OK, I will come to the title topic, but before that, let me get one thing out of the way.
It’s been more than a week that I got my second dose of the Covid-19 vaccine.
The systems at the vaccination center were, once again, excellent. In fact, they have improved and streamlined the processes even further from the last time [^]! Also, once again, there were no issues or side effects at all.
… Yes, knowing that you are fully vaccinated does bring a sense of relief. … A big thanks to all the doctors, nurses, and also the entire administrative machinery, yes including the people from government!…
Bliss. (Actually, “blisses”.):
All this while, I have been working on QM, mostly on the topic of quantum mechanical angular momentum (which topic, in case you don’t know, includes the quantum mechanical spin). It’s been hard work, though mostly involving thinking in the head, and not writing down something in LaTeX or doing simulations.
In this process of thinking things through, pieces began to fall together in place, and simultaneously, pieces also began to fall apart!
I began to see how the QM textbooks fall short, so short in fact—even if you ignore all the misleading parts in them (which have been inserted because of the mainstream QM’s attempt to adhere to the Copenhagen interpretation—though no one knows precisely what it is, anyway!)… Descriptions in the standard treatments are so incomplete.
At the same time, I also began to see how my own ideas related to the spin were … err… wrong, meaningless, arbitrary, whimsical, nonsensical, etc. Yes, actually!
Ignorance, they say, is bliss. But what they say isn’t, as usual, sufficient. The complete statement is this: Realization of such bliss isn’t automatic. It requires hard work on your part! Knowing precisely what you are ignorant of, requires tremendous hard work. Only then can you enjoy the bliss because only then can you realize that you do carry ignorance.
Towards the end of the last week, I entered the aforementioned state. As a consequence, I can now confidently say the following.
I don’t know anything about the QM spin.
… It would be wrong to say that my new approach doesn’t handle the spin right. The correct statement here would be to say that my new approach, as of today, doesn’t even handle the quantum mechanical spin.
Why? Let me explain.
Here is the hierarchy of the topics. To understand any given node, you have to understand all its sub-nodes.
- Quantum Mechanical Spin
- Relativistic QM
- Special Theory of Relativity
- Electrodynamics in its entirety
- Special Theory of Relativity
- Non-Relativistic QM
- Relativistic QM
Now, as you know, I don’t know electro-dynamics. Ergo….
But it was only about a week ago that I began to fully realize the implications of the above structure. All along, I was thinking that I could at least offer some conjectures, if not qualitative statements, for the QM spin. (I wans’t even aiming for quantitative treatment.) But all my conjectures began to turn out, first, insufficient to explain different scenarios I imagined, and then, soon later, plain wrong.
So, now, I’ve came to realize that I can’t any longer do even just that—I mean, plain conjecturing. My ignorance is that total, complete, and full. A state of a lovely, dark, and thick… bliss!
But my enlightenment didn’t stop just there. Soon later, as I began to think about what I was thinking (regarding the quantum mechanical spin), one thing suddenly came in sharp focus.
… It’s true that when it comes to the quantum mechanical spin, not just me but the entire community of physicists is blissful. Utterly so. … Now, what needs to be highlighted here is the method which physicists use while staying blissful.
Physicists have nothing more than some purely formal (and mostly algebraic) similarity, elevated to the exalted status of postulates, when it comes to the theorization for the spin. Their entire theoretical structure for the spin rests, purely, on similarities with some parts of the theoretical structure for the orbital angular momentum. And it was this similarity which now suddenly came in the spotlight of my mind. And with it, I began to trace everything back, to see if there is any further sources of bliss still left for me.
And sure enough, I found some more. I realized that I am quite blissful about many aspects regarding the orbital angular momentum too!
Here is how. Once again, what else, a tree. (In theoretical physics and indeed in any knowledge, we don’t do graphs in their fully generality. We do only the directed acyclic graphs. [Ask me some time why. There is a neat epistemological point about it.]) So, anyway, here is the tree:
- Orbital Angular Momentum in QM
- Interaction of the electron with an external magnetic field
- Magnetic fields
- Interaction of the electron with an external magnetic field
OK, can you see my point? … No? … Well, let me spell it out for you.
At the most fundamental level, I gather, magnetism itself is a relativistic effect!
Therefore, the following bit is true:
Even in order to understand the non-relativistic QM well, you still first have to understand the relativity theory.
That was the second source of the bliss for me. Two blisses, back to back, right within one week! … Ah!
No one tells you that—the second source of the bliss. Professors of computer science are in fact busy attempting teaching “QM” without using Schrodinger’s equation. Yet, the aforementioned bit is true. The founders of QM were all well versed with the relativity theory, and of course, with the Maxwell-Lorentz electrodynamics (which precedes the relativity theory). But as a metallurgical-materials and then mechanical-software engineer, I haven’t ever studied this topic of electrodynamics (to the required depth and breadth). Hence…
But why is magnetic field required here at all? Well, that’s because, in experiments, you don’t measure the orbital angular momentum directly. You only measure the magnetic dipole moment of the electron, and then infer information about its from the former. And, experimentally measuring involves interaction of the electron with external magnetic fields.
That’s the reason why I have now realized such bliss about many aspects of the orbital angular momentum too, not just the spin angular momentum!
But why so late?
Why did the “blisses” come so late in my studies and research?
Well, the topics themselves are like that. You have to read the books again and again. It’s only in multiple passes that you spot the fine points, a few in each pass. That is, if such points are mentioned at all in the books.
Let me give you a concrete example. Here is an excerpt from Eisberg and Resnick, one of the best sources for an introduction to the non-relativistic QM. See for yourself how this passage (2e, p. 269) goes:
The ratio of to does not depend on the size of the orbit or on the orbital frequency. By making a calculation similar to the one above for an elliptical orbit, it can be shown that is independent of the shape of the orbit. That this ratio is completely independent of the details of the orbit suggests its value might not depend on the details of the mechanical theory used to evaluate it, and this is actually the case. Upon evaluation of quantum mechanically (which cannot be done here because the electromagnetic theory required is too sophisticated), and dividing by the quantum mechanical expression , the ratio of to is found to have the same value that we obtained. Granting this, the student will accept that the correct quantum mechanical expressions for the magnetic and component of the orbital magnetic dipole moment are
The minus sign in the last equation reflects the fact that the vector is antiparallel to the vector .
If you are studying QM in the pure self-studies mode, without any teacher to clarify matters or friends for discussions, then it does so happen that you don’t realize the full import of some fine points—like the one given in the parentheses above.
Here is another example, this time literally printed in the fine print in the book [ibid., p. 270]:
To simplify the discussion in subsequent sections, we shall frequently speak of the precession of a quantum mechanical magnetic dipole moment in a magnetic field, although to be strictly correct we should speak of the cyclic changes in the expectation values of its perpendicular components.
…But yes, at least Eisberg and Resnick do tell you many such things. Many other books don’t even bother to mention most of such things, not even via footnotes.
And then, there are tens of books to be looked into anyway, with each book giving its own share of such points! And, also committing its own share of misleading statements or even some small mistakes too. For example, even such careful authors as Eisberg and Resnick say [ibid. p. 269] that associated with the torque on a magnetic dipole is a potential energy of orientation. Strictly speaking, yes, there is an energy associated with the orientation of the dipole, but no, it’s not a potential energy proper—or so I gather. [I don’t quite understand this point, but I read something to this effect in Sears and Zemansky. Is it that you can’t take the gradient of this energy and obtain a force? Who knows. …Hmmm… Yet another [minor] bliss!]
Points like these may look minor, but at least some of them can become important when what you are aiming at is not an “A” grade in a university course but a proper answer to the measurement problem, which, in turn, requires a more or less complete reformulation of QM. It’s then that all such fine points begin to matter.
And then, you figure out something.
And then, come those moments of bliss!
I wasn’t so stupid to have theorized a sharp vector for the dipole moment in my new approach. The idea of vectors itself is rooted in classical mechanics; it doesn’t apply “as is” in QM, and I knew that. Yet, I must say that I had definitely underestimated what would be required to simulate the Larmour precession [^].
But why would I want to simulate the Larmour precession? After all, this topic doesn’t enter into QM postulates, does it?
Well, even if I don’t actually simulate it, I would have to know how a simulation can be done. And the reason for that, in turn, is because solving the measurement problem requires understanding, at least in schematic terms, how the new description would work out in various experimental arrangements for measuring various variables (or at least a few salient ones among them).
OK, what else? Did I reach a similar bliss regarding the measurement problem too?
Well, the answer, at least at this point of time, seems to be: no!
One last minute addition: Why “seems”? … Well, the point is this. I don’t understand electrodynamics, and hence, the magnetic field. However, the moving electron has a magnetic field associated with it—its own. The non-relativistic theory completely ignores this “internal” magnetic field. I can just hope, but don’t know for sure, that neglecting this field doesn’t matter much in the mechanism which I am proposing for solving the measurement problem… Hence the “seems”.
Let me leave this post on that note.
I could be away from blogs and all for quite some time… In the meanwhile, take care, and bye for now…
A song I like:
(Fusion) “Passages — Offering”
Composers: Ravi Shankar and Philip Glass
[I used to have the high-quality cassette of this album, back in the mid-1980s. It was released in India by Magnasound. A good quality audio is here [^]. See if you too like it…]