Ontologies in physics—6: A basic problem: How the mainstream QM views the variables in Schrodinger’s equation

1. Prologue:

From this post, at last, we begin tackling quantum mechanics! We will be covering those topics from the physics and maths of it which are absolutely necessary from developing our own ontological viewpoint.

We will first have a look at the most comprehensive version of the non-relativistic Schrodinger equation. (Our approach so far has addressed only the non-relativistic version of QM.)

We will then note a few points concerning the way the mainstream physics (MSMQ) de facto approaches it—which is remarkably different from how engineers regard their partial differential equations.

In the process, we will come isolate and pin down a basic issue concerning how the two variables \Psi and V from Schrodinger’s equation are to be seen.

We regard this issue as a problem to be resolved, and not as just an unfamiliar kind of maths that needs no further explanation or development.

OK. Let’s get going.


2. The N-particle Schrodinger’s equation:

Consider an isolated system having 3D infinite space in it. Introduce N number of charged particles (EC Objects in our ontological view) in it. (Anytime you take arbitrary number of elementary charges, it’s helpful to think of them as being evenly spread between positive and negative polarities, because the net charge of the universe is zero.) All the particles are elementary charges. Thus, -|q_i| = e for all the particles. We will not worry about any differences in their masses, for now.

Following the mainstream QM, we also imagine the existence of something in the system such that its effect is the availability of a potential energy V.

The multi-particle time-dependent Schrodinger equation now reads:

i\,\hbar \dfrac{\partial \Psi(\vec{R},t)}{\partial t} = - \dfrac{\hbar^2}{2m} \nabla^2 \Psi(\vec{R},t) + V(\vec{R},t)\Psi(\vec{R},t)

Here, \vec{R} denotes a set of particle positions, i.e., \vec{R} = \lbrace \vec{r}_1, \vec{r}_2, \vec{r}_3, \dots, \vec{r}_N \rbrace. The rest of the notation is standard.


3. The mainstream view of the wavefunction:

The mainstream QM (MSMQ) says that the wavefunction \Psi(\vec{R},t) exists not in the physical 3-dimensional space, but in a much bigger, abstract, 3N-dimensional configuration space. What do they mean by this?

According to MSQM, a particle’s position is not definite until it is measured. Upon a measurement for the position, however, we do get a definite 3D point in the physical space for its position. This point could have been anywhere in the physical 3D space spanned by the system. However, measurement process “selects” one and only one point for this particle, at random, during any measurement process. … Repeat for all other particles. Notice, the measured positions are in the physical 3D.

Suppose we measure the positions of all the particles in the system. (Actually, speaking in more general terms, the argument applies also to position variables before measurement concretizes them to certain values.)

Suppose we now associate the measured positions via the set \vec{R} = \lbrace \vec{r}_1, \vec{r}_2, \vec{r}_3, \dots, \vec{r}_N \rbrace, where each \vec{r}_i refers to a position in the physical 3D space.

We will not delve into the issue of what measurement means, right away. We will simply try to understand the form of the equation. There is a certain issue associated with its form, but it may not become immediately apparent, esp. if you come from an engineering background. So, let’s make sure to know what that issue is:

Following the mainstream QM, the meaning of the wavefunction \Psi is this: It is a complex-valued function defined over an abstract 3N-dimensional configuration space (which has 3 coordinates for each of the N number of particles).

The meaning of any function defined over an abstract 3ND configuration space is this:

If you take the set of all the particle positions \vec{R} and plug them into such a function, then it evaluates to some single number. In case of the wavefunction, this number happens to be a complex number, in general. (Remember, all real numbers anyway are complex numbers, but not vice-versa.) Using the C++ programming terms, if you take real-valued 3D positions, pack them in an STL vector of size N, and send the vector into the function as an argument, then it returns just one specific complex number.)

All the input arguments (the N-number of 3D positions) are necessary; they all taken at once produce the value of the function—the single number. Vary any Cartesian component (x, y, or z) for any particle position, and \Psi will, in general, give you another complex number.

Since a 3D space can accommodate only 3 number of independent coordinates, but since all 3N components are required to know a single \Psi value, it can only be an abstract entity.

Got the argument?

Alright. What about the term V?


4. The mainstream view of V in the Schrodinger equation:

In the mainstream QM, the V term need not always have its origin in the electrostatic interactions of elementary point-charges.

It could be any arbitrary source that imparts a potential energy to the system. Thus, in the mainstream QM, the source of V could also be gravitational, magnetic, etc. Further, in the mainstream QM, V could be any arbitrary function; it doesn’t have to be singularly anchored into any kind of point-particles.

In the context of discussions of foundations of QM—of QM Ontology—we reject such an interpretation. We instead take the view that V arises only from the electrostatic interactions of charges. The following discussion is written from this viewpoint.

It turns out that, speaking in the most fundamental and general terms, and following the mainstream QM’s logic, the V function too must be seen as a function that “lives” in an abstract 3ND configuration space. Let’s try to understand a certain peculiarity of the electrostatic V function better.

Consider an electrostatic system of two point-charges. The potential energy of the system now depends on their separation: V = V(\vec{r}_2 - \vec{r}_1) \propto q_1q_2/|\vec{r}_2 - \vec{r}_1|. But a separation is not the same as a position.

For simplicity, assume unit positive charges in a 1D space, and the constant of proportionality also to be 1 in suitable units. Suppose now you keep \vec{r}_1 fixed, say at x = 0.0, and vary only \vec{r}_2, say to x = 1.0, 2.0, 3.0, \dots, then you will get a certain series of V values, 1.0, 0.5, 0.33\dots, \dots.

You might therefore be tempted to imagine a 1D function for V, because there is a clear-cut mapping here, being given by the ordered pairs of \vec{r}_2 \Rightarrow V values like: (1.0, 1.0), (2.0, 0.5), (3.0, 0.33\dots), \dots. So, it seems that V can be described as a function of \vec{r}_2.

But this conclusion would be wrong because the first charge has been kept fixed all along in this procedure. However, its position can be varied too. If you now begin moving the first charge too, then using the same \vec{r}_2 value will gives you different values for V. Thus, V can be defined only as a function of the separation space \vec{s} = \vec{r}_2 - \vec{r}_1.

If there are more than two particles, i.e. in the general case, the multi-particle Schrodinger equation of N particles uses that form of V which has N(N-1) pairs of separation vectors forming its argument. Here we list some of them: \vec{r}_2 - \vec{r}_1, \vec{r}_3 - \vec{r}_1, \vec{r}_4 - \vec{r}_1, \dots, \vec{r}_1 - \vec{r}_2, \vec{r}_3 - \vec{r}_2, \vec{r}_4 - \vec{r}_2, \dots, \vec{r}_1 - \vec{r}_3, \vec{r}_2 - \vec{r}_3, \vec{r}_4 - \vec{r}_1, \dots, \dots. Using the index notation:

V = \sum\limits_{i=1}^{N}\sum\limits_{j\neq i, j=1}^{N} V(\vec{s}_{ij}),

where \vec{s}_{ij} = \vec{r}_j - \vec{r}_i.

Of course, there is a certain redundancy here, because the s_{ij} = |\vec{s}_{ij}| = |\vec{s}_{ji}| = s_{ji}. The electrostatic potential energy function depends only on s_{ij}, not on \vec{s}_{ij}. The general sum formula can be re-written in a form that avoids double listing of the equivalent pairs of the separation vectors, but it not only looks a bit more complicated, but also makes it somewhat more difficult to understand the issues involved. So, we will continue using the simple form—one which generates all possible N(N-1) terms for the separation vectors.

If you try to embed this separation space in the physical 3D space, you will find that it cannot be done. You can’t associate a unique separation vector for each position vector in the physical space, because associated with any point-position, there come to be an infinity of separation vectors all of which have to be associated with it. For instance, for the position vector \vec{r}_2, there are an infinity of separation vectors \vec{s} = \vec{a} - \vec{r}_2 where \vec{a} is an arbitrary point (standing in for the variable \vec{r}_1). Thus, the mapping from a specific position vector \vec{r}_2 to potential energy values becomes an 1: \infty mapping. Similarly for \vec{r}_1. That’s why V is not a function of the point-positions in the physical space.

Of course, V can still be seen as proper 1:1 mapping, i.e., as a proper function. But it is a function defined on the space formed by all possible separation vectors, not on the physical space.

Homework: Contrast this situation from a function of two space variables, e.g., F = F(\vec{x},\vec{y}). Explain why F is a function (i.e. a 1:1 mapping) that is defined on a space of position vectors, but V can be taken to be a function only if it is seen as being defined on a space of separation vectors. In other words, why the use of separation vector space makes the V go from a 1:\infty mapping to a 1:1 mapping.


5. Wrapping up the problem statement:

If the above seems a quizzical way of looking at the phenomena, well, that precisely is how the multi-particle Schrodinger equation is formulated. Really. The wavefunction \Psi is defined on an abstract 3ND configuration space. Really. The potential energy function V is defined using the more abstract notion of the separation space(s). Really.

If you specify the position coordinates, then you obtain a single number each for the potential energy and the wavefunction. The mainstream QM essentially views them both as aspatial variables. They do capture something about the quantum system, but only as if they were some kind of quantities that applied at once to the global system. They do not have a physical existence in the 3D space-–even if the position coordinates from the physical 3D space do determine them.

In contrast, following our new approach, we take the view that such a characterization of quantum mechanics cannot be accepted, certainly not on the grounds as flimsy as: “That’s just how the math of quantum mechanics is! And it works!!” The grounds are flimsy, even if a Nobel laureate or two might have informally uttered such words.

We believe that there is a problem here: In not being able to regard either \Psi or V as referring to some simple ontological entities existing in the physical 3D space.

So, our immediate problem statement becomes this:

To find some suitable quantities defined on the physical 3D space, and to use them in such a way, that our maths would turn out to be exactly the same as given for the mainstream quantum mechanics.


6. A preview of things to come: A bit about the strategy we adopt to solve this problem:

To solve this problem, we begin with what is easiest to us, namely, the simpler, classical-looking, V function. Most of the next post will remain concerned with understanding the V term from the viewpoint of the above-noted problem. Unfortunately, a repercussion would be that our discussion might end up looking a lot like an endless repetition of the issues already seen (and resolved) in the earlier posts from this series.

However, if you ever suspect, I would advise you to keep the doubt aside and read the next post when it comes. Though the terms and the equations might look exactly as what was noted earlier, the way they are rooted in the 3D reality and combined together, is new. New enough, that it directly shows a way to regard even the \Psi field as a physical 3D field.

Quantum physicists always warn you that achieving such a thing—a 3D space-based interpretation for the system-\Psi—is impossible. A certain working quantum physicist—an author of a textbook published abroad—had warned me that many people (including he himself) had tried it for years, but had not succeeded. Accordingly, he had drawn two conclusions (if I recall it right from my fallible memory): (i) It would be a very, very difficult problem, if not impossible. (ii) Therefore, he would be very skeptical if anyone makes the claim that he does have a 3D-based interpretation, that the QM \Psi “lives” in the same ordinary 3D space that we engineers routinely use.

Apparently, therefore, what you would be reading here in the subsequent posts would be something like a brand-new physics. (So, keep your doubts, but hang on nevertheless.)

If valid, our new approach would have brought the \Psi field from its 3N-dimensional Platonic “heaven” to the ordinary physical space of 3 dimensions.

“Bhageerath” (भगीरथ) [^] ? … Well, I don’t think in such terms. “Bhageerath” must have been an actual historical figure, but his deeds obviously have got shrouded in the subsequent mysticism and mythology. In any case, we don’t mean to invite any comparisons in terms of the scale of achievements. He could possibly serve as an inspiration—for the scale of efforts. But not as an object of comparison.

All in all, “Bhageerath”’s deed were his, and they anyway lie in the distant—even hazy—past. Our understanding is our own, and we must expend our own efforts.

But yes, if found valid, our approach will have extended the state of the art concerning how to understand this theory. Reason good enough to hang around? You decide. For me, the motivation simply has been to understand quantum mechanics right; to develop a solid understanding of its basic nature.

Bye for now, take care, and sure join me the next time—which should be soon enough.


A song I like:

[The official music director here is SD. But I do definitely sense a touch of RD here. Just like for many songs from the movie “Aaraadhanaa”, “Guide”, “Prem-Pujari”, etc. Or, for that matter, music for most any one of the movies that the senior Burman composed during the late ’60s or early ’70s. … RD anyway was listed as an assistant for many of SD’s movies from those times.]

(Hindi) “aaj ko junali raat maa”
Music: S. D. Burman
Singer: Lata Mangeshkar, Mohammad Rafi
Lyrics: Majrooh Sultanpuri


History:
— First published 2019.10.13 14:10 IST.
— Corrected typos, deleted erroneous or ill-formed passages, and improved the wording on home-work (in section 4) on the same day, by 18:29 IST.
— Added the personal comment in the songs section on 2019.10.13 (same day) 22:42 IST.