for engineering mechanics. It definitely takes less amount of time to generate a good grasp of the subject for most students. (To get to the best possible grasp, you have no choice but to look up the history and fill in the details from such sources, purely on your own.)

So, what you do, for courses in engineering mechanics, using the “combination” approach, is this:

You begin with the separate scalar component equations, and list them once. But you actually use them only in the simplest cases like motion of particles in $1D$ and $2D$ (e.g. the parabolic path of a projectile, the uniform circular motion, etc.). Then, soon enough, you take a jump of approximately one–two centuries, and immediately introduce the idea of vectors right at this stage. You don’t get into all the complications of the concept, like the distinction between a true vector and a pseudo-vector, right at this stage. And, you certainly don’t give a formal definition of vector spaces at all. But, you do begin deploying the idea of vectors in calculations. And, you do so using only a simplified (or “curtailed”) sense of the term “vector” (which is: as a directed line segment). Then, as the physical systems to be analyzed become more and more complex, you also go on expanding and clarifying the idea of vectors and operations with them further. You introduce the scalar and cross products of vectors, and explain the necessity of their differences in reference to different aspects of physical phenomena, then you go on to the calculus of vector-valued functions, etc…

So, all in all, you are going back and forth in history a bit, but without necessarily creating too much confusion about the proper hierarchical relations among the physical concepts. Going back and forth in this way is OK, up to an extent, if the jumping around is kept limited to the concepts and techniques of mathematics. However, I would definitely say, such jumping around does not work for the concepts and contents of physics as such.

Now, an interesting fact here is that a lot of ideas from physics also have a heavy methodological, even mathematical, context to them. For example, ideas like: the variational principles (taught initially as the “energy” principles), operators, and of course, the spin in QM.

Teaching such topics too can become more efficient using the “combination” approach, but then, doing so requires a teacher who is comparatively more skillful, and also, much more careful.

The combination approach might be characterized, using a slightly more fundmental terminology, as the following:

Start with the phenomenological knowledge, and use induction to introduce certain important facts that are also generally applicable. Then, translate these preliminary ideas into more formal concepts. These new, formal concepts might themselves be of a far greater methodological scope, but start using them anyway, without pondering over all aspects of the expansion in scope which such a formalization implicitly brings in. Then start working out simple mathematical manipulations, while using the greatly generalized formalism, but only in the simpler contexts, and thereby make students comfortable with the rules of manipulations as well as the hieroglyphics (i.e. symbols). Then progressively go on fleshing up the meaning behind the symbolism as the student understanding deepens (and his facility in using the rules and symbols improves). And, all through this activity, always keep on dropping small bits of physical insights (or at least some hints) which show where the “floor” of the “ocean” lies. Do that frequently.

It works. Provided that, what the teacher is aiming at is only a more systematic treatment, i.e., if, emphatically, he has not sold out his soul to deduction as such. Not all teachers or textbook writers are of the latter kind.

Now, the trouble with today’s teachers (and text-book writers) is that they just don’t know how to stop short of being an outright slave to deduction—and in the process, they also pull their students in the same vortex. Their short-coming is especially evident as the physics gets more and more complicated and therefore more and more abstract, starting right from the fields idea in EM, and up to the modern physics of special relativity and QM.

[Aside: IMO, not even that great teacher—Feynman—could manage this challenge right, always. In evidence, see his deductive treatment in the Volume 2 of his Lectures, and compare it not just with Resnick and Halliday, or Sears and Zemansky, but also with Purcell (recently updated by Morin). I don’t know about you, but I would always go in for the latter three as my primary sources for learning. Once you have already learnt the topics, then Feynman does become good—especially for the occasional insights that are hard to find elsewhere (and more occasionally, for your cocktail party points). Personally, I have avoided The Feynman Lectures’ second volume (after a rapid but careful browsing some three decades ago). Aside over.]

So, yes, using some degree of a deductively oriented organization can lead to efficiency in terms of classroom time. Actually, the gain is in terms of generating an averagely good sort of competency, in the least possible time, for more number of students. But such methodology also is a bit more demanding on the main teacher.

And, this problem becomes an order of magnitude worse when it comes to teaching quantum mechanics.

That’s why, when one runs into a mildly-deductive treatment of QM that’s also done competently, one not only appreciates it, one also wants to applaud it. In this post, I will mention two such examples.

2. “Notes on Quantum Mechanics” by Prof. Daniel Schroeder:

This is a set of notes for an introductory UG course on QM at the Weber State University, available here [^].

The organization and presentation style followed in these notes is such that a definite slant towards deduction is very easy to make out. Yet, the writing is such that the notes remain very easy to follow—even in the absence of a class-room teacher, i.e., even for a “pure” self-study mode (without any video recordings and all).

I had noticed these notes some couple of years ago or so, but in the sea of all the material available on the ‘net for QM, I had come to postpone reading through it, back then. Then, some time later, I somehow came to forget about these notes.

Recently, I checked out Prof. Schroeder’s Web pages after a while; found a new version of these notes; and immediately downloaded the freely available PDF [ (PDF) ^]. A quick browsing later, I now decided to keep the MIT course-work on a further hold, and instead to go through these notes first.

Turns out that it was a very good decision to make.

By now, I have gone through the angular momentum- and spin-related parts of this book. (I skipped most of the initial parts of the book, simply because I knew those topics pretty well.) I have found the treatment of the QM spin here to be outstanding.

It in fact is the best introduction to spin among all notes and books I have seen so far.

Yes, the treatment of the spin here is, IMO, better than that in Townsend’s text-book. Reason: Schroeder’s notes are short, more readable, and the problems too are “doable”. In contrast, Townsend’s book is big in size, too big in fact. (I think it’s 800 pages long. I have merely browsed through it once or twice, but have not properly read through even one section completely. (TBD later!))

The treatment of spin in Schroeder’s notes also is better than that in Eisberg and Resnick (and many other text-books). An important reason: Schroeder’s notes have a distinctly modern “flavour”, and so, you can so easily transition from the introductory QM to reading the special-purpose books and literature (say on the QC) without much effort.

Another plus point:

These notes are the only source I know of which shows how to “implement” two-state systems using the spatial wavefunctions (i.e. without at all using the spin).

[Aside: To tell you the truth, I had independently figured out something like this—two-state systems using only $\Psi(x,t)$ some time ago, as also the fact that entanglement can be explained in reference to the spin-less particles too.

… The earliest memory I have about thinking of entanglement with only spatial wavefunctions, i.e., without involving any spin at all, goes back at least to November 2014, when I was teaching engineering courses in Mumbai; and then, a highlight also occurred around Diwali-time in 2017. … But my ideas were rather “conceptual” in nature. Actually, my ideas were relatively vague, though they were not quite “floating” abstractions. And remember, all my studies and research in QM has been purely on a part-time basis, except for the last one year (since the Covid-19 began).

Anyway, when I saw Schroeder’s paper, “Entanglement isn’t just for spin” [^] (which was may be in 2018 or so), I remember, how I had marveled at it. Now, coming back to the present, to these notes, the marvel repeated. I mean, it was pleasant to read a description which was also physics-wise fully correct! Aside over.]

The only “downside” (if it can be called that) which I found with Schroeder’s notes is this:

There is no coverage of topics like the “total” wavefunction (spatial + spinor) for the many-particle systems. … May be it was not practical for them to cover this topic during their regular university coursework. However, an additional chapter dealing with the details of this topic would have been very helpful.

Added attraction: The simulation applets written by the author himself. (His HTML5 code is clean!)

All in all:

Strongly recommended, especially for the topic of the spin.

If you come from a BS in CS sort of a background, but have never studied QM beyond the Modern Physics courses, and still, if you have somehow grown very enthusiastic about the QC, and are championing it around, then, for God’s sake, let me dispense away this completely gratuitous and unsought advice to you:

Don’t even consider opening your mouth to champion the QC until you have have already mastered this book, cover-to-cover, complete with solving the section-end problems too. (As to for my opinion about the pre-requisites required for this book, see the section 4. below.)

And yes, I mean it!

3. “Quirky Quantum Concepts: The Anti-Textbook” by Prof. Eric Michelsen:

I’ve forgotten the track of where I gathered about Prof. Michelsen’s background (even if I did it within the last fortnight!). Anyway, here it is, in brief. Michelsen started out as an engineer. He spent quite some time (“decades”) in engineering industry (IIRC, in electronics / semiconductors). Then, I gather, he also founded start-ups in software. Then, he turned to research, and did a PhD in lasers and astrophysics, from UCSD. He is now a professor at UCSD [^].

… In short, I might say that I am sort of like him (or he is sort of like me), minus his practical success. [See the endnote at the end of this section.]

…Anyway, to come back to the “Anti-Textbook” by Prof. Michelsen….

Looks like I had downloaded an old PDF for the draft of this book too, though I don’t seem to have much of a recollection of my initial impressions about it. May be I had downloaded it around the same time as Schroeder’s notes were (i.e. in 2018 or so). I guess I must have downloaded this book, looked at some equations, and closed the PDF, saying to myself that I would have to return to it for a careful look later on. Then, as usual, I must have come to forget it along with Schroeder’s notes too.

Anyway, so… A couple of weeks ago or so, I once again downloaded the latest copy of Michelsen’s draft. (The book has been published by Springer, but the draft version is still available for free, here [ (PDF) ^] ).

I am still going through it. However, by now, I have read significant parts from: chapter 2 (“Riding the Wave: More on Wave Mechanics”), chapter 4 (“Matrix Mechanics”), and some initial parts from chapter 5 (“Angular Momentum”). In particular, I’ve not yet completed the portion on the spin, and haven’t even begun with the next chapter (which is on the many-particle systems).

My impression?

I am plain astounded at the richness of the insights offered here, full stop. This is one of the best resources for understanding the subtle aspects of QM.

Again, sometimes, I was even stunned to find the same insights as I myself had come to develop, independently. … Not so mysterious! The very approach of engineers is like that. Engineers (I mean: people who have worked as engineers for long enough of time so as to internalize the peculiar approach of engineering) do tend to think in a subtly (but definitely) different way than “pure” physicists do. (And we won’t even mention mathematicians here!) That’s why, it shouldn’t come as a surprise that when two engineers think deeply about the same new subject, there is a considerably similarity in terms of all: how they approach it, what they find interesting in it, and what they choose to highlight or take up for detailed considerations.

Of course, with his further formal training in physics (at the level of a PhD in physics), Michelsen has a much better knowledge of the mainstream QM than I do. He certainly has far more insights to offer on the more advanced aspects of the mainstream QM. These are difficult topics, and my studies of QM itself are relatively much more limited. I am not even aware of some of the topics whose quirkiness he notes. Yet, since his thinking retains the characteristic fold of an engineer’s thought processes, I have not found major difficulty in getting his points—even if these are quite quirky!

So, all in all, I think I can say this about my impression of this book (at this point of time):

I can always understand what Michelsen is saying, and often times, I also find myself having already worked through to precisely the same (or very similar) conclusions. However, I don’t always anticipate all his insights pertaining to the peculiarities of the mainstream QM.

But, yes, one way or the other, I find that his book is packed with insights. Even if you are not an engineer, you should benefit tremendously from this book. … Don’t take my word for it. Just go through the book and see for yourself. … OK. Let me copy-paste just one insight (just to help concretize this point); the following excerpt is from the draft copy page 49 (i.e. p. 51 of the PDF file):

Observable Operators Do Not Produce States Resulting From Measurements:

The mathematical result of an observable operator acting on a state is very different from the state resulting from actually measuring that observable.

Many people confuse the result of an observable operator on a state with the act of measuring that observable. These are very different things!

Note that the act of measurement is a nonlinear operation on the wave function; it can not be represented by a linear operator acting on the wave function. Recall that the whole point of a linear operator is to produce a superposition of results based on the superposition that composes the given function (or ket). In contrast, the consequence of a measurement is to choose one specific state out of a superposition of eigenstates.

A measurement eliminates a superposition, in favor of a more definite state. Therefore, a measurement is not a linear operation on the state; it is inherently nonlinear.

[Emphasis in bold original.]

See, see, why I am so impressed with this book? (And if you can’t figure out the reason, then check out my Outline document, here [(PDF) ^].)

… The entire book is filled with such nuggets.

No, this book is not at all induction-primary; it’s not even a historically sequenced presentation. In fact, this book isn’t even your usual text-book. Read the Preface to see why the author himself calls it an “Anti-Textbook”. …So, yes, this book is quirky! But yes, it’s quite rich in insights too.

So far, all that I’ve done is to rapidly read through the aforementioned chapters. But I don’t think that I’ve had enough opportunity to ponder over every subtle point, every nuance. I will have to read through the remaining parts, and then, I will have to return and re-read some parts again (may be 2–3 times).

The whole book is a kind of a teaser, as it were, to me. (Yes, my hard-copy is full of underlines, margin notes, scribblings, and all.)

Yes, this book is going to keep me engaged for quite some time to come.

However, no, do not bring up some points from this book for discussion with me. Not right away. I still have to learn a lot, and I am definitely quite a distance away from mastering the pre-requisite contents. I am also not likely to attempt mastering it any time soon. Reason: Many of these topics are not relevant to the research on Foundations of QM as such, even though the book deals with many advanced or subtle aspects of the mainstream QM, in a very admirable way.

As to me: First, I have to complete the first version of my document on my new approach to QM. Before that, I’ve to complete the MIT course 08.05 (i.e., watching videos). Before that, I’ve to complete the second half of MIT 08.04 too.

So, the bottomline is this:

Bottomline: One of the best books for really learning the subtleties of QM that I’ve ever come across, at such a level that it should be accessible even to the undergraduate students.

But make sure that you have completed the pre-requisites.

[Endnote to section 4: Why do I say that I am like Dr. Michelsen, minus his practical success? … Well, some of you should know the background behind that statement already, but in case you’ve just begun visiting this blog in the more recent times, there is a story about me and UCSD (which doesn’t come up on my CV)…:

After my failure in the PhD qualifiers at UAB in 1993, I was admitted to the mechanics program at UCSD, for the academic year beginning Fall ’93. … Not bad for a guy with a “mere” 8.25/10.00 CGPA at IITM and a 3.16/04.00 GPA at UAB. … OTOH, in fact, people at UCSD were (very) impressed by me. Reason? A literature review document which I had (on my own) attached to my application. It was on the micro-mechanics of fracture in ceramic composites. … Ummm, yes, the document was pretty good. So, they had decided to have me on board as soon as the funding arrives (which was around the May–June 1993 times).

However, roughly around the same time, even their on-going funding got cut down. So, formally, they said that they will keep the offer open for me for at least a year (which, eventually, they did), and informally, they called me to discuss the situation in all its detail with me. (By they, I mean the professors, not the staff at the graduate school.) Some of their on-going PhD students, supported previously via the funded projects which were now cancelled, began working in Pizza Hut, I gathered. I was willing to follow the suit. But starting completely afresh just on Pizz-Hut, i.e., without any prior savings, was not at all practical, someone (an IITian) doing his PhD there told me. The professors confirmed this assessment too, during the several telephonic discussions I had with them. … Anyway, all in all, I had to let go their offer, and return to India (which was in the last week of August ’93).

… BTW, my admission letter at UCSD was signed by Prof. Nemat-Nasser, the then Chair of the program. He and his colleagues had already brought this mechanics PhD program to a very good reputation; it was already ranked within the top 10 US-based programs or so. For comparison: My earlier program at UAB was ranked 60+ at that time (within about 70 PhD programs, in all, in the USA). … BTW, eventually, Prof. Nemat-Nasser also went on to receive the Timoshenko Medal. As to this medal …Well, yes, you may think of it as the “Nobel” of applied mechanics [^] [^]. … So, that’s the part related to UCSD

… But coming back to the other aspects of practical success: As to my later experience in software, in particular, in the SF Bay Area… Well, ask me some other time, preferably in private (so I can be a bit free-er in my… err… expressions). Footnote over. ]

4. Revised recommended sequence for learning QM through self-studies alone:

If you want to study QM through the self-study mode (i.e. completely on your own, without any personal guidance from any one), then condensing down everything (including whatever I have said about this topic in the past, and now revising it), here is my advice in a nut-shell:

Resnick and Halliday / Sears and Zemansky + if necessary, Purcell (updated by Morin) $\Rightarrow$ Beiser (Modern Physics) $\Rightarrow$ First half of McQuarry (Quantum Chemistry), up to and including the Helium atom + augmented readings for the same topics from Atkins (Molecular Chem.) $\Rightarrow$ Schroeder’s notes $\Rightarrow$ Eisberg and Resnick, Alastair I. M. Rae, and may be an occasional look into others like Griffiths / Gasiorowicz $\Rightarrow$ Michelsen.

For the last two stages, you can start with Michelsen and “dip back” into Griffiths etc. as the need arises. Also, consider watching the video series by ViaScience (mentioned in my earlier post here [^]) any time after you are past the books on quantum chemistry (including the He atom).

[Aside: Once my viewing of the MIT course-work (08.04 and 08.05) is over, it’s possible that I will revise the above sequence. So far, I’m half-way through 08.04 (2013 version), and I’m impressed with it. However, I don’t think that I am going to include the MIT course work in the shortest sequence presented above, and I don’t think I am eventually going to drop anything from the above sequence either. So, there. Aside over.]

Addendum: If your interest is in the Foundations of QM, rather than in QM itself or the QC, then my advice would depend on your background.

If you come with a background in physical / engineering sciences, then go through the above mentioned sequence at least up to and including Schroeder’s notes. Then, follow this sequence:

Travis Norsen (Foundations of QM) $\Rightarrow$ Tim Maudlin (Philosophy of Physics: Quantum Theory), both augmented with David Harriman (Induction in Physics).

(BTW, I have only browsed through some initial parts of Maudlin’s book, but I can definitely recommend it without any reservations.)

If you come with a background in other sciences or philosophy, then follow this sequence:

David Harriman (Induction in Physics) + Tim Maudlin (Philosophy of Physics: Quantum Theory), in any order $\Rightarrow$  Travis Norsen (Foundations of QM).

If you cannot understand the physics part of Harriman’s or Maudlin’s book even after a second or a third reading, then I would suggest: Quit pursuing Foundations of QM; this field is not for you. The field of Foundations of QM has a far greater basis in physics rather than philosophy—regardless of what other people might have led you to believe.

But if you still must persist with this field (Foundations of QM) at any cost, then quit pursuing all philosophical and popular science books on this topic (including those by Bell, and Bohmians), and instead, begin with the first sequence (given above), right from Resnick & Halliday etc., and going up to (and including) Schroeder’s notes. Once you are through with it (which should take at least a couple of years, may be 3–4 years), then once again check out Maudlin. If you can follow it right on the first read, then you may follow either of the two sequences given for the Foundations of QM.

Good luck!

5. A new, personal, long-term ambition:

Now that I had accidentally “re-discovered” the two gems mentioned above (the notes by Schroeder and Michelsen), I chastened myself a bit. Then, straining my memory, I remembered about

Malcom Longair’s book: “Quantum Concepts in Physics: An Alternative Approach to the Understanding of Quantum Mechanics”.

I had bought this book way back [^]. However, I have never been able to take it up for a systematic and comprehensive reading. … All that I’ve been able to do is sometimes to “take a dip” into it, may be for some 2–3 pages at a time, only then to toss it aside once again. “No right time for this book!” That’s what I’ve been saying to myself, invariably…

The last time I checked Longair’s book was, may be, 3+ years ago. It certainly was at least months before February 2019, which is when I wrote the Outline document [^].

A few days ago, I picked up this book once again. It was the first time I was touching it after the Outline document was posted. Presently, I came to a conclusion.

But before telling you the conclusion, let me ask: Remember Dr. Jennifer Coopersmith’s book “The Lazy Universe”? I had mentioned it pretty recently, just a few posts ago, here [^]. In the preface to her book, Coopersmith says that her book is like a simplified version of Lanczos’ book on the variational calculus. Now, Lanczos’ book is the Ultimate One, when it comes to the calculus of variations (i.e. the “energy” principles). And Coopersmith seems to have handled the simplification very well…

Now, coming back to Longair’s book and my long-term ambition.

I would like to write a simplified account of QM, based on Longair’s book.

The write up would be in a text-book like manner—complete with some solved problems and some review questions.

I would like to work at it slowly, one section at a time, and also irregularly, as my free time permits. I would like to post each section (or sub-section) in a GitHub repo or so. It will be, by intention, a long-term, and also irregular, hobby project.

But why rewrite a book if the original itself is so great?

Well, there are multiple reasons for that:

Yes, Longair has done a very admirable, scholarly, job in his book. However, he also gives greater detail of the initial analyses of experimental results, events and personalities than what a modern UG student of QM could possibly handle. If the goal is to simplify the presentation, one could omit many such details—precisely because Longair’s book is there!

Another point. Just the way the teacher has to exercise great care when presenting anything with a deduction-oriented approach (e.g. vector mechanics), similarly, the teacher also has to exercise a great care when presenting anything with a historically-oriented approach. Reason: Following the historical sequence helps in achieving a focus on the inductive roots of concepts and ideas. However, the former does not automatically ensure the latter. Isolation of the inductive roots is a separate task by itself.

With my enhanced understanding of induction (as brought about by David Harriman’s book “Induction in Physics”), I think that I can have a good shot at simplification. …

Let me be clear: I wouldn’t be explaining, let alone proving, how this or that development does have inductive roots; that’s not my goal. But I would like to present the physics points in such a way that their inductive roots become easier to grasp (even if they don’t become inescapable). I might not always do a great job for this aim; it’s too lofty for me. But I do think that I stand a good chance in converting the description from a mainly historically oriented account to one that is (i) simplified and (ii) highlights the nature of the respective inductive generalizations performed by the physicists at various stages of the main development. I think I can do that, to an acceptable degree.

One more point: I also think that, in the process of developing my new approach to QM, I’ve achieved a good clarity regarding what the mainstream QM theory is trying to say. This greater clarity, brought about by my new approach, should help in my goal, even if the explicit concern remains only with the development of the mainstream theory.

OK, so… I will begin working on it some time after the upcoming document on my new approach is done. I will work at it purely at leisure, purely as a hobby, and by intention, without any explicit plan… I think one should have some long-term hobby project like that going at any time…

6. But when will I start writing my planned document on the new approach?

Oh well. You tell me. … I’ve already told you the status as of now, and also the plan. Here it is—the plan—once again:

A few days more for going through Schroeder and Michelsen  $\Rightarrow$ II half of MIT 08.04 (videos) $\Rightarrow$ MIT 08.05 $\Rightarrow$ Some rest $\Rightarrow$ Thinking as to how my new approach holds up—if it does! And, if it does hold—and I see no reason why it shouldn’t hold, including for the spin–then (and only then) $\Rightarrow$ Some planning for writing $\Rightarrow$ Start actually writing.

So, whaddaya think? When will I begin writing (that goddman document on that goddamn new approach of mine)? …

Well… Want to consult astrologers / tarot card readers / psychics?  … Fine by me! Just let me know what they think (if they do), and then, also, what you think, if you do, after you have heard from them. … Or may be, want to consult some AI program? may be after you implementing a rough-and-ready one? Fine by me, again! … Or, perhaps, want to put to a practical use some certifiably random RNG (random number generator)? simply on the grounds that QM is supposed to be fundamentally random, and physics is universal? Or simpler still: want to toss a coin a few times? … Once again, fine by me. Whatever floats your boat! … From my sides, I’m all ears…

As to me, from my side, I will come back with a status update some time after watching the videos for the MIT 08.04 course is over, and watching for 08.05 is already in progress… That is, may be after two weeks or so (unless I have some brief update to post or so)…

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

Two songs I like:

Actually, I’ve two songs this time around. … Let me first give the credits and the links for both:

(Hindi) हम प्यार में जलने वालों को… (“ham pyaar mein jalane waalon ko…”)
Singer: Lata Mangeshkar
Lyrics: Rajinder Krishan

A good audio for the original version is here [^]. There are other versions, including a so-called “revival” series version, but I won’t bother to give you the links to them.

The second song I have in mind is this:

(Hindi) फिर तेरी कहानी याद आयी… (“phir teri kahaani yaad aayee…”)
Singer: Lata Mangeshkar

A good quality audio is here [^].

Both are Hindi film songs based on Indian classical music. Both are sung by Lata. Both have been composed by highly acclaimed music composers. The two songs also have a certain unmistakable kind of a similarity in terms of certain turns of the tune, certain phrases of melody, so to speak. Both are serious kind of songs, evoking a sombre kind of a mood, one that borders on sadness but in a somewhat abstract sense. Further, people usually describe both these songs in quite superlative terms, and generally speaking, I quite agree with such assessments too.

However, personally, I happen to like one of them a bit more than the other. The question is: Which one? And, why, i.e., for what aspects / reasons? (And we consider only the audio aspects of the two songs here.) …

… I will let you think for a bit about it, and only then tell you my answer, in briefest possible terms, say in one sentence or two at the most, say via an update to this post (which may occur after 2–3 days or so). … In the meanwhile, happy guessing and / or consulting (once again) astrologers / tarot card readers / psychics / whoever, or even using (or implementing) an AI, or using RNGs / tossing coins. … Optionally, thinking too!…

BTW, if the songs are new to you, see if you enjoy any of them, or both…

Bye for now and take care…

History:
— 2021.05.31 20:01 IST: First published
— 2021.06.01 19:39 IST: Added a sub-section in section 4, covering a recommended sequence for Foundations of QM. Also, generally streamlined content, with some minor additions throughout.
— 2021.06.02 13:22 IST: Some more streamlining and fixing of typo’s. Now, I am done with this post.

# No entanglement is possible in one-particle QM systems. [A context-specific reply touching on superposition and entanglement.]

Update alert!: Several addenda have been inserted inline on 21st and 22nd May 2021, IST.

Special Note: This post is just a reply to a particular post made by Dr. Roger Schlafly at his blog.

Those of you who’ve come here to check out the general happenings from my side, please see my previous post (below this one); I posted it just a couple of days ago.

1. Context for this post:

This is an unplanned post. In fact, it’s a reply to an update to a post by Dr. Roger Schlafly. His own post can be found here [^]. Earlier, I had made a couple of comments below that post. Then, later on, Schlafly added an update to the same post, in order to clarify how he was thinking like.

As I began writing a reply to that update, at his blog, my write-up became way too big. Also, I couldn’t completely avoid LaTeX. So, I decided to post my reply here, with a link to be noted at Schlafly’s blog too. …

… I could’ve, perhaps, shortened my reply and posted it right at Schlafly’s blog. However, I also think that the points being discussed here are of a more general interest too.

Many beginners in QM carry exactly the same or very similar kind of misconceptions concerning superposition and entanglement. Further, R&D investments in the field of Quantum Computers have grown very big, especially in the recent few years. Many of the QC enthusiasts come with a CS background and almost nothing on the QM side. In any case, a lot of them seem to be carrying similar misconceptions. Even pop-sci write-ups about quantum computing show a similar lack of understanding—all too often.

Hence this separate, albeit very context-specific, post. … This post does not directly focus on the difference between superposition and entanglement (which will take a separate post/document). However, it does touch upon many points concerning the two related, but separate, phenomena. [Done!

2. What Dr. Schlafly said in his update:

Since Schlafly’s update is fairly stand-alone, let me copy-paste it here for ease of reference. However, it’s best if you also go through the entirety of his post, and also the earlier replies, for the total context.

Anyway, the update Schlafly noted is this:

Update: Reader Ajit suggests that I am confusing entanglement with superposition. Let me explain further. Consider the double-slit experiment with electrons being fired thru a double-slit to a screen, and the screen is divided into ten regions. Shortly before an electron hits the screen, there is an electron-possibility-thing that is about to hit each of the ten regions. Assuming locality, these electron-possibility-things cannot interact with each other. Each one causes an electron-screen detection event to be recorded, or disappears. These electron-possibility-things must be entangled, because each group of ten results in exactly one event, and the other nine disappear. There is a correlation that is hard to explain locally, as seeing what happens to one electron-possibility-thing tells you something about what will happen to the others. You might object that the double-slit phenomenon is observed classically with waves, and we don’t call it entanglement. I say that when a single electron is fired, that electron is entangled with itself. The observed interference pattern is the result.

Let me cite some excerpts from this passage as we go along…

3.1. I will state how the mainstream QM (MSQM) conceptualizes the scenario Schlafly describes, and leave any comments from the viewpoint of my own new approach, for some other day (after my document is done)…

So, let’s get going with MSQM (I mean the non-relativistic version, unless otherwise noted):

3.2.

Excerpt:

“Consider the double-slit experiment with electrons being fired thru a double-slit to a screen, and the screen is divided into ten regions.”

To simplify our discussion, let’s assume that the interference chamber forms an isolated system. Then we can prescribe the system wavefunction $\Psi$ to be zero outside the chamber.

(MSQM can handle open systems, but doing so only complicates the maths involved; it doesn’t shed any additional light on the issues under the discussion. OTOH, MSQM agrees that there is no negative impact if we make this simplification.)

So, let’s say that we have an isolated system.

Electrons are detected at the screen in spatially and temporally discrete events. In MSQM, detectors are characterized classically, and so, these can be regarded as being spatially finite. (The “particle” aspect.)

Denote the time interval between two consecutive electron detection events as $T$. In experiment, such time-durations (between two consecutive detections) appear to be randomly distributed. So, let $T$ be a random variable. The PDF (probability distribution function) which goes with $T$ can be reasonably modeled with a distribution having a rapidly decaying and long tail. For bosons (e.g. photons), the detection events are independent and so can be modeled with a Poisson distribution. However, for electrons (fermions), the Poisson distribution won’t apply. Yet, when the electron “gas” is so thin as to have just a few electrons in a volume that is $\gg$ the scale of the wavelength of electrons as in the experiment, the tail of PDF is very long—indefinitely long.

That’s why, when you detect some electron at the screen, you can never be $100\ \%$ sure that the next electron hadn’t already been emitted and hadn’t made its way into the interference chamber.

Practically, however, observing that the distribution decays rapidly, people consider the average (i.e. expectation) value for the time-gap $T$, and choose some multiple of it that is reasonably large. In other words, a lot of “screening” is effected (by applying an opposite potential) after the electron gun, before the electrons enter the big interference chamber proper (Five sigma? I don’t know the criterion!)

Thus, assuming a large enough a time-gap between consecutive events, we can make a further simplifying assumption: There is only one electron in the chamber at a time.

3.3.

Excerpt:

“Shortly before an electron hits the screen, there is an electron-possibility-thing that is about to hit each of the ten regions.”

In the MSQM, before the lone electron hits the screen, the state of the electron is described by a wavefunction of the form: $\Psi(\vec{x},t)$.

If, statistically, there are two electrons in the chamber at the same time (i.e. a less effective screening), then the assumed system wavefunction would have the form:

$\Psi(\vec{x}_1, \vec{x}_2, t)$,

where $\vec{x}_1$ and $\vec{x}_2$ are not the positions of the two electrons, but the two $3D$ vector coordinates of the configuration space (i.e. six degrees of spatial freedom in all).

Should we assume some such a thing?

If you literally apply MSQM to the universe, then in principle, all electrons in the universe are always interacting with each other, no matter how far apart. Further, in the non-relativistic QM, all the interactions are instantaneous. In the relativistic QM the interactions are not instantaneous, but we need not consider relativity here, simply because the chamber is so small in extent. [I am not at all sure about this part though! I don’t have any good intuition about relativity; in fact I don’t know it! I should have just said: Let’s ignore the relativistic considerations, as a first cut!]

So, keeping out relativity, the electron-to-electron interactions are modeled via the Coulomb force. This force decays rapidly with distance, and hence, is considered negligibly small if the distance is of the order of the chamber (i.e., practically speaking, the internal cavity of a TEM (transmission electron microscope)).

Aside: In the scenarios where the interaction is not negligibly small, then the two-particle state $\Psi(\vec{x}_1, \vec{x}_2, t)$ cannot be expressed as a tensor product of two one-particle states $\Psi_1(\vec{x}_1,t) \otimes \Psi_2(\vec{x}_2,t)$. In other words, entanglement between the two electrons can no longer be neglected.

Let us now assume that in between emission and absorption there is only one electron in the chamber.

Now, sometimes, it can so happen that, due to some statistical fluke, there may be two (or even three, four…) electrons in the chamber. However, we now have a stronger argument for assuming that there is always only one particle in the chamber, when detection occurs. Reason: We are now saying is that the magnitude of the interaction between the two electrons (the one which was intended to be in the chamber, and the additional one(s) which came by fluke) is so small that these interactions can be assumed to be zero. We can make that assumption simply because the electrons are so far apart in the TEM chamber—as compared to their wavelengths as realized in this experiment.

So, at this point, we assume that a wavefunction of the form $\Psi(\vec{x},t)$ applies.

Note, the configuration space now has a single variable vector $\vec{x}$, and so, there is no problem interpreting it as the coordinate of the ordinary physical space. So, we can say that wavefunction (which describes a wave—a distributed entity) is, in this case, defined right over the physical space (the same space as is used in NM / EM). Note: We still aren’t interpreting this $\vec{x}$ as the particle-position of the electron!

3.4.

Excerpt:

“Assuming locality, these electron-possibility-things cannot interact with each other.”

The wavefunction for the lone electron $\Psi(\vec{x},t)$ always acts as a single entity over the entire $3D$ domain at the same time. (The “wave” aspect.)

The wavefunction has support all over the domain, and the evolution of each of the energy eigenstates comprising it occurs, by Fourier theory, at all points of space simultaneously.

In short: The wavefunction evolution is necessarily “global”. That’s how the theory works—I mean, the classical theory of Fourier’s.

[Addendum made on 2021.05.21: BTW, there can be no interaction between the energy eigen-states comprising the total wavefunction $\Psi(\vec{x},t)$  because all eigenfunctions of a given basis are always orthogonal to each other. Addendum over.]

3.5.

“Each one causes an electron-screen detection event to be recorded, or disappears.”

Great observation! I mean this part: “or disappears”. Most (may be $99.9999\,\%$ or more, including some PhD physicists) would miss it!

OK.

Assume that the detector efficiency is $100\ \%$.

Assuming a less-than-perfect detector-efficiency doesn’t affect the foundational arguments in any way; it only makes the maths a bit more complicated. Not much, but a shade more complicated. Like, by a multiplying factor of the square-root of something… But why have any complications if we can avoid them?

[Addendum made on 2021.05.21: Clarification: May be, I mis-interpreted Schlafly’s write up here. He could easily be imagining here that there are ten components in the total wavefunction of a single electron, and that only one component remains and the other disappear. OTOH, I took the “disappearing” part to be the electron itself, and not the components entering into that superposition which is the system wavefunction $\Psi(\vec{x},t)$. … So, please read these passages accordingly. The explanation I wrote anyway has covered decomposing the system wavefunction $\Psi(\vec{x},t)$ into two different eigenbases: (i) the total energy (i.e. the Hamiltonian) operator, and (ii) the position operator. Addendum over.]

3.6.

Excerpt:

“These electron-possibility-things must be entangled, because each group of ten results in exactly one event, and the other nine disappear.”

Well…

Bohr insisted that the detector be described classically (i.e. using the ideas of classical EM), by insisting on his Correspondence principle. (BTW, Correspondence is not the same idea as the Complementarity principle. (BTW, IMO, the abstract idea of the Correspondence principle is good, though not how it is concretely applied, as we shall soon touch upon.))

This is the reason why the MSQM does not describe the ten detectors at the screen quantum mechanically, to begin with.

MSQM also cannot. Even if we were to describe the ten detectors quantum mechanically, problems would remain.

According to MSQM, the quantum-mechanical system would now consist of {1 electron + 10 detectors (with all their constituent quantum mechanical particles)}.

This entire huge system would be described via a single wavefunction. Just keep adding $\vec{x}_i$, as many of them as needed. Since there no longer is a classical-mechanical detector in the description, the system would forever go oscillating, with its evolution exactly as dictated by the Schrodinger evolution. Which implies that there won’t be this one-time big change of a detection event, in such a description. MSQM cannot accomodate an irreversible change in the state of the {the 1e + 10 detectors} system. By postulation, it’s linear. (Show some love to Bohr, Dirac, and von Neumann, will you?)

Following the lead supplied by Bohr (and all Nobel laureates since), the MSQM models our situation as the following:

There is a single quantum-mechanically described electron. It is described by a wavefunction which evolves according to the Schrodinger equation. Then, there are those 10 classical detectors that do not quantum mechanically interact with the electron (the system wavefunction) at all, for any and all instants, until the detection event actually happens.

Then, the detection event happens, and it occurs at one and only one detector. Which detector in particular? “At random”. What is the mechanism to describe it? Blank-out!

But let’s continue with the official view (i.e. MSQM)…

The detection event has two parts: (1) The randomly “chosen” detector irreversibly changes its always classical state, from “quiscent” to “detected”. At the same time, (2) the quantum-mechanical wavefunction “collapses” into that particular eigenfunction of the position operator which has the associated eigenvalue of that Dirac’s delta which is situated at the detector (which happened to undergo the detection event).

What is a collapse? Refer to the above. It refers to a single eigenfunction remaining from among a superposition of all eigenfunctions that were there. (The wave was spread out, i.e. having an infinity of Dirac’s delta positions; after the collapse, it became a single Dirac’s delta.)

What happened to the other numerous (here, an infinity of) eigenfunctions that were not selected? Blank out.

What is the mechanism for the collapse? Blank out. (No, really!)

How much time does it take for the detection event to occur? Blank out. (No, really!)

To my limited knowledge, MSQM is actually silent about the time lapse. Even Bohr himself, I think, skirted around the issue in his more official pronouncements. However, he also never gave up the idea of those sudden “quantum jumps”—which idea Schrodinger hated.

So, MSQM is silent on the time taken for collapse. But people (especially the PhD physicists) easily rush in, and will very confidently tell you: “infinitesimally small”. Strictly speaking, that’s their own interpretation. (Check out the QM Postulates document [^], or the original sources.)

One more point.

Carefully note: There were no ten events existing prior to a detection anywhere in the above description. That’s why the question of the nine of them then disappearing simply cannot arise. MSQM doesn’t describe the scenario the way Schlafly has presented (and many people believe it does)—at all.

IMO, MSQM does that with good reason. You can’t equate a potential event with an actual event.

Perhaps, one possible source of the confusion is this: People often seem to think that probabilities superpose. But it’s actually only the complex amplitudes (the wavefunctions) that superpose.

[Addendum made on 2021.05.21: Clarification: Even if we assume that by ten things we mean ten components of the wavefunction and not ten events, the rest of the write-up adequately indicates the decomposition of $\Psi(\vec{x},t)$ into eigenbasis of the Hamiltonian (total energy) operator as well the position operator. Addendum over.]

3.7.

Excerpt:

“There is a correlation that is hard to explain locally, as seeing what happens to one electron-possibility-thing tells you something about what will happen to the others.”

There are no ten events in the first place; there is only one. So, there is no correlation to speak of.

[Addendum made on 2021.05.21:  Clarification. Just in case the ten things refer to the ten components (not a complete eigenbasis, but components in their own right, nevertheless) of the wavefunction and not ten events, there still wouldn’t be correlations to speak of between them, because all of them would collapse to a single Dirac’s delta at the time of the single detection event. Addendum over.]

That’s why, we can’t even begin talking of any numerical characteristics (or relative “strengths”) of the so-supposed correlations. Not in single-particle experiments.

In one-particle situations, we can’t even address issues like: Whether the correlations are of the same strengths as what QM predicts (as for the entangled particles); or they are weaker than what QM predicts (which is what happens with the predictions made using some NM- / EM-inspired “classical” models of the kind Bell indicated, i.e., with the NM / EM ontologies), or they are stronger than what QM predicts. (Experiments say that the correlations are not stronger either!)

Correlations become possible once you have at least two electrons at the same time in a system.

Even if, in MSQM, the two electros have a single wavefunction governing their evolution, the configuration space then has two 3D vectors as independent variables. That’s how the theory changes (in going from one particle to two particles).

As to experiments: There is always only one detection event per particle. Also, all detection events must occur—i.e. all particles must get detected—before the presence or absence of entanglement can be demonstrated.

One final point. Since all particles in the universe are always interconnected, they are always interacting. So, the “absence of entanglement” is only a theoretical abstraction. The world is not like that. When we say that entanglement is absent, all that we say is that the strength of the correlation is so weak that it can be neglected.

BTW, even in the classical theories like the Newtonian gravity, and even the Maxwell-Lorentz EM, all particles in the universe are always interconnected. In Newtonian gravity, the interactions are instantaneous. In EM (and even in GR for that matter), the interactions are time-delayed, but the amount of delay for any two particles a finite distance apart is always finite, not infinite.

So, the idea of the universe as being fully interconnected is not special to QM.

One classical analog for the un-entangled particles is this: Kepler’s law says that each planet moves around the Sun in a strictly elliptical orbit. If we model this empirical law with the Newtonian mechanics, we have to assume that the interactions in between the planets are to be neglected (because they are relatively so small). We also neglect the interactions of the planets with everything else in the universe like the distant stars and galaxies. In short, each planet independently interacts with the Sun and only with the Sun.

So, even in classical mechanics, for the first cut in our models, for simplification, we do neglect some interactions even if they are present in reality. Such models are abstractions, not reality. Ditto, for the un-entangled states. They are abstractions, not reality.

4. But what precisely is the difference?

This section (# 4.) is actually a hurriedly written addendum. It was not there in my comment/reply. I added it only while writing this post.

I want to make only this point:

All non-trivial entangled states are superposition states. But superposition does not necessarily mean entanglement. Entanglement is a special kind of a superposition.

Here is a brief indication of how it goes, in reference to a concrete example.

Consider the archetypical example of an entangled state involving the spins of two electrons (e.g., as noted in this paper [^], which paper was noted in Prof. (and Nobel laureate) Franck Wiczek’s Quanta Mag article [^]). Suppose the spin-related system state is given as:

$|\Psi_{\text{two electrons}}\rangle = \tfrac{1}{\sqrt{2}} \left(\ |\uparrow \downarrow\rangle \ +\ |\downarrow \uparrow\rangle \ \right)$               [Eq. 1].

The state of the system, noted on the left hand-side of the above equation, is an entangled state. It consists of a a linear superposition of the following two states, each of which, taken by itself, is un-entangled:

$|\uparrow \downarrow\rangle = |\uparrow\rangle \otimes |\downarrow\rangle$,           [Eq. 2.1]

and

$| \downarrow \uparrow \rangle = |\downarrow\rangle \otimes |\uparrow\rangle$           [Eq. 2.2].

The preceding two states are un-entangled because as the right hand-sides of the above two equations directly show, each can be expressed—in fact, each is defined—as a tensor product of two one-particle states, which are: $|\uparrow\rangle$, and $|\downarrow\rangle$. Thus, the states which enter into the superposition themselves are factorizable into one-particle states; so, they themselves are un-entangled. But once we superpose them, the resulting state (given on the left hand-side) turns out to be an entangled state.

So, the entangled state in this example is a superposition state.

Let’s now consider a superposition state that is not also an entangled state. Simple!

$|\Psi_{\text{one particle}}\rangle = \tfrac{1}{\sqrt{2}} \left(\ |\uparrow\rangle + |\downarrow\rangle\ \right)$            [Eq. 3].

This state is in a superposition of two states; it is a spin-related analog of the single-particle double-slit interference experiment.

So, what is the essential difference between entangled states from the “just” superposition states?

If the “total” state of a two- (or more-) particle system can be expressed as a single tensor product of two (or more) one-particle states (as in Eqs. 2.1 and 2.2], i.e., if the total state is “separable”/”factorizable” into one-particle states, then it is an independent i.e. un-entangled state.

All other two-particle states (like that in Eq. 1) are entangled states.

Finally, all one-particle states (including the superpositions states as in Eq. 3) are un-entangled states.

One last thing:

The difference between the respective superpositions involved in the two-particle states vs. one-particle states is this:

The orthonormal eigenbasis vectors for two-particle states themselves are not one-particle states.

The eigenvectors for any two-particle states (including those for the theoretically non-interacting particles), themselves are, always, two-particle states.

But why bother with this difference? I mean, the one between superpositions of two-particle states vs. superpositions of one-particle states?

Recall the postulates. The state of the system prior to measurement can always be expressed as a superposition of the eigenstates of any suitable operator. Then, in any act of measurement of an observable, the only states that can at all be observed are the eigenstates of the operator associated with that particular observable. Further, in any single measurement, one and only one of these eigenstates can ever be observed. That’s what the postulates say (and every one else tells you anyway).

Since every eigenfunction for a two-particle system is a two-particle state, what a theoretically single measurement picks out is not a one-particle state like $|\uparrow\rangle$ or $|\downarrow\rangle$, but a two-particle state like $|\uparrow\downarrow\rangle$ or $|\downarrow\uparrow\rangle$. Only one of them, but it’s a two-particle state.

So, the relevant point (which no one ever tells you) is this:

A theoretically (or postulates-wise) single measurement, on a two-particle system, itself refers to two distinct observations made in the actual experiment—one each for the two particles. For an $N$-particle systems, $N$ number of one-particle detections are involved—for what the theory calls a single measurement!

In entanglement studies, detectors are deliberately kept as far apart as they can manage. Often, the detectors are on the two opposite sides of the initial (source) point. But this need always be the case. The theory does not demand it. The two detectors could be spatially anywhere (wherever the spatial part of the total wavefunction is defined). The detectors could be right next to each other. The theory is completely silent about how far the detectors should be.

In short:

All that the theory says is:

Even for an $N$-particle system, the state which is picked out in a single measurement itself is one of the eigenstates (of the operator in question).

But you are supposed to also know that:

Every eigenstate for such a system necessarily is an $N$-particle state.

Hence the implication is:

For a single observation during an actual experiment, you still must make $N$ number of separate observation events, anyway!

So…

There are $N$ number of particles and $N$ number of events. But the theory is still going to conceptualize it as a single measurement of a single eigenfunction.

Every one knows it, but no one tells you—certainly not in textbooks / lecture notes / tutorials / YouTube videos / blogs / Twitter / FaceBook / Instagram / whatever. [Yes, please feel challenged. Please do bring to my notice any source which tells it like it is—about this issue, I mean.]

For a more general discussion of the mathematical criterion for un-entangled (or factorizable) vs. entangled states (which discussion also is simple enough, i.e. not involving the most general case that can arise in QM), then check out the section “Pure states” in the Wiki on “Quantum entanglement”, here [^].

And, another, last-last, thing!:

Yes, the states comprising the eigenbasis of any two non-interacting particles always consist of tensor-product states (i.e. they are separable, i.e. non-entangled).

However, when it comes to interacting particles: Especially for systems of large number of particles that interact, and talking of their “total” wavefunctions (including both: the spatial Schrodinger wavefunctions defined over an infinite spatial domain, and their spinor functions), I am not sure if all their eigenvectors for all observables are always represent-able as tensor-product states or not. … I mean to say, I am not clear whether the Schmidt decomposition always applies or not. My studies fall short. The status of my knowledge is such that I am unable to take a definitive position here (for uncountably infinite-dimensional Hilbert spaces of very large number of particles). May be there is some result that does prove something one way or the other, but I am not sure.

That’s why, let me now stop acting smart, and instead turn back to my studies!

Best,
–Ajit

5. To conclude this post…

….Phew!… So, that was (supposed to be) my “comment” i.e. “reply”. …Actually, the first draft of my “reply” was “only” about 1,500 words long. By the time of publication, this post has now become more than 3,300 word long…

If there is any further correspondence, I plan to insert it too, right here, by updating this post.

… I will also update this post if (and when!) I spot any typo’s or even conceptual / mathematical errors in my reply. [Always possible!] Also, if you spot any error(s), thanks in advance for letting me know.

OK, take care and bye for now…

[No songs section this time around. Will return with the next post.]

History:
— 2021.05.20 21:00 IST: Originally published
— 2021.05.21 17:17 IST: Added some clarifications inline. Streamlined a bit. Corrected some typo’s.
— 2021.05.22 13:15 and then also 22:00 IST: Added one more inline explanation, in section 4. Also added a confession of ignorance about relativity, and a missing normalization constant. …Now I am going to leave this post in whatever shape it is in; I am done with it…

# Trying to be a student…

This post is going to be brief. Further, for quite some time, it’s only this post that you may expect to see here. Reason: I am not going to be blogging, because I am going to be “studying”….

… Ummm… But that’s like jumping ahead. So let me begin at the beginning.

The mood i.e. “frame of mind”…

In my last post, I had said that I am going to be writing the new document (on my new approach to QM) by April-end, and post it by the first week of May.

Well, as you can see, it didn’t happen.

Writing a document like that requires a certain “mood”… Actually, “mood” is not the correct word. “Mood” is a bit too light a word, also not very specific, and perhaps even careless. … Ummm… “Mental frame”? Or, a “right frame of the mind”? Yeah…. That seems to be correct…. There is a precedent of sorts, and let me share it with you…

Even while writing the Outline document on my new approach [^], way back in Feb. 2019, it had required some effort. I had begun writing it some 2–3 weeks earlier. I kept of revising my writing back then, and these revisions often happened not with the in-place editing of an already existing file, but by starting with a completely fresh LaTeX document. I might have written several such versions, may be 8 or 10 of them, and many of these files were left half-done. … I was writing a lot (or so it seemed), but nothing satisfactory was coming out. … All these pieces were like a research paper, with completely formed sentences and all…

Then, one day in the first week of February 2019, I thought of adopting an outline (or bulleted list) kind of an approach. It helped, but still didn’t actually work.

Finally, one day, I hit the right “frame of mind”, and somehow, I started writing almost effortlessly. … All the points that I had thought of jotting down, now began coming out in a right, condensed sort of, form. … Yet, I still kept working with a fresh LaTeX document every time. (I had used a source-code version control system while writing my PhD thesis, but for the Outline document project, there were too many fresh starts, and so, renamed documents worked just as fine.)

The version which was eventually uploaded on the ‘net was, thus, may be 15th or 20th LaTeX file! Much of its content came directly from an earlier file which had been “Save As…”-ed to a different name. I edited some parts of it the night before (on 10th February 2019), and thought I was done. In the morning, however, I found myself in the right frame once again, and I ended up changing or refining a lot of points (I mean positions taken too, not just the words or the phrasing). I finished the document on the evening, and immediately afterwards uploaded it (on 11th February 2019).

So, yes, the right frame of mind is necessary. At least for me.

The mood strikes, but…

I did find myself in an exactly a similar kind of a frame of mind, which occurred on the morning of 24th April. … I was sipping my cup of tea in the balcony then, as usual. (Yes, a cup of the tea [^].) … So many points were now more or less automatically falling into the place, forming a chain (or well connected “graph”) of sorts, and I in fact knew that if I don’t at least jot these points down (or at least the scheme of organization / presentation which was occurring to me at that time), then it’s going to be difficult to reconstruct it later.

However, I had no time at hand, at that time. I had an appointment for my Covid-19 vaccine shot the same morning. So, I had to wind up this rare moment of inspiration, get ready, and rush to the private hospital where I had made my prior appointment. …

I got my first jab of the vaccine…

… Once at the venue, however, there was a bad surprise. … While waiting in the queue, a decision arrived from somewhere “above” that all the physical vaccines were being diverted to some other centers. Only those people who had come for the second shot would be entertained. People who had come for the first shot, even if they might have made confirmed appointments using the CoWIN portal, had to go home.

Naturally, I got irritated.

And somehow, after returning home, I couldn’t, somehow, reconstruct that scheme of writing, that flow of thoughts, which had so naturally occurred to me on the morning of the say day! … A few elements I could recall, but a lot of them had gone. There was nothing new to the elements that were gone, but crucially, it was the “flow” or the organization scheme that had gone by now. (I did attempt jotting down something. It didn’t work.)

The cancelled appointment also meant checking out the CoWin portal again and again, for any availability of any slots at any center. The vaccines were in a short supply and so, the portal had to be checked some 3–4 times a day.

Somehow, I was lucky enough to land a slot immediately later, on the 26th April, at a different vaccination center (another private hospital). … There was only one vaccination center in the entire Pune city that had now become available (at least over the 5–6 visits to the Co-WIN portal which I had made on the 24th and 25th). I took it immediately. Only 55 jabs were allotted to this lone center, and they all were gone within a matter of just 15 minutes or so.

However, during this visit, everything went through very smoothly. I really appreciated the arrangements and processes at this second center. May be because it was a much smaller hospital, but my experience was great.

Only the people with prior appointments were allowed in; others were courteously but firmly sent back. All the necessary documents were checked, but without letting you feel like you had arrived at the immigration counter at the Mumbai airport. … Most time was spent only in the waiting area, but it was arranged in a clean and open part of the space in the parking lot. The location meant that there was shade, but the place being open from all sides, the ambient air-flow was good too—an important consideration.

The vaccination itself was conducted in a room inside the hospital. People were called in, in batches of 10. Everything was well organized, and all the processes were performed punctually and carefully, following all the protocols and precautions. Yes, post-vaccination counseling was given too. And yes, every one at the hospital was very courteous and helpful (not just the doctors and the nurses, but also the rest of the staff, including the security staff). …

All in all, no walk-in appointments meant no crowding. Everything was streamlined, on time, and efficient. We were done right within 1.5 hours… People left the center feeling not just relieved (at having got the jab, finally), but actually in a distinctly light and happy mood, smiling and waving hands generally as they stepped out. Many people spontaneously felt like expressing their appreciation to the doctors or the staff, and so did I. …

Later…

By this time you have forgotten that we were talking about my plan to write the document, right?

That exactly is what happened to me, too!

After returning home from the vaccination center, I began checking all the available reports / articles on the vaccines—their types, mechanisms of action, efficacy, availability, side effects, optimum gap between the first and the second doses, and all that stuff. … I got engrossed in browsing on vaccines and the Covid-19.

Another major factor was that the second wave had already begun hitting new high’s every day all over the nation, and it was impossible to ignore all this development. … So, day after day, I kept on browsing on news, Twitter feeds, blogs, and all that…

During my browsing, I also read a bit on other topics. Even politics and all, simply because it was impossible to ignore it. But not much, come to think of it. The reason is: I don’t enjoy it. I even get irritated. A lot. And pretty quickly too. … Come to think of it, reading on current politics is, in many respects, comparable to reading through Bohr’s original writings.

OK, so, the rest of April was gone more or less just like that!

Getting back on the line…

Some time in between, the month of May had already arrived, and with that, the already missed deadline now became far too glaring even for me to ignore. … So, I also began trying picking up the threads on the QM side. … But no, that special mental frame was still not to be had. …

Of course, there never is any point in waiting for inspiration to strike you. You gotta move, you gotta start kicking, whether it arrives or not. … So, that’s what I began doing.

In my last post here, I had mentioned about the MIT course sequence. I had said: “Wish I had the time to systematically pursue this entire three-course sequence!”

Now that the deadline had anyway been missed, and despite trying again and again (may be 3 times) I wasn’t succeeding in getting myself to start writing the fresh document (not even in a raw or draft form), I wondered: Why not give the MIT course-work a try?

So, immediately, I did that.

Becoming a student…

To cut a long story short, I have decided to watch through at least the first two courses of the MIT UG sequence on QM, right away!

I have downloaded the videos for the MIT 08.04 Spring 2013 course (delivered by Prof. Allan Adams et al. [^]), as also the videos for the MIT 08.04 Spring 2016 version (by Prof. Barton Zwiebach [^]).

I have just begun systematically watching the Spring 2013 course (i.e. the one by Prof. Adams et al.). I am going in the sequence presented, not skipping any lecture—whether I know the stuff or not.

… I am not exactly taking notes, but I do jot down a bit on the fly. … I am not going to attempt solving even a single problem though! So, more or less, it’s going to be just watching these lectures, but attentively.

Once this QM-I course gets over, I may either continue the studies with the next course in the sequence (viz. QM-II i.e. MIT 08.05 [^]), or I may decide to also go through the 2016 version of QM-I too.

For the time being, I’ve decided to leave QM-III alone. So, it’s going to be just QM-I and QM-II.

I haven’t drawn up any time-table. Reason: I always tend to be too irregular to follow any time-table even if I draw up one. … However, I do expect to / hope to / will have to finish the 08.04 and 08.05 videos within, say, 3–4 weeks or so (say by May-end). The earlier, the better.

Simultaneously, I would also keep myself open about beginning on the document-writing project—I mean the one on my new approach. … Why, if inspiration strikes me, I will immediately start writing down the document too. However, even if that happens, one thing is for certain. I am going to include the QM spin, whenever I write the document. Another thing is for certain. I am going to finish watching all the videos of QM-I and QM-II before finishing the document.

OK. That’s about it! I don’t have anything else to blog about. (But, turns out, this post also hasn’t come out as very brief either.)

If you don’t see this blog updated, then know that I didn’t care to post anything because I was busy with the above mentioned course-work.

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

A couple of songs I like:

Since I will be away from the blog for a while, let me jot down two songs this time around.

(Marathi) सर्वात्मका सर्वेश्वरा (“sarvaatmakaa sarveshwaraa”)
Singer and Music: Pt. Jitendra Abhisheki