Ontologies in physics—10: Objects in QM. Aetherial fields in QM. Particle-in-a-box.

0. Prologue:

The last time we saw the context for, and the scheme of the inductive derivation of, the Schrodinger equation. In this post, we will see the ontology which it demands—the kind of ontological objects there have to be, so that the physical meaning of the Schrodinger equation can be understood correctly.

I wrote down at least 2 or 3 different ways of presentations of the topics for this post. However, either the points weren’t clear enough, or the discussion was going too far away, and I was losing the focus on ontology per say.

That’s why, I have decided to first present the ontology of QM without any justification, and only then to explain why assuming this particular ontology, rather than any other, makes sense. In justifying this ontology, we will have to note the salient peculiarities regarding the mathematical nature of Schrodinger’s equation, as also many relevant quantum mechanical features.

In this post, we will deal with only one-particle quantum systems.

So, let’s get going with the actual ontology first.


1. Our overall view of the QM ontology:

1.1. Introductory remarks:

To specify an ontology of physics is to state the basic types of objects there have to exist in the physical reality, and the basic ways in which they interact, so that the given theory of physics makes sense—the physical phenomena the theory subsumes are identified with appropriate concepts, causal relations, laws, and so, an understanding can be developed for applications, for building new systems that make use of the subsumed phenomena. The basic purpose of physics is to develop understanding so that it can be put to use to build better systems—structures, engines, machines, circuits, devices, gadgets, etc.

Accordingly, we will first give a list of the type of objects that must exist in the physical world so that the quantum mechanical phenomena can be completely described using them. The theory we will assume is Schrodinger’s non-relativistic quantum mechanics of multiple particles, including phenomena like entanglement, but without including the quantum mechanical spin. However, in this post, we will cover those aspects that can be understood (or at least touched upon) using only the single-particle quantum systems.

1.2. The list of objects in our QM ontology:

The list of our QM ontological objects is this:

  • The EC Objects of electrons and protons.
  • A special category of objects called neutrons.
  • The aether filling all of the 3D space where other objects are not, and certain field-conditions present in it; the all-connecting aspect of the physical universe.
  • The photon as a certain kind of a transient condition in the aether, i.e., a virtual object.

Let’s see all of them in detail, one by one, but beginning with the aether first.


2. The aether:

Explaining the concept of the aether and its necessity are full-fledged topics by themselves, and we have already said a lot about the ontology of this background object in the previous posts. So, we will note just a few indicative characteristics of the aether here.

Our idea of the QM aether is exactly the same as that of the EM aether of Lorentz. The only difference is that the aether, when used in QM, the aether is seen as supporting not only the electrostatic fields but also one more type of a field, the complex-valued quantum mechanical field.

To note some salient points about the aether:

  • The aether has no such inertia that it shows up in the electrostatic or quantum-mechanical phenomena. So, in this sense, the aether is non-inertial in nature.
  • It exists in all parts of space where the other QM ontological objects (of electrons, protons and neutrons) are not.
  • It exchanges electrostatic as well as additional quantum-mechanical forces with the electrons and protons, but always by direct contact alone.
  • Apart from the electrostatic and quantum-mechanical forces, there are no other forces that enter into our ontological description. Thus, there is no drag-force exerted by the aether on the electrons, protons or neutrons (basically because the Lorentz aether is not a mechanical aether; it is not an NM-Ontological object). In the non-relativistic QM, we also ignore fields like magnetic, gravitational, etc.
  • All parts of the aether always remains stationary, i.e., no CV of itself translates in space at any time. Even if there is any actual translation going on in the aether, the quantum mechanical phenomena are unable to capture it, and so, a capacity to translate does not enter our ontology.
  • However, unlike in the EM theory, when it comes to QM, we have to assume that there are other motions in aether. In QM, the aether does come to carry a kinetic energy too, whereas in EM, the kinetic energy is a feature of only the massive EC Objects. So, the aether is stationary—but that’s only translation-wise. Yet, even in the absence of net displacements, it does force (and is forced by) the elementary charged objects of the electrons and protons.

We will note further details regarding the fields in the aether as we progress.


3. Electrons and protons:

The view of electrons and protons which we take in the QM ontology is exactly the same as that in the ontology of electrostatics; so see the previous posts in this series for details not again repeated here.

Electrons and protons are seen as elementary point-particles having, within the algebraic sign, the same amount of electrostatic charge e. They set up certain 3D field conditions in the non-inertial aether, but acting in pairs. We may sometimes informally call them as point-charges, but it is to be kept in mind that, strictly speaking, in our view, we do not regard the charge to be an attribute of the point-particle, but only of the aether.

For two arbitrary EC objects (electrons or protons) q_i and q_j forming a pair, there are two fields which simultaneously exist in the 3D aether. None can exist without the other. These fields may be characterized as force-fields or as potential energy fields.

In the interest of clarity in the multi-particle situations, we will now expand on the notation presented earlier in this series. Accordingly,

\vec{\mathcal{F}}(q_i|q_j) is the 3D force field which exists everywhere in the aether. It gives the Coulomb force that q_j experiences from the aether at its instantaneous position \vec{r}_j via direct contact (between the aether and itself). Thus, in this notation, q_j is the forced charge, and q_i is the field-producing charge. Quantitatively, this force-field is given by Coulomb’s law:

\vec{\mathcal{F}}(q_i|q_j) = \dfrac{1}{4\,\pi\,\epsilon_0}\dfrac{q_i q_A}{r_{iA}^2} \hat{r}_{iA}, where q_A = q_j.

Similarly, \vec{\mathcal{F}}(q_i|q_j) is the aetherial force-field set up by q_j and felt by q_i in the same pair, and is given as:

\vec{\mathcal{F}}(q_j|q_i) = \dfrac{1}{4\,\pi\,\epsilon_0}\dfrac{q_j q_A}{r_{jA}^2} \hat{r}_{jA}, where q_A = q_i.

The fields are singular at the location of the forcing charge, but not at the location of the forced charge. Due to the divergence theorem, a given charge does not experience its own field.

There is no self-interaction problem either, because the EC Object (the point-charge) is ontologically a different object from both the aether and the NM objects. Only an NM Object could possibly explode under the self-field, primarily, because an NM Object is a composite. However, an EC Object (of an electron or a proton) is not an NM Object—it is elementary, not composite.

Notice that the specific forces at the positions of the q_i and q_j are equal in magnitude and opposite in directions. However, these two vectors act on two different objects, and therefore they don’t cancel each other. The two vectors also act at two different locations. In any case, in going from these two vectors to the two vector fields, it’s misleading to keep thinking in terms of one force-field as being the opposite of the other! Their respective anchoring locations (i.e. the two singularities) themselves are different, and they have the same signs too!! They are the same 1/(r^2) fields, but spatially shifted so as to anchor into the two charges of a pair.

When there are N number of elementary charged particles in a system, then a given charge q_j will experience the force fields produced by all the other (N-1) number of charges at its position. We can list them all before the pipe | symbol. For instance, \vec{\mathcal{F}}(q_1, q_3, q_4|q_2) is the net field that q_2 feels at its position \vec{r}_2; it equals the sum of the three force-fields produced by the other three charges because of the three pairs in which they act:
\vec{\mathcal{F}}(q_1, q_3, q_4|q_2) = \vec{\mathcal{F}}(q_1|q_2) + \vec{\mathcal{F}}(q_3|q_2) + \vec{\mathcal{F}}(q_4|q_2).

The charges always act pairs-wise; hence there always are pairs of fields; a single field cannot exist. Therefore, any analysis that has only one field (e.g., as in the quantum harmonic oscillator problem or the H atom problem), must be regarded as only a mathematical abstraction, not an existent.

The two fields of a given specific pair both are of the same algebraic sign: both + or both -. However, a given charge q_j may come to experience fields of arbitrary signs—depending on the signs of the other q_i‘s forming those particular pairs.

The electrons and protons thus affect each other via the intervening aether.

In electrostatics as well as in non-relativistic QM, the interaction between charges are via direct contact. However, the two fields of any arbitrary pair of charges shift instantaneously in space—the entirety of a field “moves” when the singular point where it is anchored, moves. Thus, there is no action-at-a-distance in this ontology. However, there are instantaneous changes everywhere in space.

A relativistic theory of QM would include magentic fields and their interactions with the electric fields. It is these interactions which together impose the relativistic speed limit of v < c for all material particles. However, such speed-limiting interaction are absent in the non-relativistic QM theory.

The electron and protons have the same magnitude of charge, but different masses.

The Coulombic force should result in accelerations of both the charges in a pair. However, since the proton is approx. 1846 times more massive than the electron, the actual accelerations (and hence net displacements over a finite time interval) undergone by them are vastly different.

There is a viewpoint (originally put forth by Lorentz, I guess) which says that since the entire interaction proceeds through the aether, there is no need to have massive particles of charge at all. This argument in essence says: We took the attribute of the electric charge away from the particle and re-attributed it to the aether. Why not do the same for the mass?

Here we observe that mass can be regarded as an attribute of the interactions of two *singular* fields in the aether. We tentatively choose to keep the instantaneous location of the attribute of the mass only at the distinguished point of the singularity. In short, we have both particles and the aether. If the need be, we will revisit this aspect of our ontology later on.

The electrostatic aetherial fields can also be expressed via two physically equivalent but mathematically different formulations: vector force-fields, and scalar energy-fields—also called the “potential” energy fields in the Schrodinger QM.

Notation: The potential energy field seen by q_j due to q_i is now on noted, and given, as:

V(q_i|q_j) = \dfrac{1}{4\,\pi\,\epsilon_0}\dfrac{q_i\,q_A}{r_{iA}},

where q_A = q_j, and similarly for the other field of the pair, viz., V(q_j|q_i)

See the previous posts from this series for a certain reservation we have for calling them the potential energy fields (and not just internal energy fields). In effect, what we seem to have here is an interesting scenario:

When we have a pair of charges in the physical 3D space (say an infinite domain), then we have two singular fields existing simultaneously, as noted above. Moving the two charges from their definite positions “to” infinity makes the system devoid of all energy. When they are present at definite positions, their singular fields of V noted above imply an infinite amount of energy within the volume of the system. However, since the system-boundaries for a system of charged point-particles can be defined only at the point-locations where they are present, the work that can be extracted from the system is finite—even if the total energy content is infinite. In short, we have a situation that addition of two infinities results in a finite quantity.

Does this way of looking at the things provide a clue to solve the problem of cancelling infinities in the re-normalization problem? If yes, and if none has put forth a comparably clear view, please cite this work.


4. Neutrons:

Neutrons are massive objects that do not participate in electrostatic interactions.

From very basic, ontological, viewpoint, they could have presented very tricky situations to deal with.

For instance: When an EC Object (i.e., an electron or a proton) moves through the aether, there is no force over and above the one exerted by the Coulombic field on it. But EC Objects are massive particles. So, a tempting conclusion might be to say that the aether exerts no drag force at all on any massive object, and hence, there should be no drag force on the motion of a free neutron either.

I am not clear on such points. But I have certain reservations and apprehensions about it.

It is clear that the aforementioned tempting conclusion does not carry. It is known that the aether does not exert drag on the EC Objects. But an EC Object is different from a chargeless object of the neutron. Even a forced EC Object still has a field singularly anchored in its own position; it is just that in experiencing the forces by the field, the component of its own singular field plays no part (due to the divergence theorem). But the neutron, being chargeless object, has no singular field anchored in its position at all. It doesn’t have a field that is “silent” for its own motions. Since for a forced particle, the forces are exerted by the aether in its vicinity, I am not clear if the neutron should behave the same. May be, we could associate a pair of equal and opposite (positive and negative) fields anchored in the neutron’s position (of arbitrary q_N strength, not elementary), so that it again is chargeless, but can be seen to be interacting with the aether. If so, then the neutron could be seen as a special kind of an EC Object—one which has two equal and opposite aetherial-fields associated with it. In that case, we can be consistent and say that the neutron will not experience a drag force from the aether for the same reason the electron or the proton does not. I am not clear if I should be adopting such a position. I have to think further about it.

So, overall, my choice is to ignore all such issues altogether, and regard the neutrons, in the non-relativistic QM, as being present only in the atomic nucleus at all times. The nucleus itself is regarded, abstractly, as a charged point-particle in its own right.

Thus, effectively, we come regard the nuclear neutrons as just additions of constant masses to the total mass of the protons, and consider this extra-massive positively charged composite as the point-particle of the nucleus.


5. In QM, there is an aetherial field for the kinetic energy:

As stated previously, in addition to the electrostatic fields (mathematically expressed as force-fields or as energy-fields), in QM, the aether also comes to carry a certain time-varying field. The energy associated with these fields is kinetic in nature. That is to say, there should be some motion within the aether which corresponds to this part of the total energy.

We will come to characterize these motions with the complex-valued \Psi(x,t) field. However, as the discussion below will clarify, the wavefunction is only a mathematically isolated attribute of the physically existing kinetic energy field.

We will see that the motion associated with the quantum mechanical kinetic energy does not result in the net displacement of a CV. (It may be regarded as the motion of time-varying strain-fields.)

In our ontology, the kinetic energy field (and hence the field that is the wavefunction) primarily “lives” in the physical 3D space.

However, when the same physics is seen from a higher-level, abstract, mathematical viewpoint, the same field may also be seen as “living” in an abstract 3ND configuration space. Adopting such an abstract view has its advantages in simplifying some of the mathematical manipulations at a more abstract level. However, make a note that doing so also risks losing the richness of the concept of the physical fields, and with it, the opportunity to tackle the unusual features of the quantum mechanical theory right.


6. Photon:

In our view, the photon is a neither a spatially discrete particle nor even a condition that is permanently present in the aether.

A photon represents a specific kind of a transient condition in the aetherial quantum mechanical fields which comes to exist only for some finite interval of time.

In particular, it refers to the difference in the two field-conditions corresponding to a change in the energy eigenstates (of the same particle).

In the last sentence, we should have added: “of the same particle” without parentheses; however, doing so requires us to identify what exactly is a particle when the reference is squarely being made to field conditions. A proper discussion of photons cannot actually be undertaken until a good amount of physics preceding it is understood. So, we will develop the understanding of this “particle” only slowly.

For the time being, however, make a note of the fact that:

In our view, all photons always are “virtual” particles.

Photons are attributes of real conditions in the aether, and in this sense, they are not virtual. But they are not spatially discrete particles. They always refer to continuous changes in the field conditions with time. Since these changes are anchored into the positions of the positively charged protons in the atomic nuclei, and since the protons are point-particles, therefore, a photon also has at least one singularity in the electrostatic fields to which its definition refers. (I am still not clear whether we need just one singularity or at least two.) In short, photon does have point-position(s) as the reference points. Its emission/absorption events cannot be specified without making reference to definite points. In this sense, it does have a particle character.

Finally, one more point about photons:

Not all transient changes in the fields refer to photons. The separation vectors between charges are always changing, and they are always therefore causing transient changes in the system wavefunction. But not all such changes result in a change of energy eigenstates. So, not all transient field changes in the aether are photons. Any view of QM that seeks to represent every change in a quantum system via an exchange of photons is deeply suspect, to say the least. Such a view is not justified on the basis of the inductive context or nature of the Schrodinger equation.

We will now develop the context required to identify the exact ontological nature of the quantum mechanical kinetic energy fields.


7. The form of Schrodinger’s equation points to an oscillatory phenomenon:

Schrodinger’s equation (SE) in 1D formulation reads:

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

BTW, when we say SE, we always mean TDSE (time-dependent Schrodinger’s equation). When we want to refer specifically to the time-independent Schrodinger’s equation, we will call it by the short form TISE. In short, TISE is not SE!

Setting constants to unity, the SE shows this form:
i\,\dfrac{\partial \Psi(x,t)}{\partial t} =\ -\, \dfrac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t).

Its form is partly comparable to the following two real-valued PDEs:

heat-diffusion equation with internal heat generation:
\dfrac{\partial T(x,t)}{\partial t} =\ \dfrac{\partial^2 T(x,t)}{\partial x^2} + \dot{Q}(x,t),

and the wave equation:
\dfrac{\partial^2 u(x,t)}{\partial t^2} =\ \dfrac{\partial^2 u(x,t)}{\partial x^2} + V(x,t)u(x,t),

Yet, the SE is different from both.

  • Unlike the diffusion equation, the SE has the i sticking out on the left-hand side, and a negative sign (think of it as (i)(i) on the first term on the right hand-side. That makes the solution of SE complex—literally. For quite a long time (years), I pursued this idea, well known to the Monte Carlo Quantum Chemistry community, that the SE is the diffusion equation but in imaginary time it. Turns out that this idea, while useful in simplifying simulation techniques for problems like determining the bonding energy of molecules, doesn’t really help throw much light on the ontology of QM. Indeed, it serves to get at the right ontology more difficult.
  • As to the wave equation, it too has only a partial similarity to SE. We mentioned the last time the main difference: In the wave PDE, the time differential is to the second order, whereas in the SE, it is to the first order.

The crucial thing to understand here is (and I got it from Lubos Motl’s blog or replies on StackExchange or so) that even if the time-differential is to the first-order, you still get solutions that oscillate in time—if the wave variable is regarded as being full-fledged complex-valued.

The important lesson to be drawn: The Schrodinger equation gives the maths of some kind of a vibratory/oscillatory system. The term “wavefunction” is not a misnomer. (Under the diffusion equation analogy, for some time, I had wondered if it shouldn’t be called “diffusionfunction”. That way of looking at it is wrong, misleading, etc.)

So, to understand the physics and ontology of the SE better, we need to understand vibrations/oscillations/waves better. I don’t have the time to do it here, so I refer you to David Morin’s online draft book on waves as your best free resource. A good book also seems to be the one by Walter Fox Smith’s “Waves and Oscillations, a Prelude to QM” though I haven’t gone through all its parts (but what exactly is his last name?). A slightly “harder” book but excellent, at the UG level, and free, comes from Howard Georgi. Mechanical engineers could equally well open their books on vibrations and FEM analysis of the same. For real quick notes, see Allan Bower’s UG course notes on this topic as a part of his dynamics course at the Brown university.


8. Ontology of the quantum mechanical fields:

8.1. Schrodinger’s equation has complex-valued fields of energies:

OK. To go back to Schrodinger’s equation:

i\,\hbar \dfrac{\partial \Psi(x,t)}{\partial t} =\ -\, \dfrac{\hbar^2}{2m} \dfrac{\partial^2\Psi(x,t)}{\partial x^2} + V(x,t)\Psi(x,t) = (\text{a real-valued constant}) \Psi(x,t).

As seen in the last post, the scheme of derivation of the SE makes it clear that these terms have come from: the total internal energy, the kinetic energy, and the potential energy, respectively. Informally, we may refer to them as such. However, notice that whereas V(x,t) by itself is a field, what appears in the SE is the term of V(x,t) multiplifed by \Psi(x,t), which makes all the energies complex-valued. Further, since \Psi(x,t) is a field, all energies in the SE also are fields.

If you wish to have real-valued fields of energies, then you have no choice but to divide all the terms in the SE by \Psi(x,t). That’s what we indicated in the last post too. However, note, complex-valued fields cannot still be got rid of; they still enter the calculations.

8.2. Potential energy fields only come from the elementary point-charges:

The V(x,t) field itself is the same as in the electrostatics:

V(x,t) = \dfrac{1}{2} \dfrac{1}{4\,\pi\,\epsilon_0} \sum\limits_{i}^{N}\sum\limits_{j\neq i; j=1}^{N} \dfrac{q_i\,q_j}{r_iA},
where |q_i| = |q_j| = -e, with e being the fundamental electronic charge.

In our QM ontology we postulate that the above equation is logically complete as far as the potential energy field of QM is concerned.

That is to say, in the basic ontological description of QM, we do not entertain any other sources of potentials (such as gravity or magnetism). Equally important, we also do not entertain arbitrarily specified values for potentials (such as the parabolic potential well of the quantum harmonic oscillator, or the well with the sharply vertical walls of the particle-in-a-box model). Arbitrary potentials are mere mathematical abstractions—approximate models—that help us gain insight into some aspects of the physical phenomena; they do not describe the quantum mechanical reality in full. Only the electrostatic potential that is singularly anchored into elementary charge positions, does.

At least in the basic non-relativistic quantum mechanics, there is no scope to accommodate magnetism. The gravity, being too weak, also is best neglected. Thus, the only potentials allowed are the singular electrostatic ones.

We shall revisit this issue of the potentials after we solve the measurement problem. From our viewpoint, the mainstream QM’s use of arbitrary potentials of arbitrary sources is fine too, as the linear formulation of the mainstream QM turns out to be a limiting case of our nonlinear formulation.

8.3. What physically exists is only the complex-valued internal energy field:

Notice that according to our QM ontology, what physically exists is only the single field of the complex-valued total internal energy field.

Its isolation into different fields like the potential energy field, the kinetic energy field, the momentum field, or the wavefunction field, etc. are all mathematically isolated quantities. These fields do have certain direct physical referents, but only as aspects or attributes of the total internal energy field. They do have a physical existence, but their existence is not independent of the total internal energy field.

Finally, note that the total internal energy field itself exists only as a field condition in the aether; it is an attribute of the aether; it cannot exist without the aether.


9. Implications of the complex-valued nature of the internal energy field:

9.1. System-level attributes to spatial fields—real- vs. complex-valued functions:

Consider an isolated system—say the physical universe. In our notation, E denotes the aspatial global attribute of its internal energy. Think of a perfectly isolated box for a system. Then E is like a label identifying a certain quantity of joule slapped on to it. It has no spatial existence inside the box—nor outside it. It’s just a device of book-keeping.

To convert E into a spatially identifiable object, we multiply it by some field, say F(x,t). Then, E F(x,t) becomes a field.

If F(x,t) is real-valued, then \int\limits_{\Omega_\text{small CV}} \text{d}\Omega_\text{small CV}\, E\,F(x,t) gives you the amount of E present in a small CV (which is just a part of the system, not the whole). To fix ideas, suppose you have a stereo boom-box with two detachable speakers. Then, the volume of the overall boombox is a sum of the volumes of each of its three parts. The volume is a real-valued number, and so, the total volume is the simple sum of its parts V = V_1 + V_2 + V_3. Ditto for the weights of these parts. Ditto, for the energy in a volumetric part of a system if the energy forms a real-valued field.

Now, when the field is complex-valued, say denoted as \tilde{F}(x,t), then the volume integral still applies. \int\limits_{\Omega_\text{small CV}} \text{d}\Omega_\text{small CV}\, E\,\tilde{F}(x,t) still gives you the amount of the complex valued quantity E\tilde{F}(x,t) present in the CV. But the fact that \tilde{F} is complex-valued means that there actually are two fields of E inside that small CV. Expressing \tilde{F}(x,t) = a(x,t) + i b(x,t), there are two real-valued fields, a(x,t) and b(x,t). So, the energy inside the small CV also has two energy components: E_R = E a(x,t) and E_I = E b(x,t), which we call “real” and “imaginary”. Actually, physically, they both are real-valued. However, the magnitude of their net effect |E \tilde{F}(x,t)| != E_R + E_I. Instead, it follows the Pythagorean theorem all the way to the positive sign: |E \tilde{F}| = |\sqrt{E_R^2 + E_I^2}|. (Aren’t you glad you learnt that theorem!)

If you take it in a naive-minded way, then E can be greater or smaller than E_R + E_I, and so things won’t sum up to |E \tilde{F}|—conservation seems to fail.

But in fact, energy conservation does hold. It’s just that it follows a further detailed law of combining the two field components within a given CV (or the entire system).

In QM, the wavefunction \Psi(x,t) plays the role of \tilde{F} given above. It brings the aspatial energy E from its Platonic mathematical “heaven” and, further, being a field itself, also distributes it in space—thereby giving a complex-valued field of E.

We do not know the physical mechanism which manipulates the real and imaginary parts \Psi_R(x,t) and \Psi_I(x,t) so that they come to obey the Pythogorean theorem. But we know that unless we have \Psi(x,t) as complex-valued, the book-keeping of the system’s energy does not come out right—in QM, that is.

Since the product E_{\text{sys}}\Psi(x,t) can come up any time, and since what ontologically exists is a single quantity, not a product of two, it’s better to have a different notation for it. Accordingly, define:

\tilde{E}(x,t) = E_{\text{sys}}\,\Psi(x,t)

9.2. In QM, the conserved quantity itself is complex-valued:

Note an important difference between pre-quantum mechanics and QM:

The energy conservation principle for the classical (pre-quantum) mechanics says that E_{\text{sys}} = \int\limits_{\Omega} \text{d}\Omega E(x,t) is conserved.
The energy conservation principle for quantum mechanics is that \tilde{E}_{\text{sys}} = \int\limits_{\Omega} \text{d}\Omega \tilde{E}(x,t) is conserved.

No one says it. But it is there, right in the context (the scheme of derivation) of the Schrodinger equation!

For the cyclic change, we started from the classical conservation statement:
\oint \text{d}E_{\text{sys}} = 0 = \oint \text{d}T_{\text{sys}} + \oint \text{d}\Pi_{\text{sys}}

Or, in differential terms (for an arbitrary change, not cyclic):
\text{d}E_{\text{sys}} = 0 = \text{d}T_{\text{sys}} + \text{d}\Pi_{\text{sys}}.

Or, integrating over the end-points of an arbitrary process,
E_{\text{sys}} = \text{a constant (real-valued) number}.

We then multiplied both sides by \Psi(x,t) (remember the quizzical-looking multiplication from the last post?), and only then got to Schrodinger’s equation. In effect, we did:
\text{d}E_{\text{sys}}\Psi(x,t) = 0 = \text{d}T_{\text{sys}}\Psi(x,t) + \text{d}\Pi_{\text{sys}}\Psi(x,t).

That’s nothing but saying, using the notation introduced just above, that:
\text{d}\tilde{E}(x,t) = 0 = \text{d}\tilde{T}(x,t) + \text{d}\tilde{\Pi}(x,t).

Or, integrating over the end-points of an arbitrary process and over the system volume,
\tilde{E}_{\text{sys}} = \text{a constant complex number}.

So, what’s conserved is not E but \tilde{E}.

The aspatial, global, thermodynamic number for the total internal energy is the complex number \tilde{E}_{\text{sys}} in QM. QM by postulation comes with two coupled real-valued fields together obeying the algebra of complex numbers.


10. Consequences of conservation of complex-valued energy of the universe:

10.1. There is a real-valued measure of quantum-mechanical energy which is conserved too:

In QM, is there a real-valued number that gets conserved too? if not by postulate then at least by consequence?

Answer: Well, yes, there is. But it loses the richness of the physics of complex-numbers.

To obtain the conserved real-valued number, we follow the same procedure as for “converting” a complex number to a real number, i.e., extracting a real-valued and essential feature of a complex number. We take its absolute magnitude. If \tilde{E}_{\text{sys}} is a constant complex number, then obviously, |\tilde{E}_{\text{sys}}| is a constant number too. Accordingly,

|\tilde{E}_{\text{sys}}| = |\sqrt{\tilde{E}_{\text{sys}}\,\tilde{E}_{\text{sys}}^{*}}| = \text{another, real-valued, constant}.

But obviously, a statement of this kind of a constancy has lost all the richness of QM.

10.2. The normalization condition has its basis in the energy conservation:

Another implication:

Since |\tilde{E}_{\text{sys}}| itself is conserved, so is |\tilde{E}_{\text{sys}}|^2 too.

[An aside to experts: I think we thus have solved the curious problem of the arbitrary phase factors in quantum mechanics, too. Let me know if you disagree.]

It then follows, by definitions of \tilde{E}_{\text{sys}}, \tilde{E} and \Psi(x,t), that

\int\limits_{\Omega}\text{d}\Omega\,\Psi(x,t)\Psi^{*}(x,t) = 1

Thus, the square-normalization condition follows from the energy conservation principle.

We believe this view places the normalization condition on firm grounds.

The mainstream QM (at least as presented in textbooks) makes reference to (i) Born’s postulate for the probability of finding a particle in an elemental volume, and (ii) conservation of mass for the system (“the electron has to be somewhere in the system”).

In our view, the normalization condition arises because of conservation of energy alone. Conservation of mass is a separate principle, in our opinion. It applies to the attribute of mass of the EC Object of elementary charges. But not to the aetherial field of \Psi. Ontologically, the massive EC Objects and the aether are different entities. Finally, the probabilistic notions of particle position have no relevance in deriving the normalization condition. You don’t have to insert the measurement theory before imposing the normalization condition. Indeed, the measurement postulate comes way later.

Notice that the total complex-valued number for the energy of the universe remains constant. However, the time-dependence of \Psi(x,t) implies that the aether, and hence the universe forever remains in a state of oscillatory motions. (In the nonlinear theory, the system remains oscillatory, but the state evolutions are not periodic. Mark the difference between these two ideas.)

10.3. The wavefunction of the universe is always in energy eigenstates.

Another interesting consequence of the energy conservation principle is this:

Consider these two conclusions: (i) The universe is an isolated system; hence, its energy is conserved. (ii) There is only one aether object in the universe; hence, there is only one universal wavefunction.

A direct consequence therefore is this:

For an isolated system, the system wavefunction always remains in energy eigenstates. Hence, every state assumed by the universal wavefunction is an energy eigenstate.

Take a pause to note a few peculiarities about the preceding statement.

No, this statement does not at all reinforce misconceptions (see Dan Styer’s paper, here: [^][Preprint PDF ^])

The statement refers to isolated systems, including the universe. It does not refer to closed or open systems. When matter and/or energy can cross system boundaries, a mainstream-supposed “wavefunction” of the system itself may not remain in an energy eigenstate. Yet, the universe (system plus environment) always remains in some or the other energy eigenstate.

However, the fact that the universal wavefunction is always in an energy eigenstate does not mean that the universe always remains in a stationary state. Notice that the V(x,t) itself is time-dependent. So, the time-changes in it compel the \Psi to change in time too. (In the language of mainstream QM: The Hamiltonian operator is time-dependent, and yet, at any instant, the state of the universe must be an energy eigenstate.)

In our view, due to nonlinearity, V(x,t) also is an indirect function of the instantaneous \Psi(x,t). Will cover the nonlinearity and the measurement problem the next time. (Yes, I am extending this series by one post.)

Of course, at any instant, the integral over the domain of the algebraic sum of the kinetic and the potential energy fields is always going to come to the single number which is: the aspatial attribute of the total internal energy number for the isolated system.

10.4. The wavefunction \Psi(x,t) is ontic, but only indirectly so—it’s an attribute of the energy field, and hence of the aether, which is ontic:

So, is the wavefunction ontic or epistemic? It is ontic.

An attribute does not have a physical existence independent of, or as apart from, the object whose attribute it is. However, this does not mean that an attribute does not have any physical existence at all. Saying so would be a ridiculously simple error. Objects exist, and they exist as identities. The identity of an object refers to all its attributes—known and unknown. So, to say that an object exists is also to say that all its attributes exist (with all their metaphysically existing sizes too). It is true that blueness does not exist without there being a blue object. But if a blue object exist, obviously, its blueness exists in the reality out there too—it exists with all the blue objects. So, “things” such as blueness are part of existence. Accordingly, the wavefunction is ontic.

Yet, the isolation (i.e. identification) of the wavefunction as an attribute of the aether does require a complex chain of reasoning. Ummm… Yes, literally complex too, because it does involve the complex-valued SE.

The aether is a single object. There are no two or more aethers in the universe—or zero. Hence, there is only a single complex-valued field of energy, that of the total internal energy. For this reason, there is only one wavefunction field in the universe—regardless of the number of particles there might be in it. However, the system wavefunction can always be mathematically decomposed into certain components particular to each particle. We will revisit this point when we cover multi-particle quantum systems.

10.5. The wavefunction \Psi(x,t) itself is dimensionless:

In our view, the wavefunction, i.e., \Psi(x,t) itself is dimensionless. We base this conclusion on the fact that while deriving the Schrodinger equation, where \Psi(x,t) gets introduced, each term of the equation is regarded as an energy term. Since each term has \Psi(x,t) also appearing in it (and you cannot get rid of the complex nature of the Schrodinger equation merely by dividing all terms by it), obviously, the multiplying factor of \Psi(x,t) must be taken as being dimensionless. That’s how we in fact have proceeded.

The mainstream view is to assign the dimensions of \dfrac{1}{\sqrt{\text{(length)}^d}}, where d is the dimensionality of the embedding space. This interpretation is based on Born’s rule and conservation of matter; for instance, see here [^].

However, as explained in the sub-section 10.2., we arrive at the normalization condition from the energy conservation principle, and not in reference to Born’s postulate at all.

All in all, \Psi(x,t) is dimensionless. It appears in theory only for mathematical convenience. However, once defined, it can be seen as an attribute (aspect) of the complex-valued internal energy field (and its two components, viz. the complex-valued kinetic- and potential-energy fields). In this sense, it is ontic—as explained in the preceding sub-section.


11. Visualizing the wavefunction and the single particle in the PIB model:

11.1. Introductory remarks:

What we will be doing in this section is not ontology, strictly speaking, but only physics and visualization. PIB stands for: Particle-In-a-Box. Study this model from any textbook and only then read further.

The PIB model is unrealistic, but pedagogically useful. It is unrealistic because it uses a potential energy distribution that is not singularly anchored into point-particle positions. So, the potential energy distribution must be seen as a mathematically convenient abstraction. PIB is not real QM, in short. It’s the QM of the moron, in a way—the electron has no “potential” inside the well.

11.2. The potential energy function used in the model:

The model says that there is just one particle in a finite interval of space, and its V(x,t) always stays the same at all times. So, it uses V(x) in place of V(x,t).

The V(x) is defined to be zero everywhere in the domain except at the boundary-points, where the particle is supposed to suddenly acquire an infinite potential energy. Yes, the infinitely tall walls are inside the system, not outside it. The potential energy field is the potential energy of a point-particle, and unless it were to experience an infinity of potential energy while staying within the finite control volume of the system, no non-trivial solution would at all be possible. (The trivial solution for the SE when V(x) = 0 is that \Psi(x,t) = 0—whether the domain is finite or infinite.) In short, the “side-walls” are included in the shipped package.

If the particle is imagined to be an electron, then why does its singular field not come into picture? Simple: There is only one electron, and a given EC Object (an elementary point-charge) never comes to experience its own field. Thus, the PIB model is unrealistic on another ground: In reality, force-fields due to charges always come in pairs. However, since we consider only one particle in PIB, there are no singular force-fields anchored into a moving particle’s position, in it, at all.

Yes, forces do act on the particle, but only at the side-walls. At the boundary points, it is a forced particle. Everywhere else, it is a free particle. Peculiar.

The domain of the system remains fixed at all times. So, the potential walls remain fixed in space—before, during, and after the particle collides with them.

The impulse exerted on the particle at the time of collision at the boundary is theoretically infinite. But it lasts only for an infinitesimally small patch of space (which is represented as the point of the boundary). Hence, it cannot impart an infinity of velocity or displacement. (An infinitely large force would have to act over a finite interval of space and time before it could possibly result in an infinitely large velocity or displacement.)

OK. Enough about analysis in terms of forces. To arrive at the particular solution of this problem using analytical methods (as with most any other advanced problem), energy-analytical methods are superior. So, we go back to the energy-based analysis, and Schrodinger’s equation.

11.3. TDSE as a continuous sequence of TISE’s:

Note that you can always apply the product ansatz to \Psi(x,t), and thereby split it into two functions:

\Psi(x,t) = \chi(x)\tau(t),

where \chi(x) is the space-dependent part and \tau(t) is the time-dependent part.

No one tells you, but it is true that:

Even when the Hamiltonian operator is time-dependent, you can still use the product ansatz separately at every instant.

It is just that doing so is not very useful in analytical solution procedures, because both the \chi(x) and \tau(t) themselves change in time. Therefore, you cannot take a single time-dependent function \tau(t) as applying at all times, and thereby simplify the differential equation. You would have to progress the solution in time—somehow—and then again apply the product ansatz to obtain new functions of \chi(x) and \tau(t) which would be valid only for the next instant in the continuous progression of such changes.

So, analytical solution procedures do not at all benefit from the product ansatz when the Hamiltonian operator is time-dependent.

However, when you use numerical approaches, you can always progress the solution in time using suitable methods, and then, whatever \Psi(x,t)\big|_{t_n} you get for the current time t_n, you can regard it as if it were solving a TISE which was valid for that instant alone.

In other words, the TDSE is seen as being a continuous progression of different instantaneous TISE’s. Seen this way, each \Psi(x,t)\big|_{t_n} can be viewed as representing an energy eigenstate at every instant.

Not just that, but since there is no heat in QM, the adiabatic approximation always applies. So, for an isolated system or the physical universe:

For an isolated system or the physical universe, the time-dependent part \tau(t) of \Psi(x,t) may not be the same function at all times. Yet, it always progresses through a continuous progression of different \chi(x) and \tau(t)‘s.

We saw in the sub-section 10.3. that the universal wavefunction must always be in energy eigenstates. We had reached that conclusion in reference to energy conservation principle and the uniqueness of the aether in the universe. Now, in this sub-section, we saw a more detailed meaning of it.

11.4. PIB anyway uses time-independent potential energy function, and hence, time-independent Hamiltonian:

When V(x) is time-independent, the time-dependent part \tau(t) stays the same for all times. Using this fact, the SE reduces to one and the same pair of \chi(x) and \tau(t). So, the TISE in this case is very simple to solve. See your textbooks on how to solve the TISE for the PIB problem.

However, make sure to

work through any solution using only the full-fledged complex variables.

The solutions given in most text-books will prove insufficient for our purposes. For instance, if \tau(t) is the time-dependent part of the solution of TISE, then don’t substitute \tau(t) = \cos \omega t in place of the full-fledged \tau = e^{-i\omega t}.

Let the \tau(t) acquire imaginary parts too, as it evolves in time.

The reason for this insistence on the full complex numbers will soon become apparent.

11.5. Use the full-fledged 3D physical space:

To visualize this solution, realize that as in EM so also in QM, even if the problem is advertised as being 1D, it still makes sense to see this one dimension as an aspect of the actually existing 3D physical space. (In EM, you need to go “up” to 3D because the curl demands it. In QM, the reason will become apparent if you do the homework given below.)

Accordingly, we imagine two infinitely large parallel planes for the system boundaries, and the aether filling the space in between them. (Draw a sketch. I won’t. I would have, in a real class-room, but don’t have the enthusiasm to draw pics while writing mere blog-posts. And, whatever happened to your interest in visualization rather than in “math”?) The planes remain fixed in space.

Now, pick up a line passing normally through the two parallel planes. This is our x-axis.

11.6. The aetherial momentum field:

Next, consider the aetherial momentum field, defined by:

\vec{p}(x,t) =\ i\,\hbar\,\nabla\Psi(x,t).

This definition for the complex-valued momentum field is suggested by the form of the complex-valued quantum mechanical kinetic energy field. It has been derived in analogy to the classical expression T = \dfrac{p^2}{2m}.

In our PIB model, this field exists not just on the chosen line of the x-axis, but also everywhere in the 3D space. It’s just that it has no variation along the y– and z-axes.

11.7. Gaining physical clarity (“intuition”) with analysis in terms of forces, first:

In the PIB model, when the massive point-particle of the electron is at some point \vec{r}_j, then it experiences a zero potential force (except at the boundary points).

So, electrostatically speaking, the electron (i.e. the singularity at the EC Object’s position) should not move away from the point where it was placed as part of IC/BCs of the problem. However, the existence of the momentum field implies that it does move.

To see how this happens, consider the fact that \Psi(x,t) involves not just the space-dependent part \chi(x), but also the time-dependent part \Theta(t). So,

The total wavefunction \Psi(\vec{r}_j, t) is time-dependent—it continuously changes in time. Even in stationary problems.

Naturally, there should be an aetherial force-field associated with the aetherial momentum field (i.e. the aetherial kinetic energy field) too. It is given by:

\vec{F}_{T}(x,t) = \dfrac{\partial}{\partial t} \vec{p}_{T}(x,t) = \dfrac{\partial}{\partial t} \left[ i\,\hbar\,\nabla\Psi(x,t) \right],

where the subscript T denotes the fact these quantities refer to their conceptual origins in the kinetic energy field. These _T quantities are over and above those due to the electrostatic force-fields. So, if V were not to be zero in our model, then there would a force-field due to the electrostatic interactions as well, which we might denote as \vec{F}_{V}, where the subscript _V denotes the origin in the potentials.

Anyway, here V(x) = 0 at all internal points, and so, only the quantity of force given by \vec{F}_{T}(\vec{r}_j,t) would act on our particle when it strays at the location \vec{r}_j. Naturally, it would get whacked! (Feel good?)

The instantaneous local acceleration for the elemental CV of the aether around the point \vec{r}_j is given by \vec{a}_{T}(\vec{r}_j,t) = \dfrac{1}{m} \dfrac{\partial \vec{p}_{T}(\vec{r}_j,t)}{\partial t}.

This acceleration should imply a velocity too. It’s easy to see that the velocity so implied is nothing but

\vec{v}_{T}(\vec{r}_j,t) = \dfrac{1}{m} \vec{p}_{T}(\vec{r}_j,t).

Yes, we went through a “circle,” because we basically had defined the force on the basis of momentum, and we had given the more basic definition of momentum itself on the basis of the kinetic energy fields.

11.8. Representing complex-valued fields as spatial entities is logically consistent with everything we know:

Notice that all the fields we considered in the force-based analysis: the momentum field, the force-field, the acceleration field, and the velocity field are complex-valued. This is where the 3D-ness of our PIB model comes handy.

Think of any arbitrary yz-planes in the domain as representing the mathematical Argand-plane. Then, the \Psi(x,t) field at an arbitrary point \vec{r}_j would be a phasor of constant length, but rotating in the same yz-plane at a constant angular velocity, given by the time-dependent part \tau(t).

Homework: Write a Python simulation to show an animation of a few representative phasors for a few points in the domain, following the above convention.

11.9. Time evolution, and the spatial directions of the \Psi(x,t)-based vector fields:

Consider the changes in the \Psi(x,t) field, distributed in the physical 3D space.

Consider that as \tau(t) evolves in time, even if the IC had only a real-valued function like \cos t specified for it, considering the full-fledged complex-valued nature of \tau(t), it would soon enough (with the passage of an infinitesimal amount of time), acquire a so-called “imaginary” component.

Following our idea of representing the real- and imaginary-components in the y– and z-axes, the \Psi(x,t) field no longer remains confined to a variation along the x-axis alone. It also has variations along the plane normal to the x-axis.

Accordingly, the unit vectors for the grad operator, and hence for all the vector quantities (of momentum, velocity, force and acceleration) also acquire a definite orientation in the physical 3D space—without causing any discomfort to the “math” of the mainstream quantum mechanics.

Homework: Consider the case when \Psi(x,t) varies along all three spatial axes. An easy example would be that of the hydrogen atom wavefunction. Verify that the spatial representation of the vector fields (momentum, velocity, force or acceleration) proposed by us causes no harm to the the “math” of the mainstream quantum mechanics.

If doing simulations, you can integrate in time (using a suitable time-stepping technique), and come to calculate the instantaneous displacements of the particle, too. Exercise left for the reader.

Homework: Perform both analytical integration and numerical for the PIB model. Verify that your simulation is correct.

Homework: Build an animation for the motion of the point-particle of the EC Object, together with the time-variations of all the complex-valued fields: \Psi(x,t), and all the complex-valued vector fields derived from it.

11.10. Too much of homework?

OK. I’ve been assigning so many pieces for the homework today. Have I completed any one of them for myself? Well, actually not. But read on, anyway.

The locus of all possible particle-positions would converge to a point only at the boundary points (because \Psi(x,t) = 0 there. At all the internal points in the domain, the particle-position should be away from the x-axis.

That’s my anticipation, but I have not checked it. In fact, I have not built even a single numerical simulation of the sort mentioned here.

So, take this chance to prove me wrong!

Please do the homework and let me know if I am going wrong. Thanks in advance. (I have to finish this series first, somehow!)


12. What the PIB model tells about the wave-particle duality:

What happened to the world-famous wave-particle duality? If you build the animations, you would know!

There is a point-particle of the electron (which we regard as the point of the singularity in the \vec{\mathcal{F}} field), and there is an actual, 3D field of the internal energy fields—and hence of \Psi(x,t). And, assuming our hypothesis of representing phasors of the complex numbers via a spatial representation, of all the complex-valued fields—including the vector fields like displacement.

The particle motion is governed by both the potential energy-forces and the kinetic energy-forces. That is, the aetherial wavefunction “guides” etc. the particle. In our view, the kinetic energy field too forces the particle.

“Ah, smart!,” you might object. “And what happened to the Born rule? If the wavefunction is a field, then there is a probability for finding the particle anywhere—not just at the position where it is, as predicted in this model. So, your model is obviously dumb!! It’s not quantum mechanics at all!!!”

Hmmm… We have not solved the measurement problem yet, have we?

We will need to cover the many-particle QM first, and then go to the nonlinearity implied by the kinetic energy field-forces, and only then would we be able to present our solution to the measurement problem. Since I got tired of typing (this post is already ~9,500 words), I will cover it in some other post. I will also try to touch on entanglement, because it would come in the flow of the coverage.

But in the meanwhile, try to play with something.

Homework: “Invert” the time-displacement function/relationship you obtain for the PIB model, and calculate the time spent by the particle in each infinitesimally small CV of the 3D domain, during a complete round-trip across the domain. Find its x-component. See if you can relate the motion, in any way, to the probability rule given by Born (i.e., try to anticipate our next development).

Do that. This way, you will stay prepared to spot if I have made any mistakes in this post, and also if I make any further mistakes in the next—and have made any mistakes in the last post as well.

Really. I could easily have made a mistake or two. … These matters still are quite new to me, and I really haven’t worked out the maths of everything ahead of writing these posts. That’s why I say so.


13. A preview of the things to come:

I had planned to finish this series in this post. In a sense, it is over.

The most crucial ontological aspects have already been given. Starting from the comprehensive list of the QM objects, we also saw that the quantum mechanical aetherial fields are all complex-valued; that there is an additional kinetic energy field too, not just potential; and also saw our new ideas concerning how to visualize the complex-valued fields by regarding the Argand plane as a mathematical abstraction of a real physical plane in 3D. We also saw how these QM ontological objects come together in a simple but fairly well illustrative problem of the PIB. We even touched on the wave-particle duality.

So, as far as ontology is concerned, even the QM ontology is now essentially over. There might be important repercussions of the ontological points we discussed here (and, also before, in this series). But as far as I can see, these should turn out to be mostly consequences, not any new fundamental points.

Of course, a lot of physics issues still remain to be clarified. I would like to address them too.

So, while I am at it, I would also like to say something about the following topics: (i) Multi-particle quantum systems. (ii) Issue of the 3D vs. 3ND nature of the wavefunction field. (iii) Physics of entanglement. (iv) Measurement problem.

All these topics use the same ontology as used here. But saying something about them would, I hope, help understand it better. Applications always serve to understand the exact scope and the nuances of a theory. In their absence, a theory, even if well specified, still runs the risk of being misunderstood.

That’s why I would like to pick up the above four topics.

No promises, but I will try to write an “extra” post in this series, and finish off everything needed to understand the points touched upon in the Outline document (which I had uploaded at iMechanica in February this year, see here [^]). Unlike until now, this next post would be mostly geared towards QM experts, and so, it would progress rapidly—even unevenly or in a seeming “broken” manner. (Experts would always be free to get in touch with me; none has, in the 8+ months since the uploading of the Outline document at iMechanica.)

I would like it if this planned post (on the four physics topics from QM) forms the next post on this blog, but then again, as I said, no promises. There might be an interruption with other topics in the meanwhile (though I would try to keep them at the bay). Plus, I am plain tired and need a break too. So, no promises regarding the time-frame of when it might come.

OK.

So, do the homework, and think about the whole thing. Also, brush up on the topic of coupled oscillations, say from David Morin/Walter Fox Smith/Howard Georgi, or even as covered in the FEM modeling of idealized spring-mass systems. Do that, so that you are ready for the next post in this series—whenever it comes.

In the meanwhile, sure feel free to drop in a comment or email if you find that I am going wrong somewhere—especially in the maths of it or its implications. Thanks in advance.

Take care, and bye for now.


A song I like:

(Marathi) “aalee kuThoonashee kanee taaLa mrudungaachi dhoona”
Music and Singer: Vasant Ajgaonkar
Lyrics: Sopandev Chaudhari

 


History:
— First published: 2019.11.05 17:19 IST.
— Added the sub-section 10.5. and the songs section. Corrected LaTeX typos. the same day at 20:31 IST.
— Expanded the section 11. considerably, and also added sub-section titles to it. Revised also the sections 12. and 13. Overall, a further addition of approx. 1,500 words. Also corrected typos. Now, unless there is an acute need even for typo-corrections (i.e. if something goes blatantly in an opposite direction than the meaning I had in mind), I would leave this post in the shape in which it is. 2019.11.06 11:06 IST.

Ontologies in physics—7: To understand QM, you have to first solve yet another problem with the EM

In the last post, I mentioned the difficulty introduced by (i) the higher-dimensional nature of \Psi, and (ii) the definition of the electrostatic V on the separation vectors rather than position vectors.

Turns out that while writing the next post to address the issue, I spotted yet another issue. Its maths is straight-forward (the X–XII standard one). But its ontology is not at all so easy to figure out. So, let me mention it. (This upsets the entire planning I had in mind for QM, and needs small but extensive revisions all over the earlier published EM ontology as well. Anyway, read on.)


In QM, we do use a classical EM quantity, namely, the electrostatic potential energy field. It turns out that the understanding of EM which we have so painfully developed over some 4 very long posts, still may not be adequate enough.

To repeat, the overall maths remains the same. But it’s the physics—rather, the detailed ontological description—which has to be changed. And with it, some small mathematical details would change as well.

I will mention the problem here in this post, but not the solution I have in mind; else this post will become huuuuuge. (Explaining the problem itself is going to take a bit.)


1. Potential energy function for hydrogen atom:

Consider a hydrogen atom in an otherwise “empty” infinite space, as our quantum system.

The proton and the electron interact with each other. To spare me too much typing, let’s approximate the proton as being fixed in space, say at the origin, and let’s also assume a 1D atom.

The Coulomb force by the proton (particle 1) on the electron (particle 2) is given by \vec{f}_{12} = \dfrac{(e)(-e)}{r^2} \hat{r}_{12}. (I have rescaled the equations to make the constants “disappear” from the equation, though physically they are there, through the multiplication by 1 in appropriate units.) The potential energy of the electron due to the proton is given by: V_{12} = \dfrac{(e)(-e)}{r}. (There is no prefactor of 1/2 because the proton is fixed.)

The potential energy profile here is in the form of a well, vaguely looking like the letter `V’;  it goes infinitely deep at the origin (at the proton’s position), and its wings asymptotically approach zero at \pm \infty.

If you draw a graph, the electron will occupy a point-position at one and only one point on the r-axis at any instant; it won’t go all over the space. Remember, the graph is for V, which is expressed using the classical law of Coulomb’s.


2. QM requires the entire V function at every instant:

In QM, the measured position of the electron could be anywhere; it is given using Born’s rule on the wavefunction \Psi(r,t).

So, we have two notions of positions for the supposedly same existent: the electron.

One notion refers to the classical point-position. We use this notion even in QM calculations, else we could not get to the V function. In the classical view, the electronic position can be variable; it can go over the entire infinite domain; but it must refer to one and only one point at any given instant.

The measured position of the electron refers to the \Psi, which is a function of all infinite space. The Schrodinger evolution occurs at all points of space at any instant. So, the electron’s measured position could be found at any location—anywhere in the infinite space. Once measured, the position “comes down” to a single point. But before measurement, \Psi sure is (nonuniformly) spread all over the infinite domain.

Schrodinger’s solution for the hydrogen atom uses \Psi as the unknown variable, and V as a known variable. Given the form of this equation, you have no choice but to consider the entire graph of the potential energy (V) function into account at every instant in the Schrodinger evolution. Any eigenvalue problem of any operator requires the entire function of V; a single value at a time won’t do.

Just a point-value of V at the instantaneous position of the classical electron simply won’t do—you couldn’t solve Schrodinger’s equation then.

If we have to bring the \Psi from its Platonic “heaven” to our 1D space, we have to treat the entire graph of V as a physically existing (infinitely spread) field. Only then could we possibly say that \Psi too is a 1D field. (Even if you don’t have this motivation, read on, anyway. You are bound to find something interesting.)

Now an issue arises.


3. In Coulomb’s law, there is only one value for V at any instant:

The proton is fixed. So, the electron must be movable—else, despite being a point-particle, it is hard to think of a mechanism which can generate the whole V graph for its local potential energies.

But if the electron is movable, there is a certain trouble regarding what kind of a kinematics we might ascribe to the electron so that it generates the whole V field required by the Schrodinger equation. Remember, V is the potential energy of the electron, not of proton.

By classical EM, V at any instant must be a point-property, not a field. But Schrodinger’s equation requires a field for V.

So, the only imaginable solutions are weird: an infinitely fast electron running all over the domain but lawfully (i.e. following the laws at every definite point). Or something similarly weird.

So, the problem (how to explain how the V function, used in Schrodinger’s equation) still remains.


4. Textbook treatment of EM has fields, but no physics for multiplication by signs:

In the textbook treatment of EM (and I said EM, not QM), the proton does create its own force-field, which remains fixed in space (for a spatially fixed proton). The proton’s \vec{E} field is spread all over the infinite space, at any instant. So, why not exploit this fact? Why not try to get the electron’s V from the proton’s \vec{E}?

The potential field (in volt) of a proton is denoted as V in EM texts. So, to avoid confusion with the potential energy function (in joule) of the electron, let’s denote the proton’s potential (in volt) using the symbol P.

The potential field P does remain fixed and spread all over the space at any instant, as desired. But the trouble is this:

It is also positive everywhere. Its graph is not a well, it is a peak—infinitely tall peak at the proton’s position, asymptotically approaching zero at \pm \infty, and positive (above the zero-line) everywhere.

Therefore, you have to multiply this P field by the negative charge of electron e, so that P turns into the required V field of the electron.

But nature does no multiplications—not unless there is a definite physical mechanism to “convert” the quantities appropriately.

For multiplications with signed quantities, a mechanism like the mechanical lever could be handy. One small side goes down; the other big side goes  up but to a different extent; etc. Unfortunately, there is no place for a lever in the EM ontology—it’s all point charges and the “empty” space, which we now call the aether.

Now, if multiplication of constant magnitudes alone were to be a problem, we could have always taken care of it by suitably redefining P.

But the trouble caused by the differing sign still remains!

And that’s where the real trouble is. Let me show you how.

If a proton has to have its own P field, then its role has to stay the same regardless of the interactions that the proton enters into. Whether a given proton interacts with an electron (negatively charged), or with another proton (positively charged), the given proton’s own field still has to stay the same at all times, in any system—else it will not be its own field but one of interactions. It also has to remain positive by sign—even if P is rescaled to avoid multiplications.

But if V has to be negative when an electron interacts with it, and if V also has to be positive when another proton interacts with it, then a multiplication by the signs (by \pm 1 must occur. You just can’t avoid multiplications.

But there is no mechanism for the multiplications mandated by the sign conversions.

How do we resolve this issue?


5. The weakness of a proposed solution:

Here is one way out that we might think of.

We say that a proton’s P field stays just the same (positive, fixed) at all times. However, when the second particle is positively charged, then it moves away from the proton; when the second particle is negatively charged, then it moves towards the proton. Since V does require a dot product of a force with a displacement vector, and since the displacement vector does change signs in this procedure, the problem seems to have been solved.

So, the proposed solution becomes: the direction of the motion of the forced particle is not determined only by the field (which is always positive here), but also by the polarity of that particle itself. And, it’s a simple change, you might argue. There is some unknown physics to the very abstraction of the point charge itself, you could say, which propels it this way instead of that way, depending on its own sign.

Thus, charges of opposing polarities go in opposite directions while interacting with the same proton. That’s just how charges interact with fields. By definition. You could say that.

What could possibly be wrong with that view?

Well, the wrong thing is this:

If you imagine a classical point-particle of an electron as going towards the proton at a point, then a funny situation ensues while using it in QM.

The arrows depicting the force-field of the proton always point away from it—except for the one distinguished position, viz., that of the electron, where a single arrow would be found pointing towards the proton (following the above suggestion).

So, the action of the point-particle of the electron introduces an infinitely sharp discontinuity in the force-field of the proton, which then must also seep into its V field.

But a discontinuity like that is not basically compatible with Schrodinger’s equation. It will therefore lead to one of the following two consequences:

It might make the solution impossible or ill-defined. I don’t know enough about maths to tell if this could be true.  But what I can tell is this: Even if a solution is possible (including solutions that possibly may be asymptotic, or are approximate but good enough) then the presence of the discontinuity will sure have an impact on the nature of the solution. The calculated \Psi wouldn’t be the same as that for a V without the discontinuity. That’s inevitable.

But why can’t we ignore the classical point-position of the electron? Well, the answer is that in a more general theory which keeps both particles movable, then we have to calculate the proton’s potential energy too. To do that, we have to take the electric potential (in volts) P of the electron, and multiply it by the charge of the proton. The trouble is: The electric potential field of the electron has singularity at its classical position. So, classical positions cannot be dropped out of the calculations. The classical position of a given particle is necessary for calculating the V field of the other particle, and, vice-versa.

In short, to ensure consistency in the two ways of the interaction, we must regard the singularities as still being present where they are.

And with that consideration, essentially, we have once again come back to a repercussion of the idea that the classical electron has a point position, but its potential energy field in the electrostatic interaction with the proton is spread everywhere.

To fulfill our desire of having a 3D field for \Psi, we have to have a certain kind of a field for V. But V should not change its value in just one isolated place, just in order to allow multiplication by -1, because doing so introduces a very bad discontinuity. It should remain the same smooth V that we have always seen in the textbooks on QM.


6. The problem statement, in a nutshell:

So, here is the problem statement:

To find a physically realizable way such that: even if we use the classical EM properties of the electron while calculating V, and even if the electron is classically a point-particle, its V function (in joules) should still turn out to be negative everywhere—even if the proton has its own potential field (P, in volts) that is positive everywhere in the classical EM.

In short, we have to change the way we look at the physics of the EM fields, and then also make the required changes to any maths, as necessary. Without disturbing the routine calculations either in EM or in QM.

Can it be done? Well, I think the answer is “yes.”


7. A personal note:

While I’ve been having some vague sense of there being some issue “to be looked into later on” for quite some time (months, at least), it was only over the last week, especially over the last couple of days (since the publication of the last post), that this problem became really acute. I always used to skip over this ontology/physics issue and go directly over to using the EM maths involved in the QM. I used to think that the ontology of such EM as it is used in the QM, would be pretty easy to explain—at least as compared to the ontology of QM. Looks like despite spending thousands of words (some 4–5 posts with a total of may be 15–20 K words) there still wasn’t enough of a clarity—about EM.

Not if we adopt the principle, which I discovered on the fly, right while in the middle of writing this series, that nature does no multiplications without there being a physical mechanism for it.

Fortunately, the problem did become clear. Clear enough that, I think, I also found a satisfactory enough solution to it too. Right today (on 2019.10.15 evening IST).

Would you like to give it a try? (I anyway need a break. So, take about a week’s time or so, if you wish.)


Bye for now, take care, and see you the next time.


A song I like:

(Hindi) “jaani o jaani”
Singer: Kishore Kumar
Music: Laxmikant-Pyarelal
Lyrics: Anand Bakshi


History:

— Originally published: 2019.10.16 01:21 IST
— One or two typos corrected, section names added, and a few explanatory sentences added inline: 2019.10.17 22:09 IST. Let’s leave this post right in this form.

 

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.