This blog post continues from my last post, and has become overdue by now. I had promised to give my answers to the questions raised last time. Without attempting to explain too much, let me jot down the answers.
1. The rule of omitting the self-field:
This rule arises in electrostatic interactions basically because the Coulombic field has a spherical symmetry. The same rule would also work out in any field that has a spherical symmetry—not just the inverse-separation fields, and not necessarily only the singular potentials, though Coulombic potentials do show both these latter properties too.
It is helpful here to think in terms of not potentials but of forces.
Draw any arbitrary curve. Then, hold one end of the curve fixed at the origin, and sweep the curve through all possible angles around it, to get a 3D field. This 3D field has a spherical symmetry, too. Hence, gradients at the same radial distance on opposite sides of the origin are always equal and opposite.
Now you know that the negative gradient of potential gives you a force. Since for any spherical potential the gradients are equal and opposite, they cancel out. So, the forces cancel out to.
Realize here that in calculating the force exerted by a potential field on a point-particle (say an electron), the force cannot be calculated in reference to just one point. The very definition of the gradient refers to two different points in space, even if they be only infinitesimally separated apart. So, the proper procedure is to start with a small sphere centered around the given electron, calculate the gradients of the potential field at all points on the surface of this sphere, calculate the sum of the forces exerted on the domain contained inside the spherical surface by these forces, and then take the sphere to the limiting of vanishing size. The sum of the forces thus exerted is the net force acting on that point-particle.
In case of the Coulombic potentials, the forces thus calculated on the surface of any sphere (centered on that particle) turn out to be zero. This fact holds true for spheres of all radii. It is true that gradients (and forces) progressively increase as the size of the sphere decreases—in fact they increase without all bounds for singular potentials. However, the aforementioned cancellation holds true at any stage in the limiting process. Hence, it holds true for the entirety of the self-field.
In calculating motions of a given electron, what matters is not whether its self-field exists or not, but whether it exerts a net force on the same electron or not. The self-field does exist (at least in the sense explained later below) and in that sense, yes, it does keep exerting forces at all times, also on the same electron. However, due to the spherical symmetry, the net force that the field exerts on the same electron turns out to be zero.
Even if you were to include the self-field in the calculations, if the field is spherically symmetric, then the final net force experienced by the same electron would still have no part coming from its own self-field. Hence, to economize calculations without sacrificing exactitude in any way, we discard it out of considerations.The rule of omitting the self-field is just a matter of economizing calculations; it is not a fundamental law characterizing what field may be objectively said to exist. If the potential field due to other charges exists, then, in the same sense, the self-field too exists. It’s just that for the motions of the self field-generating electron, it is as good as non-existent.
However, the question of whether a potential field physically exists or not, turns out to be more subtle than what might be thought.
2. Conditions for the objective existence of electrostatic potentials:
It once again helps to think of forces first, and only then of potentials.
Consider two electrons in an otherwise empty spatial region of an isolated system. Suppose the first electron (), is at a position , and a second electron is at a position . What Coulomb’s law now says is that the two electrons mutually exert equal and opposite forces on each other. The magnitudes of these forces are proportional to the inverse-square of the distance which separates the two. For the like charges, the forces is repulsive, and for unlike charges, it is attractive. The amount of the electrostatic forces thus exerted do not depend on mass; they depend only the amounts of the respective charges.
The potential energy of the system for this particular configuration is given by (i) arbitrarily assigning a zero potential to infinite separation between the two charges, and (ii) imagining as if both the charges have been brought from infinity to their respective current positions.
It is important to realize that the potential energy for a particular configuration of two electrons does not form a field. It is merely a single number.
However, it is possible to imagine that one of the charges (say ) is held fixed at a point, say at , and the other charge is successively taken, in any order, at every other point in the infinite domain. A single number is thus generated for each pair of . Thus, we can obtain a mapping from the set of positions for the two charges, to a set of the potential energy numbers. This second set can be regarded as forming a field—in the space.
However, notice that thus defined, the potential energy field is only a device of calculations. It necessarily refers to a second charge—the one which is imagined to be at one point in the domain at a time, with the procedure covering the entire domain. The energy field cannot be regarded as a property of the first charge alone.
Now, if the potential energy field thus obtained is normalized by dividing it with the electric charge of the second charge, then we get the potential energy for a unit test-charge. Another name for the potential energy obtained when a unit test-charge is used for the second charge is: the electrostatic potential (denoted as ).
But still, in classical mechanics, the potential field also is only a device of calculations; it does not exist as a property of the first charge, because the potential energy itself does not exist as a property of that fixed charge alone. What does exist is the physical effect that there are those potential energy numbers for those specific configurations of the fixed charge and the test charge.
This is the reason why the potential energy field, and therefore the electrostatic potential of a single charge in an otherwise empty space does not exist. Mathematically, it is regarded as zero (though it could have been assigned any other arbitrary, constant value.)
Potentials arise only out of interaction of two charges. In classical mechanics, the charges are point-particles. Point-particles exist only at definite locations and nowhere else. Therefore, their interaction also must be seen as happening only at the locations where they do exist, and nowhere else.
If that is so, then in what sense can we at all say that potential energy (or electrostaic potential) field does physically exist?
Consider a single electron in an isolated system, again. Assume that its position remains fixed.
Suppose there were something else in the isolated system—-something—some object—every part of which undergoes an electrostatic interaction with the fixed (first) electron. If this second object were to be spread all over the domain, and if every part of it were able to interact with the fixed charge, then we could say that the potential energy field exists objectively—as an attribute of this second object. Ditto, for the electric potential field.
Note three crucially important points, now.
2.1. The second object is not the usual classical object.
You cannot regard the second (spread-out) object as a mere classical charge distribution. The reason is this.
If the second object were to be actually a classical object, then any given part of it would have to electrostatically interact with every other part of itself too. You couldn’t possibly say that a volume element in this second object interacts only with the “external” electron. But if the second object were also to be self-interacting, then what would come to exist would not be the simple inverse-distance potential field energy, in reference to that single “external” electron. The space would be filled with a very weird field. Admitting motion to the property of the local charge in the second object, every locally present charge would soon redistribute itself back “to” infinity (if it is negative), or it all would collapse into the origin (if the charge on the second object were to be positive, because the fixed electron’s field is singular). But if we allow no charge redistributions, and the second field were to be classical (i.e. capable of self-interacting), then the field of the second object would have to have singularities everywhere. Very weird. That’s why:
If you want to regard the potential field as objectively existing, you have to also posit (i.e. postulate) that the second object itself is not classical in nature.
Classical electrostatics, if it has to regard a potential field as objectively (i.e. physically) existing, must therefore come to postulate a non-classical background object!
2.2. Assuming you do posit such a (non-classical) second object (one which becomes “just” a background object), then what happens when you introduce a second electron into the system?
You would run into another seeming contradiction. You would find that this second electron has no job left to do, as far as interacting with the first (fixed) electron is concerned.
If the potential field exists objectively, then the second electron would have to just passively register the pre-existing potential in its vicinity (because it is the second object which is doing all the electrostatic interactions—all the mutual forcings—with the first electron). So, the second electron would do nothing of consequence with respect to the first electron. It would just become a receptacle for registering the force being exchanged by the background object in its local neighborhood.
But the seeming contradiction here is that as far as the first electron is concerned, it does feel the potential set up by the second electron! It may be seen to do so once again via the mediation of the background object.
Therefore, both electrons have to be simultaneously regarded as being active and passive with respect to each other. They are active as agents that establish their own potential fields, together with an interaction with the background object. But they also become passive in the sense that they are mere point-masses that only feel the potential field in the background object and experience forces (accelerations) accordingly.
The paradox is thus resolved by having each electron set up a field as a result of an interaction with the background object—but have no interaction with the other electron at all.
2.3. Note carefully what agency is assigned to what object.
The potential field has a singularity at the position of that charge which produces it. But the potential field itself is created either by the second charge (by imagining it to be present at various places), or by a non-classical background object (which, in a way, is nothing but an objectification of the potential field-calculation procedure).
Thus, there arises a duality of a kind—a double-agent nature, so to speak. The potential energy is calculated for the second charge (the one that is passive), in the sense that the potential energy is relevant for calculating the motion of the second charge. That’s because the self-field cancels out for all motions of the first charge. However,
The potential energy is calculated for the second charge. But the field so calculated has been set up by the first (fixed) charge. Charges do not interact with each other; they interact only with the background object.
2.4. If the charges do not interact with each other, and if they interact only with the background object, then it is worth considering this question:
Can’t the charges be seen as mere conditions—points of singularities—in the background object?
Indeed, this seems to be the most reasonable approach to take. In other words,
All effects due to point charges can be regarded as field conditions within the background object. Thus, paradoxically enough, a non-classical distributed field comes to represent the classical, massive and charged point-particles themselves. (The mass becomes just a parameter of the interactions of singularities within a field.) The charges (like electrons) do not exist as classical massive particles, not even in the classical electrostatics.
3. A partly analogous situation: The stress-strain fields:
If the above situation seems too paradoxical, it might be helpful to think of the stress-strain fields in solids.
Consider a horizontally lying thin plate of steel with two rigid rods welded to it at two different points. Suppose horizontal forces of mutually opposite directions are applied through the rods (either compressive or tensile). As you know, as a consequence, stress-strain fields get set up in the plate.
From an external viewpoint, the two rods are regarded as interacting with each other (exchanging forces with each other) via the medium of the plate. However, in reality, they are interacting only with the object that is the plate. The direct interaction, thus, is only between a rod and the plate. A rod is forced, it interacts with the plate, the plate sets up stress-strain field everywhere, the local stress-field near the second rod interacts with it, and the second rod registers a force—which balances out the force applied at its end. Conversely, the force applied at the second rod also can be seen as getting transmitted to the first rod via the stress-strain field in the plate material.
There is no contradiction in this description, because we attribute the stress-strain field to the plate itself, and always treat this stress-strain field as if it came into existence due to both the rods acting simultaneously.
In particular, we do not try to isolate a single-rod attribute out of the stress-strain field, the way we try to ascribe a potential to the first charge alone.
Come to think of it, if we have only one rod and if we apply force to it, no stress-strain field would result (i.e. neglecting inertia effects of the steel plate). Instead, the plate would simply move in the rigid body mode. Now, in solid mechanics, we never try to visualize a stress-strain field associated with a single rod alone.
It is a fallacy of our thinking that when it comes to electrostatics, we try to ascribe the potential to the first charge, and altogether neglect the abstract procedure of placing the test charge at various locations, or the postulate of positing a non-classical background object which carries that potential.
In the interest of completeness, it must be noted that the stress-strain fields are tensor fields (they are based on the gradients of vector fields), whereas the electrostatic force-field is a vector field (it is based on the gradient of the scalar potential field). A more relevant analogy for the electrostatic field, therefore might the forces exchanged by two point-vortices existing in an ideal fluid.
4. But why bother with it all?
The reason I went into all this discussion is because all these issues become important in the context of quantum mechanics. Even in quantum mechanics, when you have two charges that are interacting with each other, you do run into these same issues, because the Schrodinger equation does have a potential energy term in it. Consider the following situation.
If an electrostatic potential is regarded as being set up by a single charge (as is done by the proton in the nucleus of the hydrogen atom), but if it is also to be regarded as an actually existing and spread out entity (as a field, the way Schrodinger’s equation assumes it to be), then a question arises: What is the role of the second charge (e.g., that of the electron in an hydrogen atom)? What happens when the second charge (the electron) is represented quantum mechanically? In particular:
What happens to the potential field if it represents the potential energy of the second charge, but the second charge itself is now being represented only via the complex-valued wavefunction?
And worse: What happens when there are two electrons, and both interacting with each other via electrostatic repulsions, and both are required to be represented quantum mechanically—as in the case of the electrons in an helium atom?
Can a charge be regarded as having a potential field as well as a wavefunction field? If so, what happens to the point-specific repulsions as are mandated by the Coulomb law? How precisely is the term to be interpreted?
I was thinking about these things when these issues occurred to me: the issue of the self-field, and the question of the physical vs. merely mathematical existence of the potential fields of two or more quantum-mechanically interacting charges.
Guess I am inching towards my full answers. Guess I have reached my answers, but I need to have them verified with some physicists.
5. The help I want:
As a part of my answer-finding exercises (to be finished by this month-end), I might be contacting a second set of physicists soon enough. The issue I want to learn from them is the following:
How exactly do they do computational modeling of the helium atom using the finite difference method (FDM), within the context of the standard (mainstream) quantum mechanics?
That is the question. Once I understand this part, I would be done with the development of my new approach to understanding QM.
I do have some ideas regarding the highlighted question. It’s just that I want to have these ideas confirmed from some physicists before (or along-side) implementing the FDM code. So, I might be approaching someone—possibly you!
Please note my question once again. I don’t want to do perturbation theory. I would also like to avoid the variational method.
Yes, I am very comfortable with the finite element method, which is basically based on the variational calculus. So, given a good (detailed enough) account of the variational method for the He atom, it should be possible to translate it into the FEM terms.
However, ideally, what I would like to do is to implement it as an FDM code.
Please suggest good references and / or people working on this topic, if you know any. Thanks in advance.
A song I like:
[… Here I thought that there was no song that Salil Chowdhury had composed and I had not listened to. (Well, at least when it comes to his Hindi songs). That’s what I had come to believe, and here trots along this one—and that too, as a part of a collection by someone! … The time-delay between my first listening to this song, and my liking it, was zero. (Or, it was a negative time-delay, if you refer to the instant that the first listening got over). … Also, one of those rare occasions when one is able to say that any linear ordering of the credits could only be random.]
(Hindi) “mada bhari yeh hawaayen”
Music: Salil Chowdhury
Singer: Lata Mangeshkar