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. My reply:

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):



“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.



“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!



“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.]


“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!


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.]



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


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.]



“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.

[Addendum made on 2021.05.21:

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.

Addendum over.]

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]


| \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.

[Addendum made on 2021.05.22:

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!


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.]

Addendum over.]

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!


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.]

— 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…

Still if-ish…

1. Progress has slowed down:

Yep. … Rather, progress has been coming in the sputters.

I had never anticipated that translating my FDM code (for my new approach to QM) into a coherent set of theoretical statements is going to be so demanding or the progress so uneven. But that’s what has actually occurred.

To be able to focus better on the task at hand, I took this blog and my Twitter account off the ‘net from 26th February through 09th March.[* See the footnote below]

Yes, going off the ‘net did help.

Still, gone is that more of less smooth (or “linear”) flow of progress which I experienced in, say, mid-December 2020 through mid-January 2021 times or so, especially in January. Indeed, looking back at the past couple of weeks or so, I can say that a new pattern seems to have emerged. This pattern goes like this:

  • On day 1, I get some good idea about how to capture / encapsulate / present something, or put it in a precise mathematical form. So, I get excited. (I even feel like coming back on the ‘net and saying something.)
  • But right on day 2, I begin realizing that it doesn’t capture the truth in sufficient generality, i.e., that the insight is only partial. Or, may be, the idea even has loopholes in it, which come to the light only when I do a quick and dirty simulation about it.
  • By the time it’s day 2-end, day 3 or at most day 4, I have become discouraged, and even begin thinking of postponing everything to a June-July 2021-based schedule.
  • However, soon enough, I get some idea, hurriedly write it down…
  • …But only for the whole cycle to repeat once again!

This kind of a cycle has repeated some 3–4 times within the past 15–20 days alone.

“Tiring” isn’t the right word. “Fatigue” is.

But there is no way out. I don’t have any one to even discuss anything (though I am ready, as always, from my side.)

And, it still isn’t mid-March yet. So, I keep going back to the “drawing board.” Somehow.

[* Footnote: Curiously though, both WordPress and RevolverMaps have reported hits to this blog right in this period—even when it was not available for public viewing! … What’s going on?]

2. Current status:

In a way, persistence does seem to have yielded something on the positive side, though it has not been good enough (and, any progress that did come, has been coming haltingly).

In particular, with persistence, I kept on finding certain loop-holes in my thinking (though not in the special cases which I have implemented in code). These are not major conceptual errors. But errors, they still are. Some of these can be traced back to the June-July times last year. Funny enough, as I flip through my thoughts (and at times through my journal pages), some bits of some ideas regarding how I could possibly get out of these loop-holes, seem to have occurred, in some seed form (or half-baked form), right back to those times. …

Anyway, the current status is that I think that I am nearing completing a correct description, for the new approach, for the linear momentum operator.

This is the most important operator, because in QM, you use this operator, together with the position operators, in order to derive the operators for so many other dynamical quantities, e.g. the total energy, the angular momentum, etc. (See Shankar’s treatment, which was reproduced in the postulates document here [^].)

The biggest source of trouble for the linear momentum operator has been in establishing a mathematically precise pathway (and not just a conceptual one) between my approach and the mainstream QM. What I mean to say is this:

I could have simply postulated an equation (which I used in my code), and presented it as simply coming out of the blue, and be done with it. It would work; many people in QM have followed precisely this path. But I didn’t want to do that.

I also wanted to see if I can make the connections between my new approach and the MSQM as easy to grasp as possible (i.e., for an expert of MSQM). Making people understand wasn’t the only motive, however. I also wanted to anticipate as many objections as I could—apart from spotting errors, that is. Another thing: Given my convictions, I also have to make sure that whatever I propose, there has to be a consistent ontological “picture” which goes with it. I don’t theorize with ontology as an after-thought.

But troubles kept coming up right in the first consideration—in clearly spelling out the precise differences of the basic ideas between my approach and the MSQM.

And yes, MSQM does have a way of suddenly throwing up issues that are quite tricky to handle.

Just for this topic of linear momentum, check out, for instance, this thread at the Physics StackExchange [^] (especially, Dr. Luboš Motl’s answer), and this thread [^] (especially, Dr. Arnold Neumaier’s answer). The more advanced parts of both these threads are, frankly, beyond my capacity. Currently, I only aim for that level of rigour which is at, say, exactly and precisely the first three sentences from Motl’s answer!…

…We the engineers can happily ignore any unpleasant effects that might occur at the singular and boundary points. We simply try and see if we can get away ejecting such isolated domain points from any theoretical consideration! If something workable can still be obtained even after removing such points out of consideration, we go for it. So, that’s the first thing we check. Usually, it turns out we can isolate them out, and so we proceed to do precisely that! And that is precisely the level at which I am operating…

Even then, issues are tricky. And, at least IMO, a good part of the blame must lie with the confusions wrought by the Instrumentalist’s dogma.

… What the hell, if \Psi(x,t) isn’t an observable itself, then why does it find a place in their theory (even if only indirectly, as in Heisenberg’s formulation)? … Why can’t I just talk of a property that exists at each infinitesimal CV (control volume) \text{d}x? why must I instead take something of interest, then throw in the middle an operator (say a suitable Dirac’s delta), and then bury it all behind an integral sign? why can’t those guys (I mean the mathematical terms) break the cage of the integral sign, and come out in the open, just to feel some neat fresh air?

… Little wonder these MSQM folks live with an in-principle oscillatory universe. It’s a weird universe they have.

In their universe, Schrodinger’s cat is initially in a superposition of being alive and dead. But that’s not actually the most surprising part. Schrodinger’s cat then momentarily (or for a long but finite time) becomes full dead; but then, immediately, it “returns” from that state (of being actually dead) to once again be in a superposition of dead + alive; it spends some time in that superposition; it then momentarily (or for a long but finite time) becomes fully alive too; but only to return back into that surreal superposition…

And it is this whole cycle which goes on repeating ad infinitum.

… No one tells you. But that’s precisely what the framework of MS QM actually predicts.

MSQM doesn’t predict that once a cat does somehow become dead, it remains dead forever. And that’s because, in the MSQM, the only available mathematical machinery (which has any explanation for the quantum phenomena), in principle, predicts only infinite cycles of superposition–life–superposition–death–superposition–….

The postulates of the MS QM necessarily lead to a forever oscillatory universe! Little wonder they can’t solve the measurement problem!

One consequence of such a state of the MS QM theory is that thinking through any aspect becomes that much harder. It isn’t impossible. But hard, yes, it certainly is, where hard means: “tricky”.

Anyway, since the day before yesterday, it has begun looking like this topic (of linear momentum operator), and to the depth I mentioned above, might get over in a few days’ time. At least, that day 1–day 2–etc. pattern seems to have broken—at least for now!

If things go right at least this time round, then I might be able to finish the linear momentum operator by, say, 15th of March. Or 18th. Or 20th.

Addendum especially for Indians reading this post: No, the oscillatory universe of the MSQM people is not your usual birth-life-death-rebirth cycle as mentioned in the ancient Indian literature. The MSQM kind of “oscillations” aren’t about reincarnations of the same soul but in different bodies. In MSQM, the cat “return”s from being dead with exactly the same physical body. So, it’s not a soul temporarily acquiring one body for a brief while, and then discarding it upon its degeneration, only to get another body eventually (due to “karma” or whatever).

So, the main point is: In MSQM, Schrodinger’s cat not just manages to keep the same body, the physical laws mandate that it be exactly the same body (the same material) too! … And, the MS QM doesn’t talk of a soul anyway; it concerns itself purely with the physical aspects—which is a good thing if you ask me. (Just check the postulates document, and pick up a text book to see their typical implications.)

3. Other major tasks to be done (after the linear momentum operator):

  • Write down a brief but sufficiently accurate description of the measurement process following my new approach. This is the easiest task among all the remaining ones, because much of such a description can only be qualitative.
  • Translate my ideas for the orbital angular momentum into precise mathematical terms—something to be done, but here I guess that with almost all possible troubles having already shown up right in the linear momentum stage, the angular momentum should proceed relatively smoothly (though it too is going take quite some time).
  • And then, the biggest remaining task. Actually, many sub-tasks:
    • Study and take notes on the QM spin.
    • Think through and integrate my new approach to it.
    • Write down as much using quantitative terms as possible.

At this stage, I don’t know how long it’s going to take. However, for now, I’ve decided on the following plan for now…

4. Plan for now:

If there remain some issues with the linear momentum operator (actually, in respect of its multi-faceted usages in the MSQM, and in explaining these from the PoV of my approach including ontology), and if these still remain not satisfactorily resolved even by 15th or 18th of March (roughly, one week from now), then I will take a temporary (but long) break from QM, and instead turn my attention to Data Science.

However, if my description for \hat{p}() (i.e. the linear momentum operator) does go through smoothly during the next week, then I will immediately proceed with the remaining QM-related tasks too (i.e., only those which are listed above).

5. Bottom-line:

Expect a blog post in a week’s time or so, concerning an update with respect to the linear momentum operator and all. (I will try to keep this blog open for the upcoming week, but I guess my Twitter account is best kept closed for now—I just don’t have the time to keep posting updates there.)

In the meanwhile, take care and bye for now.

A song I like:

(Marathi) ती येते आणिक जाते (“tee yete aaNik jaate…”)
Lyrics: Aaratee Prabhu
Music: Pt. Hridaynath Mangeshkar
Singer: Mahendra Kapoor

[ Mahendra Kapoor has sung this song very well (even if he wasn’t a native Marathi speaker). Hridaynath Mangeshkar’s music, as usual, pays real good attention to words, even as also managing to impart an ingenious melodic quality to the tune—something that’s very rare for pop music in any language.

But still, frankly, this song is almost as nothing if you don’t get the lyrics of it.

And, to get the lyrics here, it’s not enough to know Marathi (the language) alone. You also have to “get” what precisely the poet must have meant when he used some word; for instance, the word “ती” (“she”). [Hint: Well, the hint has already been given. …Notice, I said “what”, and not “who”, in the preceding sentence!]

But yes, once you begin to get the subtle shades of the poetry here, then you can also begin to appreciate Hridaynath’s composition even better—you begin to see the more subtle musical phrases, the twists and turns and twirls in the tune which you had missed earlier. So, there’s a kind of a virtuous feedback circle going on here, between poetry and music… And yes, you also appreciate Mahendra Kapoor’s singing better as you go through the circle.

This song originally appeared as a part of a compilation of Aaratee Prabhu’s poems. If I mistake not (speaking purely from memory, and from a distance of several decades), the book in question was जोगवा (“jogawaa”). I had bought a copy of it during my UG days at COEP, out of my pocket-money.

We in fact had used another poem from this book as a part of our dramatics for the Firodiya Karandak. It was included on my insistence; I was a co-author of the script. As to the competition, we did win the first prize, but not so much because of the script. We won mainly because our singing and music team had such a fantastic, outstanding, class to them. Several of them later on went on to make full-time career in music…. The main judge was the late music composer Anand Modak, who later on went to win National awards too, but back then, he was at a fledgling stage of his career. But yes, talking of the script itself, in the informal chat after the prize announcement ceremony, he did mention, unprompted and on his own, that our script was good too! (Yaaaay!!) …Back then, there was no separate prize for the best script, but if there were to be one, then we would’ve probably won it. During that informal chat, the judges hadn’t bothered to even passingly mention any script by any other team!)

…Coming back to the book of poetry (Aaratee Prabhu’s), I think I still have my copy lying somewhere deep in one of the boxes, though by now, due to too many moves and all (I had also taken it to USA the first time I went there), its cover already had got dislodged from the book itself. Then, a couple of weeks ago, I saw only the title page peeping out of some bunch of unrelated and loose papers, and so, looks like, the book by now has reached a more advanced stage of disrepair! … Doesn’t matter; no one else is going to read it anyway!

A good quality audio is here [^].


2021.03.10 20:57 IST: Originally published.
2021.03.10 22.45 IST: Added links to the Physics StackExchange threads and the subsequent comments up to the mention of the measurement problem. Other minor editing. Done with this post now!
2021.03.12 18.43 IST: Some further additions, especially in section 2, including the Addendum written for Indian readers. Also, some further additions in the songs section. Some more editing. Now, am really done with this post!

Updates: RSI. QM tunnelling time.

Yes, the correct spelling of the word in the title is “tunnelling” (with a double “l”): [^].

1. Update on my RSI:

1.1. RSI :

The RSI has been waning for a few days by now. However, I am not sure if I should therefore begin my QM simulations or not. Going by how the RSI had immediately reverted its course about 8–10 days ago or so, I’ve decided to take it easy for now. This blog post itself is a “test-case” of sorts—to see how the RSI reacts.

1.2. Not quitting QM, but…:

I still have not begun simulations. It’s only after simulations that I would be able to judge whether to quit QM for a long while, or to write a paper on my new approach.

Writing documentation/paper only after conducting some simulations, might look like a lack of confidence on my part on the theoretical side. … Yes, as of now, this much is true. … Yes, by now, I’ve gathered together enough ideas about the 3D + spin with the new approach, but some elements are still to be worked through, especially those concerning the spin.

QM is complex. There is a pun here, but it was not intended. QM is complicated. And, very unintuitive. That’s why, building a completely new approach is difficult. It takes time, and thinking, and re-thinking.

2. Tunnelling time for quantum mechanical particle(s):

See the Quanta Magazine article “Quantum tunnels show how particles can break the speed of light” [^].

On 2020.10.26, I had noted on Twitter [^] that:

“This is actually a scenario that’s tough to get right. Wolchover’s coverage is v. good, but the intricacies themselves are such that I, for one, don’t have that feeling of being on top of it. Need to re-read.

A topic that rarely makes it to pop-sci level QM. Good they covered it”

Since then, I’ve re-read this Quanta Mag article some “two and a half” times.

I’ve also browsed through Prof. Aephraim Steinberg’s Web site in general (after a gap of may be 2–3 years), and his group’s page on quantum tunnelling in particular [^]. [I ignored his spelling mistake concerning “tunnelling”.]

I then rapidly looked through the arXiv version [^] of their July 2020 Nature paper [^]—the one which was covered in the above mentioned Quanta Mag article.

For the time being, let me note these comments (without explaining them):

2.1. Details of the experiment are quite complicated:

Understanding the details (even the more important ones) of this experiment is going to take a while.

2.2. But there is a video which explains the essential ideas behind this experiment:

A highly simplified version of this experiment is relatively straight-forward to understand. See this excellent German-language video with English subtitles [^] (which I found mentioned in Steinberg’s Twitter feed).

As to the video: I guess I had understood the points that have been covered in the video, and then a slight bit more too, right on the first reading of the Quanta Mag article (i.e., when I made the above mentioned tweet). However, I still had a lot of doubts / questions related to the specifics of the experimental setup. I still do.

My study of this work continues. Oh, BTW, I’ve downloaded quite a bunch of papers, including about the Hartman effect [^], e.g. this one [ (PDF) ^]. (Hartman, the first to publish the calculations even if they sounded very implausible to others due to their poor understanding of the relationship of QM and relativity principles, was an engineer!)

2.3. A SciAm article by Anil Ananthswamy:

Right as I was writing this post, I ran into Anil Ananthaswamy’s SciAm post: “Quantum tunneling is not instantaneous, physicists show” [^]. … Looks like it came in July 2020, but I had, somehow, missed it!

The Quanta Mag article covers a more comprehensive territory. It goes over the experiments done before Steinberg’s to a greater depth. In contrast, Ananthswamy’s article focuses more on Steinberg’s work, and is easier to understand. So, on the second thoughts, go through this article first.

2.4. Steinberg’s experiment is truly outstanding:

I think that Steinberg’s idea of using the Larmor precession for experimentally determining the tunnelling times is neat, exceptionally neat. Just how exceptionally neat?

Well, I still don’t understand the QM spin the way I would really like to (and that’s because I don’t know the relativity theory). It is for this reason that I request you to take my judgment with a pinch of salt.

Yet, within this explicitly stated limitation of my understanding, I still think that it would be reasonable enough to say that:

This experiment could easily get nominated for a physics Nobel.


In my opinion, this experiment is more outstanding than the famous series of experiments on testing QM entanglements, as by Aspect, Freedman and Clauser, and by others [^].

If the grapevine (i.e. opinions publicly expressed around the time of announcement of physics Nobels, over so many years by now) is anything to go by, then it’s reasonable to say that the Bell experiments must have been nominated for the physics Nobel.

If you want to know why I think the quantum tunnelling time experiment is more outstanding than the Bell test experiments, then I will try to give my reasons, but at some other time. I have to look after my wrist! Plus, I think the matter is very straight-forward. There is no room in the Copenhagen interpretation to even define something like a tunnelling time. There. Right there you have something to begin with. Also try to understand the idea behind the so called “weak measurement” experiments, and the particular advantages they bring.

2.5. The relevance of the tunnelling time experiments to my research:

Faster than light (FTL) speeds for the tunnelled particle should not surprise anyone. I don’t know why some physicists make an issue out of it.

In any case, assuming a simplified and abstract description of this experiment (as in the video mentioned above), I can say that:

My new approach  

    • is perfectly comfortable with FTL tunnelling,
    • predicts finite speeds, i.e., denies instantaneous action at a distance (IAD) for propagation of massive particles even in its present (non-relativistic) formulation.

That’s why I like this experiment. I was, in fact, looking for something on the “time taken” side, though I had somehow missed this particular experiment until the Quanta Mag ran the story.

It would be fun to develop my new approach to the point that it becomes possible to do a simulation of this experiment—at least a schematic version of it.

2.6. Should they pursue Bohmian mechanics for their simulations?

Steinberg’s group seems to have used the Bohmian mechanics for their simulations in the past. I think it’s not a good idea. See the next section.

3. Bohmian mechanics is flawed at a very basic level:

In general, by now, I have come to a definite conclusion that the Bohmian mechanics (BM) has a deep flaw in it—right at its most basic level.

So as to not stress my wrist a lot, let’s pursue this discussion in the next post (after a few days or a week).

In the meanwhile, go through this paper [^] by Prof. Travis Norsen. It’s a very well written paper; very easy to understand. It explains BM very clearly. In fact, it explains BM so clearly, in such a simplifying way, that it ends up defeating its very purpose! The author’s unstated goal here, I think, was to show that BM is reasonable. That must be the reason why he wrote this paper. But precisely because it’s so well written, you do get to understand BM very quickly. Which, in turn, makes spotting the flaws of BM so much the easier!

If you know the mainstream QM formalism well enough (especially its postulates), and if you have already thought a bit about the QM measurement problem (i.e., the “Process 1” according to von Neumann’s description of it [^]), then, it is possible to spot the essential weakness of the Bohmian mechanics just by reading only the first section (titled “Introduction”) of Norsen’s paper!

In a way, that’s why I appreciated this paper so much. In the past, I had tried to understand BM on 4–5 different occasions. But each time, I had to give up my attempt pretty soon, because I couldn’t understand the ideas like: the maths of the BM potential (after starting from geometrical optics), the physical source (if any) of that potential, etc.. … Somehow, I had not looked into this paper by Norsen all this while—the one which makes it all so easy to  understand!

So, go through this paper. We will discuss the weakness of the BM the next time. (If you know QM and are too short of patience to wait until the next post, then send me an email or leave a comment below, and I will give you an exactly one-line answer to you.)

BTW, Norsen has another paper that seeks to explain the QM spin in terms of BM; see it here [^]. I haven’t gone through it as yet, but if possible, I will try to cover it in the next post too. Or, if not in the next post, then at some other time when I discuss the QM spin.

4. My plans for the immediate future:

It was only yesterday that I began typing something in LaTeX (as in contrast to merely surfing the ‘net or tweeting). The typing was mostly a copy-paste job, plus some typing of equations in LaTeX. I pursued this activity for a couple of hours yesterday. Guess there wasn’t any noticeable worsening of the RSI today.

So, let me now try taking some notes on QM, or writing something further on my new approach to QM, or writing some Python code, from today onwards. I will be proceeding cautiously; I will not be exceeding 2–3 hours of typing per day, at least initially (over the coming few days). Let’s see how things progress.

OK, take care and bye for now.

A song I like:

(Marathi) तुझ्याच साठी कितीदा (“tujhyaach saaThee kiteedaa”)
Lyrics: N. G. Deshpande
Music: Shrinivas Khale
Singer: Krishna Kalle

[ Credits happily listed in a random order.

There are certain songs for which it doesn’t quite feel apt to say “I like this song” [so much, etc.]. A better way instead is to say this: There are some song such that, by showing how creativity and beauty can be combined with simplicity, they become some kind of a reference point for you—not just in the development of your tastes in music, but also in allowing you to grasp certain concepts like “culture” itself. And thus, it can be said that these songs have had a formative influence on you.

As far as I am concerned, this is one of such songs. I consider myself lucky to have been born at such a time that songs like these not only were being made but also were popular—at least, popular enough.

(And no, unlike many Indians/Maharashtrians who are high on culture and all, my reference points aren’t restricted to the Indian classical or semi-classical music alone. And, the set of my reference points doesn’t over-emphasize the devotional songs either. Et cetera. In fact, my referents haven’t been restricted to just the Indian songs either (as many of you might have gathered by now). …But then, matters like these is another story. Remind me some other day, when my wrist is in a better condition.)

A good quality audio for this song, appearing as a part of a collection, is here [^]. A link for a stand-alone version is here [^].


— 2020.11.08 15:39 IST: Published
— 2020.11.09 00:53 IST: Very minor revisions/additions. Am done with this post now.
— 2020.11.10 12:08 IST: Added a couple of links for the Hartman effect.