The geometry of the wavefunction, electron spin and the form factor

Our previous posts showed how a simple geometric interpretation of the elementary wavefunction yielded the (Compton scattering) radius of an elementary particle—for an electron, at least: for the proton, we only got the order of magnitude right—but then a proton is not an elementary particle. We got lots of other interesting equations as well… But… Well… When everything is said and done, it’s that equivalence between the E = m·a2·ω2 and E = m·c2 relations that we… Well… We need to be more specific about it.

Indeed, I’ve been ambiguous here and there—oscillating between various interpretations, so to speak. 🙂 In my own mind, I refer to my unanswered questions, or my ambiguous answers to them, as the form factor problem. So… Well… That explains the title of my post. But so… Well… I do want to be somewhat more conclusive in this post. So let’s go and see where we end up. 🙂

To help focus our mind, let us recall the metaphor of the V-2 perpetuum mobile, as illustrated below. With permanently closed valves, the air inside the cylinder compresses and decompresses as the pistons move up and down. It provides, therefore, a restoring force. As such, it will store potential energy, just like a spring, and the motion of the pistons will also reflect that of a mass on a spring: it is described by a sinusoidal function, with the zero point at the center of each cylinder. We can, therefore, think of the moving pistons as harmonic oscillators, just like mechanical springs. Of course, instead of two cylinders with pistons, one may also think of connecting two springs with a crankshaft, but then that’s not fancy enough for me. 🙂

V-2 engine

At first sight, the analogy between our flywheel model of an electron and the V-twin engine seems to be complete: the 90 degree angle of our V-2 engine makes it possible to perfectly balance the pistons and we may, therefore, think of the flywheel as a (symmetric) rotating mass, whose angular momentum is given by the product of the angular frequency and the moment of inertia: L = ω·I. Of course, the moment of inertia (aka the angular mass) will depend on the form (or shape) of our flywheel:

  1. I = m·a2 for a rotating point mass m or, what amounts to the same, for a circular hoop of mass m and radius a.
  2. For a rotating (uniformly solid) disk, we must add a 1/2 factor: I = m·a2/2.

How can we relate those formulas to the E = m·a2·ω2 formula? The kinetic energy that is being stored in a flywheel is equal Ekinetic = I·ω2/2, so that is only half of the E = m·a2·ω2 product if we substitute I for I = m·a2. [For a disk, we get a factor 1/4, so that’s even worse!] However, our flywheel model of an electron incorporates potential energy too. In fact, the E = m·a2·ω2 formula just adds the (kinetic and potential) energy of two oscillators: we do not really consider the energy in the flywheel itself because… Well… The essence of our flywheel model of an electron is not the flywheel: the flywheel just transfers energy from one oscillator to the other, but so… Well… We don’t include it in our energy calculations. The essence of our model is that two-dimensional oscillation which drives the electron, and which is reflected in Einstein’s E = m·c2 formula. That two-dimensional oscillation—the a2·ω2 = c2 equation, really—tells us that the resonant (or natural) frequency of the fabric of spacetime is given by the speed of light—but measured in units of a. [If you don’t quite get this, re-write the a2·ω2 = c2 equation as ω = c/a: the radius of our electron appears as a natural distance unit here.]

Now, we were extremely happy with this interpretation not only because of the key results mentioned above, but also because it has lots of other nice consequences. Think of our probabilities as being proportional to energy densities, for example—and all of the other stuff I describe in my published paper on this. But there is even more on the horizon: a follower of this blog (a reader with an actual PhD in physics, for a change) sent me an article analyzing elementary particles as tiny black holes because… Well… If our electron is effectively spinning around, then its tangential velocity is equal to ω = c. Now, recent research suggest black holes are also spinning at (nearly) the speed of light. Interesting, right? However, in order to understand what she’s trying to tell me, I’ll first need to get a better grasp of general relativity, so I can relate what I’ve been writing here and in previous posts to the Schwarzschild radius and other stuff.

Let me get back to the lesson here. In the reference frame of our particle, the wavefunction really looks like the animation below: it has two components, and the amplitude of the two-dimensional oscillation is equal to a, which we calculated as = ħ·/(m·c) = 3.8616×10−13 m, so that’s the (reduced) Compton scattering radius of an electron.

Circle_cos_sin

In my original article on this, I used a more complicated argument involving the angular momentum formula, but I now prefer a more straightforward calculation:

c = a·ω = a·E/ħ = a·m·c2/ħ  ⇔ = ħ/(m·c)

The question is: what is that rotating arrow? I’ve been vague and not so vague on this. The thing is: I can’t prove anything in this regard. But my hypothesis is that it is, in effect, a rotating field vector, so it’s just like the electric field vector of a (circularly polarized) electromagnetic wave (illustrated below).

There are a number of crucial differences though:

  1. The (physical) dimension of the field vector of the matter-wave is different: I associate the real and imaginary component of the wavefunction with a force per unit mass (as opposed to the force per unit charge dimension of the electric field vector). Of course, the newton/kg dimension reduces to the dimension of acceleration (m/s2), so that’s the dimension of a gravitational field.
  2. I do believe this gravitational disturbance, so to speak, does cause an electron to move about some center, and I believe it does so at the speed of light. In contrast, electromagnetic waves do not involve any mass: they’re just an oscillating field. Nothing more. Nothing less. In contrast, as Feynman puts it: “When you do find the electron some place, the entire charge is there.” (Feynman’s Lectures, III-21-4)
  3. The third difference is one that I thought of only recently: the plane of the oscillation cannot be perpendicular to the direction of motion of our electron, because then we can’t explain the direction of its magnetic moment, which is either up or down when traveling through a Stern-Gerlach apparatus.

I mentioned that in my previous post but, for your convenience, I’ll repeat what I wrote there. The basic idea here is illustrated below (credit for this illustration goes to another blogger on physics). As for the Stern-Gerlach experiment itself, let me refer you to a YouTube video from the Quantum Made Simple site.

Figure 1 BohrThe point is: the direction of the angular momentum (and the magnetic moment) of an electron—or, to be precise, its component as measured in the direction of the (inhomogeneous) magnetic field through which our electron is traveling—cannot be parallel to the direction of motion. On the contrary, it is perpendicular to the direction of motion. In other words, if we imagine our electron as spinning around some center, then the disk it circumscribes will comprise the direction of motion.

However, we need to add an interesting detail here. As you know, we don’t really have a precise direction of angular momentum in quantum physics. [If you don’t know this… Well… Just look at one of my many posts on spin and angular momentum in quantum physics.] Now, we’ve explored a number of hypotheses but, when everything is said and done, a rather classical explanation turns out to be the best: an object with an angular momentum J and a magnetic moment μ (I used bold-face because these are vector quantities) that is parallel to some magnetic field B, will not line up, as you’d expect a tiny magnet to do in a magnetic field—or not completely, at least: it will precess. I explained that in another post on quantum-mechanical spin, which I advise you to re-read if you want to appreciate the point that I am trying to make here. That post integrates some interesting formulas, and so one of the things on my ‘to do’ list is to prove that these formulas are, effectively, compatible with the electron model we’ve presented in this and previous posts.

Indeed, when one advances a hypothesis like this, it’s not enough to just sort of show that the general geometry of the situation makes sense: we also need to show the numbers come out alright. So… Well… Whatever we think our electron—or its wavefunction—might be, it needs to be compatible with stuff like the observed precession frequency of an electron in a magnetic field.

Our model also needs to be compatible with the transformation formulas for amplitudes. I’ve been talking about this for quite a while now, and so it’s about time I get going on that.

Last but not least, those articles that relate matter-particles to (quantum) gravity—such as the one I mentioned above—are intriguing too and, hence, whatever hypotheses I advance here, I’d better check them against those more advanced theories too, right? 🙂 Unfortunately, that’s going to take me a few more years of studying… But… Well… I still have many years ahead—I hope. 🙂

Post scriptum: It’s funny how one’s brain keeps working when sleeping. When I woke up this morning, I thought: “But it is that flywheel that matters, right? That’s the energy storage mechanism and also explains how photons possibly interact with electrons. The oscillators drive the flywheel but, without the flywheel, nothing is happening. It is really the transfer of energy—through the flywheel—which explains why our flywheel goes round and round.”

It may or may not be useful to remind ourselves of the math in this regard. The motion of our first oscillator is given by the cos(ω·t) = cosθ function (θ = ω·t), and its kinetic energy will be equal to sin2θ. Hence, the (instantaneous) change in kinetic energy at any point in time (as a function of the angle θ) is equal to: d(sin2θ)/dθ = 2∙sinθ∙d(sinθ)/dθ = 2∙sinθ∙cosθ. Now, the motion of the second oscillator (just look at that second piston going up and down in the V-2 engine) is given by the sinθ function, which is equal to cos(θ − π /2). Hence, its kinetic energy is equal to sin2(θ − π /2), and how it changes (as a function of θ again) is equal to 2∙sin(θ − π /2)∙cos(θ − π /2) = = −2∙cosθ∙sinθ = −2∙sinθ∙cosθ. So here we have our energy transfer: the flywheel organizes the borrowing and returning of energy, so to speak. That’s the crux of the matter.

So… Well… What if the relevant energy formula is E = m·a2·ω2/2 instead of E = m·a2·ω2? What are the implications? Well… We get a √2 factor in our formula for the radius a, as shown below.

square 2

Now that is not so nice. For the tangential velocity, we get a·ω = √2·c. This is also not so nice. How can we save our model? I am not sure, but here I am thinking of the mentioned precession—the wobbling of our flywheel in a magnetic field. Remember we may think of Jz—the angular momentum or, to be precise, its component in the z-direction (the direction in which we measure it—as the projection of the real angular momentum J. Let me insert Feynman’s illustration here again (Feynman’s Lectures, II-34-3), so you get what I am talking about.

precession

Now, all depends on the angle (θ) between Jz and J, of course. We did a rather obscure post on these angles, but the formulas there come in handy now. Just click the link and review it if and when you’d want to understand the following formulas for the magnitude of the presumed actual momentum:magnitude formulaIn this particular case (spin-1/2 particles), j is equal to 1/2 (in units of ħ, of course). Hence, is equal to √0.75 ≈ 0.866. Elementary geometry then tells us cos(θ) = (1/2)/√(3/4) =  = 1/√3. Hence, θ ≈ 54.73561°. That’s a big angle—larger than the 45° angle we had secretly expected because… Well… The 45° angle has that √2 factor in it: cos(45°) = sin(45°) = 1/√2.

Hmm… As you can see, there is no easy fix here. Those damn 1/2 factors! They pop up everywhere, don’t they? 🙂 We’ll solve the puzzle. One day… But not today, I am afraid. I’ll call it the form factor problem… Because… Well… It sounds better than the 1/2 or √2 problem, right? 🙂

Note: If you’re into quantum math, you’ll note ħ/(m·c) is the reduced Compton scattering radius. The standard Compton scattering radius is equal to  = (2π·ħ)/(m·c) =  h/(m·c) = h/(m·c). It doesn’t solve the √2 problem. Sorry. The form factor problem. 🙂

To be honest, I finished my published paper on all of this with a suggestion that, perhaps, we should think of two circular oscillations, as opposed to linear ones. Think of a tiny ball, whose center of mass stays where it is, as depicted below. Any rotation – around any axis – will be some combination of a rotation around the two other axes. Hence, we may want to think of our two-dimensional oscillation as an oscillation of a polar and azimuthal angle. It’s just a thought but… Well… I am sure it’s going to keep me busy for a while. 🙂polar_coordsThey are oscillations, still, so I am not thinking of two flywheels that keep going around in the same direction. No. More like a wobbling object on a spring. Something like the movement of a bobblehead on a spring perhaps. 🙂bobblehead

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The Essence of Reality

I know it’s a crazy title. It has no place in a physics blog, but then I am sure this article will go elsewhere. […] Well… […] Let me be honest: it’s probably gonna go nowhere. Whatever. I don’t care too much. My life is happier than Wittgenstein’s. 🙂

My original title for this post was: discrete spacetime. That was somewhat less offensive but, while being less offensive, it suffered from the same drawback: the terminology was ambiguous. The commonly accepted term for discrete spacetime is the quantum vacuum. However, because I am just an arrogant bastard trying to establish myself in this field, I am telling you that term is meaningless. Indeed, wouldn’t you agree that, if the quantum vacuum is a vacuum, then it’s empty. So it’s nothing. Hence, it cannot have any properties and, therefore, it cannot be discrete – or continuous, or whatever. We need to put stuff in it to make it real.

Therefore, I’d rather distinguish mathematical versus physical space. Of course, you are smart, and so you now you’ll say that my terminology is as bad as that of the quantum vacuumists. And you are right. However, this is a story that am writing, and so I will write it the way want to write it. 🙂 So where were we? Spacetime! Discrete spacetime.

Yes. Thank you! Because relativity tells us we should think in terms of four-vectors, we should not talk about space but about spacetime. Hence, we should distinguish mathematical spacetime from physical spacetime. So what’s the definitional difference?

Mathematical spacetime is just what it is: a coordinate space – Cartesian, polar, or whatever – which we define by choosing a representation, or a base. And all the other elements of the set are just some algebraic combination of the base set. Mathematical space involves numbers. They don’t – let me emphasize that: they do not!– involve the physical dimensions of the variables. Always remember: math shows us the relations, but it doesn’t show us the stuff itself. Think of it: even if we may refer to the coordinate axes as time, or distance, we do not really think of them as something physical. In math, the physical dimension is just a label. Nothing more. Nothing less.

In contrast, physical spacetime is filled with something – with waves, or with particles – so it’s spacetime filled with energy and/or matter. In fact, we should analyze matter and energy as essentially the same thing, and please do carefully re-read what I wrote: I said they are essentially the same. I did not say they are the same. Energy and mass are equivalent, but not quite the same. I’ll tell you what that means in a moment.

These waves, or particles, come with mass, energy and momentum. There is an equivalence between mass and energy, but they are not the same. There is a twist – literally (only after reading the next paragraphs, you’ll realize how literally): even when choosing our time and distance units such that is numerically equal to 1 – e.g. when measuring distance in light-seconds (or time in light-meters), or when using Planck units – the physical dimension of the cfactor in Einstein’s E = mcequation doesn’t vanish: the physical dimension of energy is kg·m2/s2.

Using Newton’s force law (1 N = 1 kg·m/s2), we can easily see this rather strange unit is effectively equivalent to the energy unit, i.e. the joule (1 J = 1 kg·m2/s2 = 1 (N·s2/m)·m2/s= 1 N·m), but that’s not the point. The (m/s)2 factor – i.e. the square of the velocity dimension – reflects the following:

  1. Energy is nothing but mass in motion. To be precise, it’s oscillating mass. [And, yes, that’s what string theory is all about, but I didn’t want to mention that. It’s just terminology once again: I prefer to say ‘oscillating’ rather than ‘vibrating’. :-)]
  2. The rapidly oscillating real and imaginary component of the matter-wave (or wavefunction, we should say) each capture half of the total energy of the object E = mc2.
  3. The oscillation is an oscillation of the mass of the particle (or wave) that we’re looking at.

In the mentioned publication, I explore the structural similarity between:

  1. The oscillating electric and magnetic field vectors (E and B) that represent the electromagnetic wave, and
  2. The oscillating real and imaginary part of the matter-wave.

The story is simple or complicated, depending on what you know already, but it can be told in an abnoxiously easy way. Note that the associated force laws do not differ in their structure:

Coulomb Law

gravitation law

The only difference is the dimension of m versus q: mass – the measure of inertia -versus charge. Mass comes in one color only, so to speak: it’s always positive. In contrast, electric charge comes in two colors: positive and negative. You can guess what comes next, but I won’t talk about that here.:-) Just note the absolute distance between two charges (with the same or the opposite sign) is twice the distance between 0 and 1, which must explains the rather mysterious 2 factor I get for the Schrödinger equation for the electromagnetic wave (but I still need to show how that works out exactly).

The point is: remembering that the physical dimension of the electric field is N/C (newton per coulomb, i.e. force per unit of charge) it should not come as a surprise that we find that the physical dimension of the components of the matter-wave is N/kg: newton per kg, i.e. force per unit of mass. For the detail, I’ll refer you to that article of mine (and, because I know you will not want to work your way through it, let me tell you it’s the last chapter that tells you how to do the trick).

So where were we? Strange. I actually just wanted to talk about discrete spacetime here, but I realize I’ve already dealt with all of the metaphysical questions you could possible have, except the (existential) Who Am I? question, which I cannot answer on your behalf. 🙂

I wanted to talk about physical spacetime, so that’s sanitized mathematical space plus something. A date without logistics. Our mind is a lazy host, indeed.

Reality is the guest that brings all of the wine and the food to the party.

In fact, it’s a guest that brings everything to the party: you – the observer – just need to set the time and the place. In fact, in light of what Kant – and many other eminent philosophers – wrote about space and time being constructs of the mind, that’s another statement which you should interpret literally. So physical spacetime is spacetime filled with something – like a wave, or a field. So how does that look like? Well… Frankly, I don’t know! But let me share my idea of it.

Because of the unity of Planck’s quantum of action (ħ ≈ 1.0545718×10−34 N·m·s), a wave traveling in spacetime might be represented as a set of discrete spacetime points and the associated amplitudes, as illustrated below. [I just made an easy Excel graph. Nothing fancy.]

spacetime

The space in-between the discrete spacetime points, which are separated by the Planck time and distance units, is not real. It is plain nothingness, or – if you prefer that term – the space in-between in is mathematical space only: a figment of the mind – nothing real, because quantum theory tells us that the real, physical, space is discontinuous.

Why is that so? Well… Smaller time and distance units cannot exist, because we would not be able to pack Planck’s quantum of action in them: a box of the Planck scale, with ħ in it, is just a black hole and, hence, nothing could go from here to there, because all would be trapped. Of course, now you’ll wonder what it means to ‘pack‘ Planck’s quantum of action in a Planck-scale spacetime box. Let me try  to explain this. It’s going to be a rather rudimentary explanation and, hence, it may not satisfy you. But then the alternative is to learn more about black holes and the Schwarzschild radius, which I warmly recommend for two equivalent reasons:

  1. The matter is actually quite deep, and I’d recommend you try to fully understand it by reading some decent physics course.
  2. You’d stop reading this nonsense.

If, despite my warning, you would continue to read what I write, you may want to note that we could also use the logic below to define Planck’s quantum of action, rather than using it to define the Planck time and distance unit. Everything is related to everything in physics. But let me now give the rather naive explanation itself:

  • Planck’s quantum of action (ħ ≈ 1.0545718×10−34 N·m·s) is the smallest thing possible. It may express itself as some momentum (whose physical dimension is N·s) over some distance (Δs), or as some amount of energy (whose dimension is N·m) over some time (Δt).
  • Now, energy is an oscillation of mass (I will repeat that a couple of times, and show you the detail of what that means in the last chapter) and, hence, ħ must necessarily express itself both as momentum as well as energy over some time and some distance. Hence, it is what it is: some force over some distance over some time. This reflects the physical dimension of ħ, which is the product of force, distance and time. So let’s assume some force ΔF, some distance Δs, and some time Δt, so we can write ħ as ħ = ΔF·Δs·Δt.
  • Now let’s pack that into a traveling particle – like a photon, for example – which, as you know (and as I will show in this publication) is, effectively, just some oscillation of mass, or an energy flow. Now let’s think about one cycle of that oscillation. How small can we make it? In spacetime, I mean.
  • If we decrease Δs and/or Δt, then ΔF must increase, so as to ensure the integrity (or unity) of ħ as the fundamental quantum of action. Note that the increase in the momentum (ΔF·Δt) and the energy (ΔF·Δs) is proportional to the decrease in Δt and Δs. Now, in our search for the Planck-size spacetime box, we will obviously want to decrease Δs and Δt simultaneously.
  • Because nothing can exceed the speed of light, we may want to use equivalent time and distance units, so the numerical value of the speed of light is equal to 1 and all velocities become relative velocities. If we now assume our particle is traveling at the speed of light – so it must be a photon, or a (theoretical) matter-particle with zero rest mass (which is something different than a photon) – then our Δs and Δt should respect the following condition: Δs/Δt = c = 1.
  • Now, when Δs = 1.6162×10−35 m and Δt = 5.391×10−44 s, we find that Δs/Δt = c, but ΔF = ħ/(Δs·Δt) = (1.0545718×10−34 N·m·s)/[(1.6162×10−35 m)·(5.391×10−44 s)] ≈ 1.21×1044 N. That force is monstrously huge. Think of it: because of gravitation, a mass of 1 kg in our hand, here on Earth, will exert a force of 9.8 N. Now note the exponent in that 1.21×1044 number.
  • If we multiply that monstrous force with Δs – which is extremely tiny – we get the Planck energy: (1.6162×10−35 m)·(1.21×1044 N) ≈ 1.956×109 joule. Despite the tininess of Δs, we still get a fairly big value for the Planck energy. Just to give you an idea, it’s the energy that you’d get out of burning 60 liters of gasoline—or the mileage you’d get out of 16 gallons of fuel! In fact, the equivalent mass of that energy, packed in such tiny space, makes it a black hole.
  • In short, the conclusion is that our particle can’t move (or, thinking of it as a wave, that our wave can’t wave) because it’s caught in the black hole it creates by its own energy: so the energy can’t escape and, hence, it can’t flow. 🙂

Of course, you will now say that we could imagine half a cycle, or a quarter of that cycle. And you are right: we can surely imagine that, but we get the same thing: to respect the unity of ħ, we’ll then have to pack it into half a cycle, or a quarter of a cycle, which just means the energy of the whole cycle is 2·ħ, or 4·ħ. However, our conclusion still stands: we won’t be able to pack that half-cycle, or that quarter-cycle, into something smaller than the Planck-size spacetime box, because it would make it a black hole, and so our wave wouldn’t go anywhere, and the idea of our wave itself – or the particle – just doesn’t make sense anymore.

This brings me to the final point I’d like to make here. When Maxwell or Einstein, or the quantum vacuumists – or I 🙂 – say that the speed of light is just a property of the vacuum, then that’s correct and not correct at the same time. First, we should note that, if we say that, we might also say that ħ is a property of the vacuum. All physical constants are. Hence, it’s a pretty meaningless statement. Still, it’s a statement that helps us to understand the essence of reality. Second, and more importantly, we should dissect that statement. The speed of light combines two very different aspects:

  1. It’s a physical constant, i.e. some fixed number that we will find to be the same regardless of our reference frame. As such, it’s as essential as those immovable physical laws that we find to be the same in each and every reference frame.
  2. However, its physical dimension is the ratio of the distance and the time unit: m/s. We may choose other time and distance units, but we will still combine them in that ratio. These two units represent the two dimensions in our mind that – as Kant noted – structure our perception of reality: the temporal and spatial dimension.

Hence, we cannot just say that is ‘just a property of the vacuum’. In our definition of as a velocity, we mix reality – the ‘outside world’ – with our perception of it. It’s unavoidable. Frankly, while we should obviously try – and we should try very hard! – to separate what’s ‘out there’ versus ‘how we make sense of it’, it is and remains an impossible job because… Well… When everything is said and done, what we observe ‘out there’ is just that: it’s just what we – humans – observe. 🙂

So, when everything is said and done, the essence of reality consists of four things:

  1. Nothing
  2. Mass, i.e. something, or not nothing
  3. Movement (of something), from nowhere to somewhere.
  4. Us: our mind. Or God’s Mind. Whatever. Mind.

The first is like yin and yang, or manicheism, or whatever dualistic religious system. As for Movement and Mind… Hmm… In some very weird way, I feel they must be part of one and the same thing as well. 🙂 In fact, we may also think of those four things as:

  1. 0 (zero)
  2. 1 (one), or as some sine or a cosine, which is anything in-between 0 and 1.
  3. Well… I am not sure! I can’t really separate point 3 and point 4, because they combine point 1 and point 2.

So we’ve don’t have a quadrupality, right? We do have Trinity here, don’t we? […] Maybe. I won’t comment, because I think I just found Unity here. 🙂

The Poynting vector for the matter-wave

In my various posts on the wavefunction – which I summarized in my e-book – I wrote at the length on the structural similarities between the matter-wave and the electromagnetic wave. Look at the following images once more:

Animation 5d_euler_f

Both are the same, and then they are not. The illustration on the right-hand side is a regular quantum-mechanical wavefunction, i.e. an amplitude wavefunction: the x-axis represents time, so we are looking at the wavefunction at some particular point in space. [Of course, we  could just switch the dimensions and it would all look the same.] The illustration on the left-hand side looks similar, but it is not an amplitude wavefunction. The animation shows how the electric field vector (E) of an electromagnetic wave travels through space. Its shape is the same. So it is the same function. Is it also the same reality?

Yes and no. The two energy propagation mechanisms are structurally similar. The key difference is that, in electromagnetics, we get two waves for the price of one. Indeed, the animation above does not show the accompanying magnetic field vector (B), which is equally essential. But, for the rest, Schrödinger’s equation and Maxwell’s equation model a similar energy propagation mechanism, as shown below.

amw propagation

They have to, as the force laws are similar too:

Coulomb Law

gravitation law

The only difference is that mass comes in one color only, so to speak: it’s always positive. In contrast, electric charge comes in two colors: positive and negative. You can now guess what comes next: quantum chromodynamics, but I won’t write about that here, because I haven’t studied that yet. I won’t repeat what I wrote elsewhere, but I want to make good on one promise, and that is to develop the idea of the Poynting vector for the matter-wave. So let’s do that now. Let me first remind you of the basic ideas, however.

Basics

The animation below shows the two components of the archetypal wavefunction, i.e. the sine and cosine:

circle_cos_sin

Think of the two oscillations as (each) packing half of the total energy of a particle (like an electron or a photon, for example). Look at how the sine and cosine mutually feed into each other: the sine reaches zero as the cosine reaches plus or minus one, and vice versa. Look at how the moving dot accelerates as it goes to the center point of the axis, and how it decelerates when reaching the end points, so as to switch direction. The two functions are exactly the same function, but for a phase difference of 90 degrees, i.e. a right angle. Now, I love engines, and so it makes me think of a V-2 engine with the pistons at a 90-degree angle. Look at the illustration below. If there is no friction, we have a perpetual motion machine: it would store energy in its moving parts, while not requiring any external energy to keep it going.

two-timer-576-px-photo-369911-s-original

If it is easier for you, you can replace each piston by a physical spring, as I did below. However, I should learn how to make animations myself, because the image below does not capture the phase difference. Hence, it does not show how the real and imaginary part of the wavefunction mutually feed into each other, which is (one of the reasons) why I like the V-2 image much better. 🙂

summary 2

The point to note is: all of the illustrations above are true representations – whatever that means – of (idealized) stationary particles, and both for matter (fermions) as well as for force-carrying particles (bosons). Let me give you an example. The (rest) energy of an electron is tiny: about 8.2×10−14 joule. Note the minus 14 exponent: that’s an unimaginably small amount. It sounds better when using the more commonly used electronvolt scale for the energy of elementary particles: 0.511 MeV. Despite its tiny mass (or energy, I should say, but then mass and energy are directly proportional to each other: the proportionality coefficient is given by the E = m·c2 formula), the frequency of the matter-wave of the electron is of the order of 1×1020 = 100,000,000,000,000,000,000 cycles per second. That’s an unimaginably large number and – as I will show when we get there – that’s not because the second is a huge unit at the atomic or sub-atomic scale.

We may refer to this as the natural frequency of the electron. Higher rest masses increase the frequency and, hence, give the wavefunction an even higher density in spacetime. Let me summarize things in a very simple way:

  • The (total) energy that is stored in an oscillating spring is the sum of the kinetic and potential energy (T and U) and is given by the following formula: E = T + U = a02·m·ω02/2. The afactor is the maximum amplitude – which depends on the initial conditions, i.e. the initial pull or push. The ωin the formula is the natural frequency of our spring, which is a function of the stiffness of the spring (k) and the mass on the spring (m): ω02 = k/m.
  • Hence, the total energy that’s stored in two springs is equal to a02·m·ω02.
  • The similarity between the E = a02·m·ω02 and the E = m·c2 formula is much more than just striking. It is fundamental: the two oscillating components of the wavefunction each store half of the total energy of our particle.
  • To emphasize the point: ω0 = √(k/m) is, obviously, a characteristic of the system. Likewise, = √(E/m) is just the same: a property of spacetime.

Of course, the key question is: what is that is oscillating here? In our V-2 engine, we have the moving parts. Now what exactly is moving when it comes to the wavefunction? The easy answer is: it’s the same thing. The V-2 engine, or our springs, store energy because of the moving parts. Hence, energy is equivalent only to mass that moves, and the frequency of the oscillation obviously matters, as evidenced by the E = a02·m·ω02/2 formula for the energy in a oscillating spring. Mass. Energy is moving mass. To be precise, it’s oscillating mass. Think of it: mass and energy are equivalent, but they are not the same. That’s why the dimension of the c2 factor in Einstein’s famous E = m·c2 formula matters. The equivalent energy of a 1 kg object is approximately 9×1016 joule. To be precise, it is the following monstrous number:

89,875,517,873,681,764 kg·m2/s2

Note its dimension: the joule is the product of the mass unit and the square of the velocity unit. So that, then, is, perhaps, the true meaning of Einstein’s famous formula: energy is not just equivalent to mass. It’s equivalent to mass that’s moving. In this case, an oscillating mass. But we should explore the question much more rigorously, which is what I do in the next section. Let me warn you: it is not an easy matter and, even if you are able to work your way through all of the other material below in order to understand the answer, I cannot promise you that the answer will satisfy you entirely. However, it will surely help you to phrase the question.

The Poynting vector for the matter-wave

For the photon, we have the electric and magnetic field vectors E and B. The boldface highlights the fact that these are vectors indeed: they have a direction as well as a magnitude. Their magnitude has a physical dimension. The dimension of E is straightforward: the electric field strength (E) is a quantity expressed in newton per coulomb (N/C), i.e. force per unit charge. This follows straight from the F = q·E force relation.

The dimension of B is much less obvious: the magnetic field strength (B) is measured in (N/C)/(m/s) = (N/C)·(s/m). That’s what comes out of the F = q·v×B force relation. Just to make sure you understand: v×B is a vector cross product, and yields another vector, which is given by the following formula:

a×b =  |a×bn = |a|·|bsinφ·n

The φ in this formula is the angle between a and b (in the plane containing them) and, hence, is always some angle between 0 and π. The n is the unit vector that is perpendicular to the plane containing a and b in the direction given by the right-hand rule. The animation below shows it works for some rather special angles:

Cross_product

We may also need the vector dot product, so let me quickly give you that formula too. The vector dot product yields a scalar given by the following formula:

ab = |a|·|bcosφ

Let’s get back to the F = q·v×B relation. A dimensional analysis shows that the dimension of B must involve the reciprocal of the velocity dimension in order to ensure the dimensions come out alright:

[F]= [q·v×B] = [q]·[v]·[B] = C·(m/s)·(N/C)·(s/m) = N

We can derive the same result in a different way. First, note that the magnitude of B will always be equal to E/c (except when none of the charges is moving, so B is zero), which implies the same:

[B] = [E/c] = [E]/[c] = (N/C)/(m/s) = (N/C)·(s/m)

Finally, the Maxwell equation we used to derive the wavefunction of the photon was ∂E/∂t = c2∇×B, which also tells us the physical dimension of B must involve that s/m factor. Otherwise, the dimensional analysis would not work out:

  1. [∂E/∂t] = (N/C)/s = N/(C·s)
  2. [c2∇×B] = [c2]·[∇×B] = (m2/s2)·[(N/C)·(s/m)]/m = N/(C·s)

This analysis involves the curl operator ∇×, which is a rather special vector operator. It gives us the (infinitesimal) rotation of a three-dimensional vector field. You should look it up so you understand what we’re doing here.

Now, when deriving the wavefunction for the photon, we gave you a purely geometric formula for B:

B = ex×E = i·E

Now I am going to ask you to be extremely flexible: wouldn’t you agree that the B = E/c and the B = ex×E = i·E formulas, jointly, only make sense if we’d assign the s/m dimension to ex and/or to i? I know you’ll think that’s nonsense because you’ve learned to think of the ex× and/or operation as a rotation only. What I am saying here is that it also transforms the physical dimension of the vector on which we do the operation: it multiplies it with the reciprocal of the velocity dimension. Don’t think too much about it, because I’ll do yet another hat trick. We can think of the real and imaginary part of the wavefunction as being geometrically equivalent to the E and B vector. Just compare the illustrations below:

e-and-b Rising_circular

Of course, you are smart, and you’ll note the phase difference between the sine and the cosine (illustrated below). So what should we do with that? Not sure. Let’s hold our breath for the moment.

circle_cos_sin

Let’s first think about what dimension we could possible assign to the real part of the wavefunction. We said this oscillation stores half of the energy of the elementary particle that is being described by the wavefunction. How does that storage work for the E vector? As I explained in my post on the topic, the Poynting vector describes the energy flow in a varying electromagnetic field. It’s a bit of a convoluted story (which I won’t repeat here), but the upshot is that the energy density is given by the following formula:

energy density

Its shape should not surprise you. The formula is quite intuitive really, even if its derivation is not. The formula represents the one thing that everyone knows about a wave, electromagnetic or not: the energy in it is proportional to the square of its amplitude, and so that’s E•E = E2 and B•B = B2. You should also note he cfactor that comes with the B•B product. It does two things here:

  1. As a physical constant, with some dimension of its own, it ensures that the dimensions on both sides of the equation come out alright.
  2. The magnitude of B is 1/c of that of E, so cB = E, and so that explains the extra c2 factor in the second term: we do get two waves for the price of one here and, therefore, twice the energy.

Speaking of dimensions, let’s quickly do the dimensional analysis:

  1. E is measured in newton per coulomb, so [E•E] = [E2] = N2/C2.
  2. B is measured in (N/C)/(m/s), so we get [B•B] = [B2] = (N2/C2)·(s2/m2). However, the dimension of our c2 factor is (m2/s2) and so we’re left with N2/C2. That’s nice, because we need to add stuff that’s expressed in the same units.
  3. The ε0 is that ubiquitous physical constant in electromagnetic theory: the electric constant, aka as the vacuum permittivity. Besides ensuring proportionality, it also ‘fixes’ our units, and so we should trust it to do the same thing here, and it does: [ε0] = C2/(N·m2), so if we multiply that with N2/C2, we find that u is expressed in N/m2.

Why is N/m2 an energy density? The correct answer to that question involves a rather complicated analysis, but there is an easier way to think about it: just multiply N/mwith m/m, and then its dimension becomes N·m/m= J/m3, so that’s  joule per cubic meter. That looks more like an energy density dimension, doesn’t it? But it’s actually the same thing. In any case, I need to move on.

We talked about the Poynting vector, and said it represents an energy flow. So how does that work? It is also quite intuitive, as its formula really speaks for itself. Let me write it down:

energy flux

Just look at it: u is the energy density, so that’s the amount of energy per unit volume at a given point, and so whatever flows out of that point must represent its time rate of change. As for the –S expression… Well… The • operator is the divergence, and so it give us the magnitude of a (vector) field’s source or sink at a given point. If C is a vector field (any vector field, really), then C is a scalar, and if it’s positive in a region, then that region is a source. Conversely, if it’s negative, then it’s a sink. To be precise, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. So, in this case, it gives us the volume density of the flux of S. If you’re somewhat familiar with electromagnetic theory, then you will immediately note that the formula has exactly the same shape as the j = −∂ρ/∂t formula, which represents a flow of electric charge.

But I need to get on with my own story here. In order to not create confusion, I will denote the total energy by U, rather than E, because we will continue to use E for the magnitude of the electric field. We said the real and the imaginary component of the wavefunction were like the E and B vector, but what’s their dimension? It must involve force, but it should obviously not involve any electric charge. So what are our options here? You know the electric force law (i.e. Coulomb’s Law) and the gravitational force law are structurally similar:

Coulomb Law

gravitation law

So what if we would just guess that the dimension of the real and imaginary component of our wavefunction should involve a newton per kg factor (N/kg), so that’s force per mass unit rather than force per unit charge? But… Hey! Wait a minute! Newton’s force law defines the newton in terms of mass and acceleration, so we can do a substitution here: 1 N = 1 kg·m/s2 ⇔ 1 kg = 1 N·s2/m. Hence, our N/kg dimension becomes:

N/kg = N/(N·s2/m)= m/s2

What is this: m/s2? Is that the dimension of the a·cosθ term in the a·ei·θ = a·cosθ − i·a·sinθ wavefunction? I hear you. This is getting quite crazy, but let’s see where it leads us. To calculate the equivalent energy density, we’d then need an equivalent for the ε0 factor, which – replacing the C by kg in the [ε0] = C2/(N·m2) expression – would be equal to kg2/(N·m2). Because we know what we want (energy is defined using the force unit, not the mass unit), we’ll want to substitute the kg unit once again, so – temporarily using the μ0 symbol for the equivalent of that ε0 constant – we get:

0] = [N·s2/m]2/(N·m2) = N·s4/m4

Hence, the dimension of the equivalent of that ε0·E2 term becomes:

 [(μ0/2)]·[cosθ]2 = (N·s4/m4)·m2/s= N/m2

Bingo! How does it work for the other component? The other component has the imaginary unit (i) in front. If we continue to pursue our comparison with the E and B vectors, we should assign an extra s/m dimension because of the ex and/or i factor, so the physical dimension of the i·sinθ term would be (m/s2)·(s/m) = s. What? Just the second? Relax. That second term in the energy density formula has the c2 factor, so it all works out:

 [(μ0/2)]·[c2]·[i·sinθ]2 = [(μ0/2)]·[c2]·[i]2·[sinθ]2 (N·s4/m4)·(m2/s2)·(s2/m2)·m2/s= N/m2

As weird as it is, it all works out. We can calculate and, hence, we can now also calculate the equivalent Poynting vector (S). However, I will let you think about that as an exercise. 🙂 Just note the grand conclusions:

  1. The physical dimension of the argument of the wavefunction is physical action (newton·meter·second) and Planck’s quantum of action is the scaling factor.
  2. The physical dimension of both the real and imaginary component of the elementary wavefunction is newton per kg (N/kg). This allows us to analyze the wavefunction as an energy propagation mechanism that is structurally similar to Maxwell’s equations, which represent the energy propagation mechanism when electromagnetic energy is involved.

As such, all we presented so far was a deep exploration of the mathematical equivalence between the gravitational and electromagnetic force laws:

Coulomb Law

gravitation law

The only difference is that mass comes in one color only, so to speak: it’s always positive. In contrast, electric charge comes in two colors: positive and negative. You can now guess what comes next. 🙂

Despite our grand conclusions, you should note we have not answered the most fundamental question of all. What is mass? What is electric charge? We have all these relations and equations, but are we any wiser, really? The answer to that question probably lies in general relativity: mass is that what curves spacetime. Likewise, we may look at electric charge as causing a very special type of spacetime curvature. However, even such answer – which would involve a much more complicated mathematical analysis – may not satisfy you. In any case, I will let you digest this post. I hope you enjoyed it as much as I enjoyed writing it. 🙂

Post scriptum: Of all of the weird stuff I presented here, I think the dimensional analyses were the most interesting. Think of the N/kg = N/(N·s2/m)= m/sidentity, for example. The m/s2 dimension is the dimension of physical acceleration (or deceleration): the rate of change of the velocity of an object. The identity comes straight out of Newton’s force law:

F = m·a ⇔ F/m = a

Now look, once again, at the animation, and remember the formula for the argument of the wavefunction: θ = E0∙t’. The energy of the particle that is being described is the (angular) frequency of the real and imaginary components of the wavefunction.

circle_cos_sin

The relation between (1) the (angular) frequency of a harmonic oscillator (which is what the sine and cosine represent here) and (2) the acceleration along the axis is given by the following equation:

a(x) = −ω02·x

I’ll let you think about what that means. I know you will struggle with it – because I did – and, hence, let me give you the following hint:

  1. The energy of an ordinary string wave, like a guitar string oscillating in one dimension only, will be proportional to the square of the frequency.
  2. However, for two-dimensional waves – such as an electromagnetic wave – we find that the energy is directly proportional to the frequency. Think of Einstein’s E = h·f = ħ·ω relation, for example. There is no squaring here!

It is a strange observation. Those two-dimensional waves – the matter-wave, or the electromagnetic wave – give us two waves for the price of one, each carrying half of the total energy but, as a result, we no longer have that square function. Think about it. Solving the mystery will make you feel like you’ve squared the circle, which – as you know – is impossible. 🙂