QM: The physical view it takes—1

So, what exactly is quantum physics like? What is the QM theory all about?

You can approach this question at many levels and from many angles. However, if an engineer were to ask me this question (i.e., an engineer with sufficiently good grasp of mathematics such as differential equations and linear algebra), today, I would answer it in the following way. (I mean only the non-relativistic QM here; relativistic QM is totally beyond me, at least as of today):

Each physics theory takes a certain physical view of the universe, and unless that view can be spelt out in a brief and illuminating manner, anything else that you talk about it (e.g. the maths of the theory) tends to become floating, even meaningless.

So, when we speak of QM, we have to look for a physical view that is at once both sufficiently accurate and highly meaningful intuitively.

But what do I mean by a physical view? Let me spell it out first in the context of classical mechanics so that you get a sense of that term.

Personally, I like to think of separate stages even within classical mechanics.

Consider first the Newtonian mechanics. We can say that the Newtonian mechanics is all about matter and motion. (Maxwell it was, I think, who characterized it in this beautifully illuminating a way.) Newton’s original mechanics was all about the classical bodies. These were primarily discrete—not quite point particles, but finite ones, with each body confined to a finite and isolated region of space. They had no electrical attributes or features (such as charge, current, or magnetic field strength). But they did possess certain dynamical properties, e.g., location, size, density, mass, speed, and most importantly, momentum—which was, using modern terminology, a vector quantity. The continuum (e.g. a fluid) was seen as an extension of the idea of the discrete bodies, and could be studied by regarding an infinitesimal part of the continuum as if it were a discrete body. The freshly invented tools of calculus allowed Newton to take the transition from the discrete bodies (billiard balls) to both: the point-particles (via the shells-argument) as well as to the continuum (e.g. the drag force on a submerged body.)

The next stage was the Euler-Lagrange mechanics. This stage represents no new physics—only a new physical view. The E-L mechanics essentially was about the same kind of physical bodies, but now a number (often somewhat wrongly called a scalar) called energy being taken as the truly fundamental dynamical attribute. The maths involved the so-called variations in a global integral expression involving an energy-function (or other expressions similar to energy), but the crucial dynamic variable in the end would be a mere number; the number would be the outcome of evaluating a definite integral. (Historically, the formalism was developed and applied decades before the term energy could be rigorously isolated, and so, the original writings don’t use the expression “energy-function.” In fact, even today, the general practice is to put the theory using only the mathematical and abstract terms of the “Lagrangian” or the “Hamiltonian.”) While Newton’s own mechanics was necessarily about two (or more) discrete bodies locally interacting with each other (think collisions, friction), the Euler-Lagrange mechanics now was about one discrete body interacting with a global field. This global field could be taken to be mass-less. The idea of a global something (it only later on came to be called a field) was already a sharp departure from the original Newtonian mechanics. The motion of the massive body could be predicted using this kind of a formalism—a formalism that probed certain hypothetical variations in the global field (or, more accurately, in the interactions that the global field had with the given body). The body itself was, however, exactly as in the original Newtonian mechanics: discrete (or spread over definite and delimited region of space), massive, and without any electrical attributes or features.

The next stage, that of the classical electrodynamics, was about the Newtonian massive bodies but now these were also seen as endowed with the electrical attributes in addition to the older dynamical attributes of momentum or energy. The global field now became more complicated than the older gravitational field. The magnetic features, initially regarded as attributes primarily different from the electrical ones, later on came to be understood as a mere consequence of the electrical ones. The field concept was now firmly entrenched in physics, even though not always very well understood for what it actually was: as a mathematical abstraction. Hence the proliferation in the number of physical aethers. People rightly sought the physical referents for the mathematical abstraction of the field, but they wrongly made hasty concretizations, and that’s how there was a number of aethers: an aether of light, an aether of heat, an aether of EM, and so on. Eventually, when the contradictions inherent in the hasty concretizations became apparent, people threw the baby with the water, and it was not long before Einstein (and perhaps Poincare before him) would wrongly declare the universe to be devoid of any form of aether.

I need to check the original writings by Newton, but from whatever I gather (or compile, perhaps erroneously), I think that Newton had no idea of the field. He did originate the idea of the universal gravitation, but not that of the field of gravity. I think he would have always taken gravity to be a force that was directly operating between two discrete massive bodies, in isolation to anything else—i.e., without anything intervening between them (including any kind of a field). Gravity, a force (instantaneously) operating at a distance, would be regarded as a mere extension of the idea of the force by the direct physical contact. Gravity thus would be an effect of some sort of a stretched spring to Newton, a linear element that existed and operated between only two bodies at its two ends. (The idea of a linear element would become explicit in the lines of force in Faraday’s theorization.) It was just that with gravity, the line-like spring was to be taken as invisible. I don’t know, but that seems like a reasonable implicit view that Newton must have adopted. Thus, the idea of the field, even in its most rudimentary form, probably began only with the advent of the Euler-Lagrange mechanics. It anyway reached its full development in Maxwell’s synthesis of electricity and magnetism into electromagnetism. Remove the notion of the field from Maxwell’s theory, and it is impossible for the theory to even get going. Maxwellian EM cannot at all operate without having a field as an intermediate agency transmitting forces between the interacting massive bodies. On the other hand, Newtonian gravity (at least in its original form and at least for simpler problems) can. In Maxwellian EM, if two bodies suddenly change their relative positions, the rest of the universe comes to feel the change because the field which connects them all has changed. In Newtonian gravity, if two bodies suddenly change their relative positions, each of the other bodies in the universe comes to feel it only because its distances from the two bodies have changed—not because there is a field to mediate that change. Thus, there occurs a very definite change in the underlying physical view in this progression from Newton’s mechanics to Euler-Lagrange-Hamilton’s to Maxwell’s.

So, that’s what I mean by the term: a physical view. It is a view of what kind of objects and interactions are first assumed to exist in the universe, before a physics theory can even begin to describe them—i.e., before any postulates can even begin to be formulated. Let me hasten to add that it is a physical view, and not a philosophical view, even though physicists, and worse, mathematicians, often do confuse the issue and call it a (mere) philosophical discussion (if not a digression). (What better can you expect from mathematicians anyway? Or even from physicists?)

Now, what about quantum mechanics? What kind of objects does it deal with, and what kind of a physical view is required in order to appreciate the theory best?

What kind of objects does QM deal with?

QM once again deals with bodies that do have electromagnetic attributes or features—not just the dynamical ones. However, it now seeks to understand and explain how these features come to operate so that certain experimentally observed phenomena such as the cavity radiation and the gas spectra (i.e., the atomic absorption- and emission-spectra) can be predicted with a quantitative accuracy. In the process, QM keeps the idea of the field more or less intact. (No, strictly speaking it doesn’t, but that’s what physicists think anyway). However, the development of the theory was such that it had to bring the idea of the spatially delimited massive body, occupying a definite place and traveling via definite paths, into question. (In fact, quantum physicists went overboard and threw it out quite gleefully, without a thought.) So, that is the kind of “objects” it must assume before its theorization can at all begin. Physicists didn’t exactly understand what they were dealing with, and that’s how arose all its mysteries.

Now, how about its physical view?

In my (by now revised) opinion, quantum mechanics basically is all about the electronic orbitals and their evolutions (i.e., changes in the orbitals, with time).

(I am deliberately using the term “electronic” orbital, and not “atomic” orbital. When you say “atom,” you must mean something that is localized—else, you couldn’t possibly distinguish this object from that at the gross scale. But not so when it is the electronic orbitals. The atomic nucleus, at least in the non-relativistic QM, can be taken to be a localized and discrete “particle,” but the orbitals cannot be. Since the orbitals are necessarily global, since they are necessarily spread everywhere, there is no point in associating something local with them, something like the atom. Hence the usage: electronic orbitals, not atomic orbitals.)

The electronic orbital is a field whose governing equation is the second-order linear PDE that is Schrodinger’s equation, and the problems in the theory involve the usual kind of IVBV problems. But a further complexity arises in QM, because the real-valued orbital density isn’t the primary unknown in Schrodinger’s equation; the primary unknown is the complex-valued wavefunction.

The Schrodinger equation itself is basically like the diffusion equation, but since the primary unknown is complex-valued, it ends up showing some of the features of the wave equation. (That’s one reason. The other reason is, the presence of the potential term. But then, the potential here is the electric potential, and so, once again, indirectly, it has got to do with the complex nature of the wavefunction.) Hence the name “wave equation,” and the term “wavefunction.” (The “wavefunction” could very well have been called the “diffusionfunction,” but Schrodinger chose to call it the wavefunction, anyway.) Check it out:

Here is the diffusion equation:

\dfrac{\partial}{\partial t} \phi = D \nabla^2 \phi
Here is the Schrodinger equation:
\dfrac{\partial}{\partial t} \Psi = \dfrac{i\hbar}{2\mu} \nabla^2 \Psi + V \Psi

You can always work with two coupled real-valued equations instead of the single, complex-valued, Schrodinger’s equation, but it is mathematically more convenient to deal with it in the complex-valued form. If you were instead to work with the two coupled real-valued  equations, they would still end up giving you exactly the same results as the Schrodinger equation. You will still get the Maxwellian EM after conducting suitable grossing out processes. Yes, Schrodinger’s equation must give rise to the Maxwell’s equations. The two coupled real-valued equations would give you that (and also everything else that the complex-valued Schrodinger’s equation does). Now, Maxwell’s equations do have an inherent  coupling between the electric and magnetic fields. This, incidentally, is the simplest way to understand why the wavefunction must be complex-valued. [From now on, don’t entertain the descriptions like: “Why do the amplitudes have to be complex? I don’t know. No one knows. No one can know.” etc.]

But yes, speaking in overall terms, QM is, basically, all about the electronic orbitals and the changes in them. That is the physical view QM takes.

Hold that line in your mind any time you hit QM, and it will save you a lot of trouble.

When it comes to the basics or the core (or the “heart”) of QM, physicists will never give you the above answer. They will give you a lot many other answers, but never this one. For instance, Richard Feynman thought that the wave-particle duality (as illustrated by the single-particle double-slit interference arrangement) was the real key to understanding the QM theory. Bohr and Heisenberg instead believed that the primacy of the observables and the principle of the uncertainty formed the necessary key. Einstein believed that entanglement was the key—and therefore spent his time using this feature of the QM to deny completeness to the QM theory. (He was right; QM is not complete. He was not on the target, however; entanglement is merely an outcome, not a primary feature of the QM theory.)

They were all (at least partly) correct, but none of their approaches is truly illuminating—not to an engineer anyway.

They were correct in the sense, these indeed are valid features of QM—and they do form some of the most mystifying aspects of the theory. But they are mystifying only to an intuition that is developed in the classical mechanical mould. In any case, don’t mistake these mystifying features for the basic nature of the core of the theory. Discussions couched in terms of the more mysterious-appearing features in fact have come to complicate the quantum story unnecessarily; not helped simplify it. The actual nature of the theory is much more simple than what physicists have told you.

Just the way the field in the EM theory is not exactly the same kind of a continuum as in the original Newtonian mechanics (e.g., in EM it is mass-less, unlike water), similarly, neither the field nor the massive object of the QM is exactly as in their classical EM descriptions. It can’t be expected to be.

QM is about some new kinds of the ultimate theoretical objects (or building blocks) that especially (but not exclusively) make their peculiarities felt at the microscopic (or atomic) scale. These theoretical objects carry certain properties such that the theoretical objects go on to constitute the observed classical bodies, and their interactions go on to produce the observed classical EM phenomena. However, the new theoretical objects are such that they themselves do not (and cannot be expected to) possess all the features of the classical objects. These new theoretical objects are to be taken as more fundamental than the objects theorized in the classical mechanics. (The physical entities in the classical mechanics are: the classical massive objects and the classical EM field).

Now, this description is quite handful; it’s not easy to keep in mind. One needs a simpler view so that it can be held and recalled easily. And that simpler view is what I’ve told you already:

To repeat: QM is all about the electronic orbital and the changes it undergoes over time.

Today, most any physics professor would find this view objectionable. He would feel that it is not even a physics-based view, it is a chemistry-based one, even if the unsteady or the transient aspect is present in the formulation. He would feel that the unsteady aspect in the formulation is artificial; it is more or less slapped externally on to the picture of the steady-state orbitals given in the chemistry textbooks, almost as an afterthought of sorts. In any case, it is not physics—that’s what he would be sure of. By that, he would also be sure to mean that this view is not sufficiently mathematical. He might even find it amusing that a physical view of QM can be this intuitively understandable. And then, if you ask him for a sufficiently physics-like view of QM, he would tell you that a certain set of postulates is what constitutes the real core of the QM theory.

Well, the QM postulates indeed are the starting points of QM theory. But they are too abstract to give you an overall feel for what the theory is about. I assert that keeping the orbitals always at the back of your mind helps give you that necessary physical feel.

OK, so, keeping orbitals at the back of the mind, how do we now explain the wave-particle duality in the single-photon double-slit interference experiment?

Let me stop here for this post; I will open my next post on this topic precisely with that question.

A Song I Like:

(Hindi) “ik ajeeb udaasi hai, meraa man_ banawaasi hai…”
Music: Salil Chowdhury
Singer: Sayontoni Mazumdar
Lyrics: (??)

[No, you (very probably) never heard this song before. It comes not from a regular film, but supposedly from a tele-film that goes by the name “Vijaya,” which was produced/directed by one Krishna Raaghav. (I haven’t seen it, but gather that it was based on a novel of the same name by Sharat Chandra Chattopadhyaya. (Bongs, I think, over-estimate this novelist. His other novel is Devadaas. Yes, Devadaas. … Now you know. About the Chattopadhyaya.)) Anyway, as to this song itself, well, Salil-daa’s stamp is absolutely unmistakable. (If the Marathi listener feels that the flute piece appearing at the very beginning somehow sounds familiar, and then recalls the flute in Hridayanath Mangeshkar’s “mogaraa phulalaa,” then I want to point out that it was Hridayanath who once assisted Salil-daa, not the other way around.) IMO, this song is just great. The tune may perhaps sound like the usual ghazal-like tune, but the orchestration—it’s just extraordinary, sensitive, and overall, absolutely superb. This song in fact is one of Salil-daa’s all-time bests, IMO. … I don’t know who penned the lyrics, but they too are great. … Hint: Listen to this song on high-quality head-phones, not on the loud-speakers, and only when you are all alone, all by yourself—and especially as you are nursing your favorite Sundowner—and especially during the times when you are going jobless. … Try it, some such a time…. Take care, and bye for now]