A further general update: The motion of a single electron in a box

I am still struggling with the conceptual issues which I mentioned in my 23rd August, 2020 post [^]. In this post, let me update you on the status in a more specific way. The conceptual trouble arises even with just the x-motion of the electron in 1D simulations. So let me note the solution of the model first.

1. The 1-dimensional Particle in a Box model:

Consider the simplest problem, a single particle in a 1D box with infinite potential walls at the ends, i.e., the famous particle-in-a-box problem (PIB for short). The analytical solution to this problem can be given via the following series of equations:

k_n = n \dfrac{\pi}{L}, where k_n is the magnitude of the wavevector \vec{k}_n, and L is the length of the box that occupies the interval x = [0.0, L].

p_n = \hbar k_n, where p_n is the magnitude of the global (aspatial, atemporal) momentum content of the system, and \hbar is the reduced Planck constant (with unit of action i.e. joule second i.e. (kilogram meter per second) times meter).

E_n = \dfrac{p_n^2}{2 m_e}, where E_n is the global (aspatial, atemporal) total internal energy content of the system, in joule, and m_e is the mass of the particle, here, electron, in kilogram.

\omega_n = \dfrac{E_n}{\hbar}, where \omega_n = 2 \pi \nu is the angular frquencey (in radian per second), and \nu is the frequency (revolution per second).

\psi(x) = A \sin(k_n x), is the space-dependent part of the wavefunction, and A = \sqrt{\dfrac{2}{L}} is the normalization constant, in \dfrac{1}{\sqrt{\text{m}}}.

\Theta(t) = e^{-i(\omega_n t)} is the time-dependent part of the wavefunction.

Now we can give the actual wavefunction as:

\Psi(x,t) = \psi(x) \Theta(t) = A \sin(k_n x) \left[ \cos(\omega_n t) - i\,\sin(\omega_n t) \right].

The unit of \Psi(x,t) and \psi(x) is \dfrac{1}{\sqrt{\text{m}^D}} where D is the dimensionality of the physical space (here, D=1).

2. What view I take of the PIB solution:

2.1. \Psi_n(x,t) field represents something that actually exists:

I have indicated (but not fully consistently stated) my new approach via the Outline document [^] and the Ontologies series of blog-posts [^], esp. # 8 [^], # 9 [^] and # 10 [^]. However, notice, there are certain changes to be made to the last part. According to my new approach \Psi_n(x,t) is a complex-valued field that physically exists in the 1D physical space.

2.2 There are electrostatic interactions going on even in the PIB model, but these are implicit:

Although the equations of the PIB model does not capture the electrostatic interactions of the electron, physically, these interactions are very much present. It’s just that the PIB model has been designed in such a way that the potential energy for the electrostatic interactions can be ignored.

In the PIB model, the said “ignorance” comes about in two ways:

Firstly, the boundary walls are (i) fixed and (ii) do not exert any electrostatic forces within the domain. That’s because the model assumes that the potential energy acquired by the electron due to the electrostatic field created by the walls drops down to zero in infinitesimally small length near the boundary points. So, in effect, electron does not acquires any potential energy from the (infinitely large) negative electric charge in the side walls.

Secondly, the introduction of the electron into the system does “create” an electrostatic potential field which is always singularly anchored into its instantaneous position. (See the Note 1 in the subsection 2.4. below.) However, since the walls always remain fixed, the potential energy of the walls due to electrons is zero—there is zero displacement and so, the work done by the forces exerted by the electron on the walls is zero. This is the reason why the PE of the walls does not at all enter into analysis. (Schrodinger’s solution for the H atom does something similar: It keeps the proton fixed. The solution then takes care of the energetics (the work done on the proton by the electron) by using the reduced mass method of classical mechanics.)

Now, why did I get into all these details?

Just to point out that

The singularity of the electrostatic potential field associated with an electron is physically still present in the domain. It’s just that the PIB model need not explicitly address the potential energy acquired by the walls due to this field, because the walls remain fixed and only at the boundary point, not in the domain.

Although the energy analysis for the PIB model does not account for it, in the new approach, the electron’s position is very much identified with it. That’s my proposition. [Accept it or reject it (or ignore it); I think about it in this way.]

2.3. How should the classical electrostatic singularity for the electron move?

The question now is, how should the singularity—i.e. the classical point-position of the electron—move? What mechanism(s) should guide its x-motion in the 1D PIB model?

As I said, following my new approach, the wavefunction \Psi(x,t) is a field in the aether. It is always complex-valued. Notice, in the PIB model, the purely space-dependent part is all real (as in the PIB equations given above), but the purely time-dependent part still remains full complex-valued. So, the \Psi(x,t) turns out to be fully complex. It’s not real-valued. Not even in the stationary states. Not even if you fill the box with a state that is a pure eigenstate of the energy operator (as in the above mentioned equations).

So, the essential issue is this:

The \Psi(x,t) is complex-valued. But there is only one x-axis for the singularity (i.e. the classical point-particle of the electron) to move. So, precisely how would it move? Following which law? which physical mechanism? And, if not a physical mechanism in real physical space, then at least an abstract mechanism that still has a sufficient closeness to the physical reality?

These questions are important, because unless we have a definitive law for the x-axis motion of the singularity within the domain, the nonlinearity in \Psi(x,t) which has been proposed by me wouldn’t work. (See the Outline document [^] for how the nonlinearity comes about.) And if the law we propose isn’t correct, then the nonlinearity wouldn’t work right. So, it’s important to have a correct law for the x-motions of the singularities of charges.

It is in this context that I did some simulation in June/July this year, and then, in August this year, I realized that while initially it looked good, a closer examination showed that it had glorious flaws: Flaw 1: Even the x-motions were coming out, in the general case, as complex-valued. I could not reduce them to real-valued ones using any plausible logic. Flaw 2: Difficult to figure out, but there was a hidden logical fallacy of circularity built into it (into my June/July simulation). I figured out this flaw only during the second thoughts, which was in August this year.

So, that’s the story in short, what I have been trying to do.

2.4. Note 1:

My further development now sheds a further light on the singularity in the electrostatic potential and potential energy fields.

The singularity in the potential energy V(x,t) turns out to be not a physical existent but only a mathematically abstract description which is obtained using the classical EM ontology.

The singularity in the potential energy field V(x,t) gets compensated by another singularity in the kinetic energy field in such a way that the field for the total internal energy is finite.

The very last bit should not be surprising, because E is finite, and \Psi(x,t) is required to be square-normalizable. In fact, neither of the above two statements should be, really speaking, come as surprising—if you treat \Psi(x,t) as an actually existing physical field in the D-dimensional physical space (and not in ND-dimensional configuration space). It’s just that the MSQM refuses to treat the wavefunction as a field in the physical space. (And the Bohmian mechanics has not been able to address the issue, I gather.)

However, it was a pleasant nugget to learn for me, when I noticed that the singularity is getting removed from my physics. For philosophic reasons, I’ve always believed that infinity is only a mathematical concept; that infinity of anything cannot physically exist.

Further, I also want to point out the following:

The idea of two singular fields cancelling each other and producing a finite field occurs very naturally, and therefore, seems to be on quite solid grounds. Contrast renormalization (which, really speaking, I don’t understand).

Please wait a little while before I adequately cover this aspect later on. For the time being, we will continue using the “fake” language as if the singularity physically exists.

To wind up:

In the description I am developing for QM, the \Psi(x,t) field (or some other complex-valued field which can be characterized using the square-normalized \Psi(x,t)) has the primacy, and so, there is no singularity in the primary description. But I continue to use the established terminology of EM ontology, just out of convenience of writing and communication. (There will always be this back and forth for some time, before I come to fully straightening out the conceptual hierarchy. Until then, using the established language of the “classical” EM seems to be a good choice.)

3. The present status—i.e., the actual update:

Now, given the above context, here is the actual update I wanted to write for this post.

I have found a way or a formula which (1) seems to carry no logical circularity, (2) is simple enough, and, crucially, (3) produces a formula for the x-motion that is real-valued (not complex-valued), even while respecting the principles of conservation of energy and momentum for the domain.

This dynamics is simple. In fact, at least outwardly, it is so [god-damn] simple that it is plain and clear that if I were not to try the more complicated ways of thinking about this problem (for the x-motion of the particle i.e. singularity), I would have got to this particular dynamics within an hour … at the most, one week. Frankly, it’s not even high-school logic, it is a middle-school-level logic. … In short, the formula is very, very, simple.

It also works—quantitatively, including units/physical dimensions.

Further, it seems to make some sense even when seen from different viewpoints—not just from the viewpoint of conservation of energy and momentum, but also from that of the correspondence of quantum mechanics to classical quantities.

Due to its simplicity, this formula also seems to hint that something physical (that is actually at work) should be very close by.

It’s just that…

with all my ability to find physical correspondents for mathematical ideas (see the Less transient page of this blog [^]), I still haven’t been able to get to a point where I can confidently say that I’ve got a good conceptual handle on this matter too.

At this juncture, I could have gone ahead and done some 2– and 3-particle simulations. After all, the “math” does work for the 1-particle case, doesn’t it?

But I being I [or is it me being me? or me being I? or I being me?] I couldn’t go further purely out of that one reason alone—viz., that the “math” works. I just cannot do that. I am constitutionally incapable of leaping ahead in that manner—for my own good or bad.

That’s why, I must spend some more time (say a week or two) before (i) I can tell myself that I don’t understand the detailed mechanism of how this simplest maths comes to be, (ii) keep this aside as a TBD (to be done) thing, and (iii) start using the goddamn simple formula as is, and see where it leads for the 2– or 3-particle systems.

So, that’s where I am, right now. I am stuck. And it’s on the conceptual side, not maths!

Within my own skill-set, I am comparatively weaker in maths, and so, if I were to get stuck somewhere in the maths, it would have been in the line of the things; something that was only to be expected. But being stuck on the conceptual side was totally unexpected—at this stage of the development anyway! [No, it’s not humiliating. Nothing is, so long as you are thinking about it. But unexpected, sure it is.]

And that was the update I had for you, for now.

… Guess, enough of writing for this post. I’ll come back, may be in a week’s time or so, either with a few links, or with a list of some of the books I’ve found useful over a period of time. At that time, I will also make sure to note whether this conceptual issue got resolved in the meanwhile, or whether I had kept it pending and proceeded ahead with the further simulations, not knowing whether they would be physically sound or not… Tough choice… Let me work through it…

A song I like:

(Marathi) चंद्र अर्धा राहिला (“chandra ardhaa raahilaa”)
Music: Yashwant Deo
Singer: Krishna Kalle
Lyrics: Madhukar Joshi

Credits happily listed in a random order. A relatively good quality audio is here [^] (and also as a part of a collection, here [^]).

Mentioning this song inevitably invites comparison to the following Hindi song—or at least, it used to, when I was young and both the songs used to be known to people. The Hindi song in question is:

(Hindi) आधा है चंद्रमा, रात आधी (“aadhaa hai chandramaa, raat aadhee”)
Singers: Mahendra Kapoor, Asha Bhosle
Music: C. Ramchandra
Lyrics: Bharat Vyas

A relatively good quality audio is here [^].

I don’t know if this Hindi film song from “Navrang” was inspired from the Marathi non-film song, or vice-versa, or whether there was no connection at all. (Some kind of a Marathi connection is easily possible. The producer-director of the film, V. Shantaram, was… err… a Marathi Manus.)

… As to me, I like both these songs.

The Hindi song has a good, flowing kind of a tune, and beautiful rendition by both the singers—they complement each other very well. Asha’s voice here is young, thin, and sharp but silky as usual. Mahendra Kapoor supplies the right contrast to her singing in a low key manner, but may be because he was rather young and not very establishedat that time (1959), his singing style here seems to be following C. Ramchandra’s instructions to the hilt. It would be easy to mistake him for C. Ramchandra.

Yes, I like them both. But in between the two, my choice is clear. The Marathi song is not just much closer to my heart, I also think that musically it’s superior. Yashwant Deo’s tune itself is more fresh and innovative, the singing by Krishan Kalle is very nuanced, and the Marathi lyrics too are outstanding… Even if you don’t know Marathi, do give it a listen too…

All in all, I would say, listen to both these songs and see whether you like any of the two or both of them; and if both, which one you like better.

… Take care and bye for now…