Quantum math: garbage in, garbage out?

This post is basically a continuation of my previous one but – as you can see from its title – it is much more aggressive in its language, as I was inspired by a very thoughtful comment on my previous post. Another advantage is that it avoids all of the math. 🙂 It’s… Well… I admit it: it’s just a rant. 🙂 [Those who wouldn’t appreciate the casual style of what follows, can download my paper on it – but that’s much longer and also has a lot more math in it – so it’s a much harder read than this ‘rant’.]

My previous post was actually triggered by an attempt to re-read Feynman’s Lectures on Quantum Mechanics, but in reverse order this time: from the last chapter to the first. [In case you doubt, I did follow the correct logical order when working my way through them for the first time because… Well… There is no other way to get through them otherwise. 🙂 ] But then I was looking at Chapter 20. It’s a Lecture on quantum-mechanical operators – so that’s a topic which, in other textbooks, is usually tackled earlier on. When re-reading it, I realize why people quickly turn away from the topic of physics: it’s a lot of mathematical formulas which are supposed to reflect reality but, in practice, few – if any – of the mathematical concepts are actually being explained. Not in the first chapters of a textbook, not in its middle ones, and… Well… Nowhere, really. Why? Well… To be blunt: I think most physicists themselves don’t really understand what they’re talking about. In fact, as I have pointed out a couple of times already, Feynman himself admits so much:

“Atomic behavior appears peculiar and mysterious to everyone—both to the novice and to the experienced physicist. Even the experts do not understand it the way they would like to.”

So… Well… If you’d be in need of a rather spectacular acknowledgement of the shortcomings of physics as a science, here you have it: if you don’t understand what physicists are trying to tell you, don’t worry about it, because they don’t really understand it themselves. 🙂

Take the example of a physical state, which is represented by a state vector, which we can combine and re-combine using the properties of an abstract Hilbert space. Frankly, I think the word is very misleading, because it actually doesn’t describe an actual physical state. Why? Well… If we look at this so-called physical state from another angle, then we need to transform it using a complicated set of transformation matrices. You’ll say: that’s what we need to do when going from one reference frame to another in classical mechanics as well, isn’t it?

Well… No. In classical mechanics, we’ll describe the physics using geometric vectors in three dimensions and, therefore, the base of our reference frame doesn’t matter: because we’re using real vectors (such as the electric of magnetic field vectors E and B), our orientation vis-á-vis the object – the line of sight, so to speak – doesn’t matter.

In contrast, in quantum mechanics, it does: Schrödinger’s equation – and the wavefunction – has only two degrees of freedom, so to speak: its so-called real and its imaginary dimension. Worse, physicists refuse to give those two dimensions any geometric interpretation. Why? I don’t know. As I show in my previous posts, it would be easy enough, right? We know both dimensions must be perpendicular to each other, so we just need to decide if both of them are going to be perpendicular to our line of sight. That’s it. We’ve only got two possibilities here which – in my humble view – explain why the matter-wave is different from an electromagnetic wave.

I actually can’t quite believe the craziness when it comes to interpreting the wavefunction: we get everything we’d want to know about our particle through these operators (momentum, energy, position, and whatever else you’d need to know), but mainstream physicists still tell us that the wavefunction is, somehow, not representing anything real. It might be because of that weird 720° symmetry – which, as far as I am concerned, confirms that those state vectors are not the right approach: you can’t represent a complex, asymmetrical shape by a ‘flat’ mathematical object!

Huh? Yes. The wavefunction is a ‘flat’ concept: it has two dimensions only, unlike the real vectors physicists use to describe electromagnetic waves (which we may interpret as the wavefunction of the photon). Those have three dimensions, just like the mathematical space we project on events. Because the wavefunction is flat (think of a rotating disk), we have those cumbersome transformation matrices: each time we shift position vis-á-vis the object we’re looking at (das Ding an sich, as Kant would call it), we need to change our description of it. And our description of it – the wavefunction – is all we have, so that’s our reality. However, because that reality changes as per our line of sight, physicists keep saying the wavefunction (or das Ding an sich itself) is, somehow, not real.

Frankly, I do think physicists should take a basic philosophy course: you can’t describe what goes on in three-dimensional space if you’re going to use flat (two-dimensional) concepts, because the objects we’re trying to describe (e.g. non-symmetrical electron orbitals) aren’t flat. Let me quote one of Feynman’s famous lines on philosophers: “These philosophers are always with us, struggling in the periphery to try to tell us something, but they never really understand the subtleties and depth of the problem.” (Feynman’s Lectures, Vol. I, Chapter 16)

Now, I love Feynman’s Lectures but… Well… I’ve gone through them a couple of times now, so I do think I have an appreciation of the subtleties and depth of the problem now. And I tend to agree with some of the smarter philosophers: if you’re going to use ‘flat’ mathematical objects to describe three- or four-dimensional reality, then such approach will only get you where we are right now, and that’s a lot of mathematical mumbo-jumbo for the poor uninitiated. Consistent mumbo-jumbo, for sure, but mumbo-jumbo nevertheless. 🙂 So, yes, I do think we need to re-invent quantum math. 🙂 The description may look more complicated, but it would make more sense.

I mean… If physicists themselves have had continued discussions on the reality of the wavefunction for almost a hundred years now (Schrödinger published his equation in 1926), then… Well… Then the physicists have a problem. Not the philosophers. 🙂 As to how that new description might look like, see my papers on viXra.org. I firmly believe it can be done. This is just a hobby of mine, but… Well… That’s where my attention will go over the coming years. 🙂 Perhaps quaternions are the answer but… Well… I don’t think so either – for reasons I’ll explain later. 🙂

Post scriptum: There are many nice videos on Dirac’s belt trick or, more generally, on 720° symmetries, but this links to one I particularly like. It clearly shows that the 720° symmetry requires, in effect, a special relation between the observer and the object that is being observed. It is, effectively, like there is a leather belt between them or, in this case, we have an arm between the glass and the person who is holding the glass. So it’s not like we are walking around the object (think of the glass of water) and making a full turn around it, so as to get back to where we were. No. We are turning it around by 360°! That’s a very different thing than just looking at it, walking around it, and then looking at it again. That explains the 720° symmetry: we need to turn it around twice to get it back to its original state. So… Well… The description is more about us and what we do with the object than about the object itself. That’s why I think the quantum-mechanical description is defective.

Wavefunctions, perspectives, reference frames, representations and symmetries

Ouff ! This title is quite a mouthful, isn’t it? 🙂 So… What’s the topic of the day? Well… In our previous posts, we developed a few key ideas in regard to a possible physical interpretation of the (elementary) wavefunction. It’s been an interesting excursion, and I summarized it in another pre-publication paper on the open arXiv.org site.

In my humble view, one of the toughest issues to deal with when thinking about geometric (or physical) interpretations of the wavefunction is the fact that a wavefunction does not seem to obey the classical 360° symmetry in space. In this post, I want to muse a bit about this and show that… Well… It does and it doesn’t. It’s got to do with what happens when you change from one representational base (or representation, tout court) to another which is… Well… Like changing the reference frame but, at the same time, it is also more than just a change of the reference frame—and so that explains the weird stuff (like that 720° symmetry of the amplitudes for spin-1/2 particles, for example).

I should warn you before you start reading: I’ll basically just pick up some statements from my paper (and previous posts) and develop some more thoughts on them. As a result, this post may not be very well structured. Hence, you may want to read the mentioned paper first.

The reality of directions

Huh? The reality of directions? Yes. I warned you. This post may cause brain damage. 🙂 The whole argument revolves around a thought experiment—but one whose results have been verified in zillions of experiments in university student labs so… Well… We do not doubt the results and, therefore, we do not doubt the basic mathematical results: we just want to try to understand them better.

So what is the set-up? Well… In the illustration below (Feynman, III, 6-3), Feynman compares the physics of two situations involving rather special beam splitters. Feynman calls them modified or ‘improved’ Stern-Gerlach apparatuses. The apparatus basically splits and then re-combines the two new beams along the z-axis. It is also possible to block one of the beams, so we filter out only particles with their spin up or, alternatively, with their spin down. Spin (or angular momentum or the magnetic moment) as measured along the z-axis, of course—I should immediately add: we’re talking the z-axis of the apparatus here.

rotation about z

The two situations involve a different relative orientation of the apparatuses: in (a), the angle is 0°, while in (b) we have a (right-handed) rotation of 90° about the z-axis. He then proves—using geometry and logic only—that the probabilities and, therefore, the magnitudes of the amplitudes (denoted by C+ and C and C’+ and C’ in the S and T representation respectively) must be the same, but the amplitudes must have different phases, noting—in his typical style, mixing academic and colloquial language—that “there must be some way for a particle to tell that it has turned a corner in (b).”

The various interpretations of what actually happens here may shed some light on the heated discussions on the reality of the wavefunction—and of quantum states. In fact, I should note that Feynman’s argument revolves around quantum states. To be precise, the analysis is focused on two-state systems only, and the wavefunction—which captures a continuum of possible states, so to speak—is introduced only later. However, we may look at the amplitude for a particle to be in the up– or down-state as a wavefunction and, therefore (but do note that’s my humble opinion once more), the analysis is actually not all that different.

We know, from theory and experiment, that the amplitudes are different. For example, for the given difference in the relative orientation of the two apparatuses (90°), we know that the amplitudes are given by C’+ = ei∙φ/2C+ = e i∙π/4C+ and C’ = ei∙φ/2C+ = e i∙π/4C respectively (the amplitude to go from the down to the up state, or vice versa, is zero). Hence, yes, wenot the particle, Mr. Feynman!—know that, in (b), the electron has, effectively, turned a corner.

The more subtle question here is the following: is the reality of the particle in the two setups the same? Feynman, of course, stays away from such philosophical question. He just notes that, while “(a) and (b) are different”, “the probabilities are the same”. He refrains from making any statement on the particle itself: is or is it not the same? The common sense answer is obvious: of course, it is! The particle is the same, right? In (b), it just took a turn—so it is just going in some other direction. That’s all.

However, common sense is seldom a good guide when thinking about quantum-mechanical realities. Also, from a more philosophical point of view, one may argue that the reality of the particle is not the same: something might—or must—have happened to the electron because, when everything is said and done, the particle did take a turn in (b). It did not in (a). [Note that the difference between ‘might’ and ‘must’ in the previous phrase may well sum up the difference between a deterministic and a non-deterministic world view but… Well… This discussion is going to be way too philosophical already, so let’s refrain from inserting new language here.]

Let us think this through. The (a) and (b) set-up are, obviously, different but… Wait a minute… Nothing is obvious in quantum mechanics, right? How can we experimentally confirm that they are different?

Huh? I must be joking, right? You can see they are different, right? No. I am not joking. In physics, two things are different if we get different measurement results. [That’s a bit of a simplified view of the ontological point of view of mainstream physicists, but you will have to admit I am not far off.] So… Well… We can’t see those amplitudes and so… Well… If we measure the same thing—same probabilities, remember?—why are they different? Think of this: if we look at the two beam splitters as one single tube (an ST tube, we might say), then all we did in (b) was bend the tube. Pursuing the logic that says our particle is still the same even when it takes a turn, we could say the tube is still the same, despite us having wrenched it over a 90° corner.

Now, I am sure you think I’ve just gone nuts, but just try to stick with me a little bit longer. Feynman actually acknowledges the same: we need to experimentally prove (a) and (b) are different. He does so by getting third apparatus in (U), as shown below, whose relative orientation to T is the same in both (a) and (b), so there is no difference there.

third apparatus

Now, the axis of is not the z-axis: it is the x-axis in (a), and the y-axis in (b). So what? Well… I will quote Feynman here—not (only) because his words are more important than mine but also because every word matters here:

“The two apparatuses in (a) and (b) are, in fact, different, as we can see in the following way. Suppose that we put an apparatus in front of which produces a pure +x state. Such particles would be split into +z and −z into beams in S, but the two beams would be recombined to give a +x state again at P1—the exit of S. The same thing happens again in T. If we follow by a third apparatus U, whose axis is in the +x direction and, as shown in (a), all the particles would go into the + beam of U. Now imagine what happens if and are swung around together by 90° to the positions shown in (b). Again, the apparatus puts out just what it takes in, so the particles that enter are in a +state with respect to S, which is different. By symmetry, we would now expect only one-half of the particles to get through.”

I should note that (b) shows the apparatus wide open so… Well… I must assume that’s a mistake (and should alert the current editors of the Lectures to it): Feynman’s narrative tells us we should also imagine it with the minus channel shut. In that case, it should, effectively, filter approximately half of the particles out, while they all get through in (a). So that’s a measurement result which shows the direction, as we see it, makes a difference.

Now, Feynman would be very angry with me—because, as mentioned, he hates philosophers—but I’d say: this experiment proves that a direction is something real. Of course, the next philosophical question then is: what is a direction? I could answer this by pointing to the experiment above: a direction is something that alters the probabilities between the STU tube as set up in (a) versus the STU tube in (b). In fact—but, I admit, that would be pretty ridiculous—we could use the varying probabilities as we wrench this tube over varying angles to define an angle! But… Well… While that’s a perfectly logical argument, I agree it doesn’t sound very sensical.

OK. Next step. What follows may cause brain damage. 🙂 Please abandon all pre-conceived notions and definitions for a while and think through the following logic.

You know this stuff is about transformations of amplitudes (or wavefunctions), right? [And you also want to hear about those special 720° symmetry, right? No worries. We’ll get there.] So the questions all revolve around this: what happens to amplitudes (or the wavefunction) when we go from one reference frame—or representation, as it’s referred to in quantum mechanics—to another?

Well… I should immediately correct myself here: a reference frame and a representation are two different things. They are related but… Well… Different… Quite different. Not same-same but different. 🙂 I’ll explain why later. Let’s go for it.

Before talking representations, let us first think about what we really mean by changing the reference frame. To change it, we first need to answer the question: what is our reference frame? It is a mathematical notion, of course, but then it is also more than that: it is our reference frame. We use it to make measurements. That’s obvious, you’ll say, but let me make a more formal statement here:

The reference frame is given by (1) the geometry (or the shape, if that sounds easier to you) of the measurement apparatus (so that’s the experimental set-up) here) and (2) our perspective of it.

If we would want to sound academic, we might refer to Kant and other philosophers here, who told us—230 years ago—that the mathematical idea of a three-dimensional reference frame is grounded in our intuitive notions of up and down, and left and right. [If you doubt this, think about the necessity of the various right-hand rules and conventions that we cannot do without in math, and in physics.] But so we do not want to sound academic. Let us be practical. Just think about the following. The apparatus gives us two directions:

(1) The up direction, which we associate with the positive direction of the z-axis, and

(2) the direction of travel of our particle, which we associate with the positive direction of the y-axis.

Now, if we have two axes, then the third axis (the x-axis) will be given by the right-hand rule, right? So we may say the apparatus gives us the reference frame. Full stop. So… Well… Everything is relative? Is this reference frame relative? Are directions relative? That’s what you’ve been told, but think about this: relative to what? Here is where the object meets the subject. What’s relative? What’s absolute? Frankly, I’ve started to think that, in this particular situation, we should, perhaps, not use these two terms. I am not saying that our observation of what physically happens here gives these two directions any absolute character but… Well… You will have to admit they are more than just some mathematical construct: when everything is said and done, we will have to admit that these two directions are real. because… Well… They’re part of the reality that we are observing, right? And the third one… Well… That’s given by our perspective—by our right-hand rule, which is… Well… Our right-hand rule.

Of course, now you’ll say: if you think that ‘relative’ and ‘absolute’ are ambiguous terms and that we, therefore, may want to avoid them a bit more, then ‘real’ and its opposite (unreal?) are ambiguous terms too, right? Well… Maybe. What language would you suggest? 🙂 Just stick to the story for a while. I am not done yet. So… Yes… What is their reality? Let’s think about that in the next section.

Perspectives, reference frames and symmetries

You’ve done some mental exercises already as you’ve been working your way through the previous section, but you’ll need to do plenty more. In fact, they may become physical exercise too: when I first thought about these things (symmetries and, more importantly, asymmetries in space), I found myself walking around the table with some asymmetrical everyday objects and papers with arrows and clocks and other stuff on it—effectively analyzing what right-hand screw, thumb or grip rules actually mean. 🙂

So… Well… I want you to distinguish—just for a while—between the notion of a reference frame (think of the xyz reference frame that comes with the apparatus) and your perspective on it. What’s our perspective on it? Well… You may be looking from the top, or from the side and, if from the side, from the left-hand side or the right-hand side—which, if you think about it, you can only define in terms of the various positive and negative directions of the various axes. 🙂 If you think this is getting ridiculous… Well… Don’t. Feynman himself doesn’t think this is ridiculous, because he starts his own “long and abstract side tour” on transformations with a very simple explanation of how the top and side view of the apparatus are related to the axes (i.e. the reference frame) that comes with it. You don’t believe me? This is the very first illustration of his Lecture on this:

Modified Stern-GerlachHe uses it to explain the apparatus (which we don’t do here because you’re supposed to already know how these (modified or improved) Stern-Gerlach apparatuses work). So let’s continue this story. Suppose that we are looking in the positive y-direction—so that’s the direction in which our particle is moving—then we might imagine how it would look like when we would make a 180° turn and look at the situation from the other side, so to speak. We do not change the reference frame (i.e. the orientation) of the apparatus here: we just change our perspective on it. Instead of seeing particles going away from us, into the apparatus, we now see particles coming towards us, out of the apparatus.

What happens—but that’s not scientific language, of course—is that left becomes right, and right becomes left. Top still is top, and bottom is bottom. We are looking now in the negative y-direction, and the positive direction of the x-axis—which pointed right when we were looking in the positive y-direction—now points left. I see you nodding your head now—because you’ve heard about parity inversions, mirror symmetries and what have you—and I hear you say: “That’s the mirror world, right?”

No. It is not. I wrote about this in another post: the world in the mirror is the world in the mirror. We don’t get a mirror image of an object by going around it and looking at its back side. I can’t dwell too much on this (just check that post, and another one who talks about the same), but so don’t try to connect it to the discussions on symmetry-breaking and what have you. Just stick to this story, which is about transformations of amplitudes (or wavefunctions). [If you really want to know—but I know this sounds counterintuitive—the mirror world doesn’t really switch left for right. Your reflection doesn’t do a 180 degree turn: it is just reversed front to back, with no rotation at all. It’s only your brain which mentally adds (or subtracts) the 180 degree turn that you assume must have happened from the observed front to back reversal. So the left to right reversal is only apparent. It’s a common misconception, and… Well… I’ll let you figure this out yourself. I need to move on.] Just note the following:

  1. The xyz reference frame remains a valid right-handed reference frame. Of course it does: it comes with our beam splitter, and we can’t change its reality, right? We’re just looking at it from another angle. Our perspective on it has changed.
  2. However, if we think of the real and imaginary part of the wavefunction describing the electrons that are going through our apparatus as perpendicular oscillations (as shown below)—a cosine and sine function respectively—then our change in perspective might, effectively, mess up our convention for measuring angles.

I am not saying it does. Not now, at least. I am just saying it might. It depends on the plane of the oscillation, as I’ll explain in a few moments. Think of this: we measure angles counterclockwise, right? As shown below… But… Well… If the thing below would be some funny clock going backwards—you’ve surely seen them in a bar or so, right?—then… Well… If they’d be transparent, and you’d go around them, you’d see them as going… Yes… Clockwise. 🙂 [This should remind you of a discussion on real versus pseudo-vectors, or polar versus axial vectors, but… Well… We don’t want to complicate the story here.]

Circle_cos_sin

Now, if we would assume this clock represents something real—and, of course, I am thinking of the elementary wavefunction eiθ = cosθ + i·sinθ now—then… Well… Then it will look different when we go around it. When going around our backwards clock above and looking at it from… Well… The back, we’d describe it, naively, as… Well… Think! What’s your answer? Give me the formula! 🙂

[…]

We’d see it as eiθ = cos(−θ) + i·sin(−θ) = cosθ − i·sinθ, right? The hand of our clock now goes clockwise, so that’s the opposite direction of our convention for measuring angles. Hence, instead of eiθ, we write eiθ, right? So that’s the complex conjugate. So we’ve got a different image of the same thing here. Not good. Not good at all. :-/

You’ll say: so what? We can fix this thing easily, right? You don’t need the convention for measuring angles or for the imaginary unit (i) here. This particle is moving, right? So if you’d want to look at the elementary wavefunction as some sort of circularly polarized beam (which, I admit, is very much what I would like to do, but its polarization is rather particular as I’ll explain in a minute), then you just need to define left- and right-handed angles as per the standard right-hand screw rule (illustrated below). To hell with the counterclockwise convention for measuring angles!

right-hand rule

You are right. We could use the right-hand rule more consistently. We could, in fact, use it as an alternative convention for measuring angles: we could, effectively, measure them clockwise or counterclockwise depending on the direction of our particle. But… Well… The fact is: we don’t. We do not use that alternative convention when we talk about the wavefunction. Physicists do use the counterclockwise convention all of the time and just jot down these complex exponential functions and don’t realize that, if they are to represent something real, our perspective on the reference frame matters. To put it differently, the direction in which we are looking at things matters! Hence, the direction is not… Well… I am tempted to say… Not relative at all but then… Well… We wanted to avoid that term, right? 🙂

[…]

I guess that, by now, your brain may suffered from various short-circuits. If not, stick with me a while longer. Let us analyze how our wavefunction model might be impacted by this symmetry—or asymmetry, I should say.

The flywheel model of an electron

In our previous posts, we offered a model that interprets the real and the imaginary part of the wavefunction as oscillations which each carry half of the total energy of the particle. These oscillations are perpendicular to each other, and the interplay between both is how energy propagates through spacetime. Let us recap the fundamental premises:

  1. The dimension of the matter-wave field vector is force per unit mass (N/kg), as opposed to the force per unit charge (N/C) dimension of the electric field vector. This dimension is an acceleration (m/s2), which is the dimension of the gravitational field.
  2. We assume this gravitational disturbance causes our electron (or a charged mass in general) to move about some center, combining linear and circular motion. This interpretation reconciles the wave-particle duality: fields interfere but if, at the same time, they do drive a pointlike particle, then we understand why, as Feynman puts it, “when you do find the electron some place, the entire charge is there.” Of course, we cannot prove anything here, but our elegant yet simple derivation of the Compton radius of an electron is… Well… Just nice. 🙂
  3. Finally, and most importantly in the context of this discussion, we noted that, in light of the direction of the magnetic moment of an electron in an inhomogeneous magnetic field, the plane which circumscribes the circulatory motion of the electron should also comprise the direction of its linear motion. Hence, unlike an electromagnetic wave, the plane of the two-dimensional oscillation (so that’s the polarization plane, really) cannot be perpendicular to the direction of motion of our electron.

Let’s say some more about the latter point here. The illustrations below (one from Feynman, and the other is just open-source) show what we’re thinking of. 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 must be perpendicular to the direction of motion. In other words, if we imagine our electron as spinning around some center (see the illustration on the left-hand side), then the disk it circumscribes (i.e. the plane of the polarization) has to comprise the direction of motion.

Of course, we need to add another detail here. As my readers will know, we do not really have a precise direction of angular momentum in quantum physics. While there is no fully satisfactory explanation of this, the classical explanation—combined with the quantization hypothesis—goes a long way in explaining this: an object with an angular momentum J and a magnetic moment μ that is not exactly parallel to some magnetic field B, will not line up: it will precess—and, as mentioned, the quantization of angular momentum may well explain the rest. [Well… Maybe… We have detailed our attempts in this regard in various posts on this (just search for spin or angular momentum on this blog, and you’ll get a dozen posts or so), but these attempts are, admittedly, not fully satisfactory. Having said that, they do go a long way in relating angles to spin numbers.]

The thing is: we do assume our electron is spinning around. If we look from the up-direction only, then it will be spinning clockwise if its angular momentum is down (so its magnetic moment is up). Conversely, it will be spinning counterclockwise if its angular momentum is up. Let us take the up-state. So we have a top view of the apparatus, and we see something like this:electron waveI know you are laughing aloud now but think of your amusement as a nice reward for having stuck to the story so far. Thank you. 🙂 And, yes, do check it yourself by doing some drawings on your table or so, and then look at them from various directions as you walk around the table as—I am not ashamed to admit this—I did when thinking about this. So what do we get when we change the perspective? Let us walk around it, counterclockwise, let’s say, so we’re measuring our angle of rotation as some positive angle. Walking around it—in whatever direction, clockwise or counterclockwise—doesn’t change the counterclockwise direction of our… Well… That weird object that might—just might—represent an electron that has its spin up and that is traveling in the positive y-direction.

When we look in the direction of propagation (so that’s from left to right as you’re looking at this page), and we abstract away from its linear motion, then we could, vaguely, describe this by some wrenched eiθ = cosθ + i·sinθ function, right? The x- and y-axes of the apparatus may be used to measure the cosine and sine components respectively.

Let us keep looking from the top but walk around it, rotating ourselves over a 180° angle so we’re looking in the negative y-direction now. As I explained in one of those posts on symmetries, our mind will want to switch to a new reference frame: we’ll keep the z-axis (up is up, and down is down), but we’ll want the positive direction of the x-axis to… Well… Point right. And we’ll want the y-axis to point away, rather than towards us. In short, we have a transformation of the reference frame here: z’zy’ = − y, and x’ = − x. Mind you, this is still a regular right-handed reference frame. [That’s the difference with a mirror image: a mirrored right-hand reference frame is no longer right-handed.] So, in our new reference frame, that we choose to coincide with our perspective, we will now describe the same thing as some −cosθ − i·sinθ = −eiθ function. Of course, −cosθ = cos(θ + π) and −sinθ = sin(θ + π) so we can write this as:

cosθ − i·sinθ = cos(θ + π) + i·sinθ = ei·(θ+π) = eiπ·eiθ = −eiθ.

Sweet ! But… Well… First note this is not the complex conjugate: eiθ = cosθ − i·sinθ ≠ −cosθ − i·sinθ = −eiθ. Why is that? Aren’t we looking at the same clock, but from the back? No. The plane of polarization is different. Our clock is more like those in Dali’s painting: it’s flat. 🙂 And, yes, let me lighten up the discussion with that painting here. 🙂 We need to have some fun while torturing our brain, right?

The_Persistence_of_Memory

So, because we assume the plane of polarization is different, we get an −eiθ function instead of a eiθ function.

Let us now think about the ei·(θ+π) function. It’s the same as −eiθ but… Well… We walked around the z-axis taking a full 180° turn, right? So that’s π in radians. So that’s the phase shift here. Hey! Try the following now. Go back and walk around the apparatus once more, but let the reference frame rotate with us, as shown below. So we start left and look in the direction of propagation, and then we start moving about the z-axis (which points out of this page, toward you, as you are looking at this), let’s say by some small angle α. So we rotate the reference frame about the z-axis by α and… Well… Of course, our ei·θ now becomes an our ei·(θ+α) function, right? We’ve just derived the transformation coefficient for a rotation about the z-axis, didn’t we? It’s equal to ei·α, right? We get the transformed wavefunction in the new reference frame by multiplying the old one by ei·α, right? It’s equal to ei·α·ei·θ = ei·(θ+α), right?

electron wave perspective changeWell…

[…]

No. The answer is: no. The transformation coefficient is not ei·α but ei·α/2. So we get an additional 1/2 factor in the phase shift.

Huh? Yes. That’s what it is: when we change the representation, by rotating our apparatus over some angle α about the z-axis, then we will, effectively, get a new wavefunction, which will differ from the old one by a phase shift that is equal to only half of the rotation angle only.

Huh? Yes. It’s even weirder than that. For a spin down electron, the transformation coefficient is ei·α/2, so we get an additional minus sign in the argument.

Huh? Yes.

I know you are terribly disappointed, but that’s how it is. That’s what hampers an easy geometric interpretation of the wavefunction. Paraphrasing Feynman, I’d say that, somehow, our electron not only knows whether or not it has taken a turn, but it also knows whether or not it is moving away from us or, conversely, towards us.

[…]

But… Hey! Wait a minute! That’s it, right? 

What? Well… That’s it! The electron doesn’t know whether it’s moving away or towards us. That’s nonsense. But… Well… It’s like this:

Our ei·α coefficient describes a rotation of the reference frame. In contrast, the ei·α/2 and ei·α/2 coefficients describe what happens when we rotate the T apparatus! Now that is a very different proposition. 

Right! You got it! Representations and reference frames are different things. Quite different, I’d say: representations are real, reference frames aren’t—but then you don’t like philosophical language, do you? 🙂 But think of it. When we just go about the z-axis, a full 180°, but we don’t touch that T-apparatus, we don’t change reality. When we were looking at the electron while standing left to the apparatus, we watched the electrons going in and moving away from us, and when we go about the z-axis, a full 180°, looking at it from the right-hand side, we see the electrons coming out, moving towards us. But it’s still the same reality. We simply change the reference frame—from xyz to x’y’z’ to be precise: we do not change the representation.

In contrast, when we rotate the apparatus over a full 180°, our electron now goes in the opposite direction. And whether that’s away or towards us, that doesn’t matter: it was going in one direction while traveling through S, and now it goes in the opposite direction—relative to the direction it was going in S, that is.

So what happens, really, when we change the representation, rather than the reference frame? Well… Let’s think about that. 🙂

Quantum-mechanical weirdness?

The transformation matrix for the amplitude of a system to be in an up or down state (and, hence, presumably, for a wavefunction) for a rotation about the z-axis is the following one:

rotation matrix

Feynman derives this matrix in a rather remarkable intellectual tour de force in the 6th of his Lectures on Quantum Mechanics. So that’s pretty early on. He’s actually worried about that himself, apparently, and warns his students that “This chapter is a rather long and abstract side tour, and it does not introduce any idea which we will not also come to by a different route in later chapters. You can, therefore, skip over it, and come back later if you are interested.”

Well… That’s how approached it. I skipped it, and didn’t worry about those transformations for quite a while. But… Well… You can’t avoid them. In some weird way, they are at the heart of the weirdness of quantum mechanics itself. Let us re-visit his argument. Feynman immediately gets that the whole transformation issue here is just a matter of finding an easy formula for that phase shift. Why? He doesn’t tell us. Lesser mortals like us must just assume that’s how the instinct of a genius works, right? 🙂 So… Well… Because he knows—from experiment—that the coefficient is ei·α/2 instead of ei·α, he just says the phase shift—which he denotes by λ—must be some proportional to the angle of rotation—which he denotes by φ rather than α (so as to avoid confusion with the Euler angle α). So he writes:

λ = m·φ

Initially, he also tries the obvious thing: m should be one, right? So λ = φ, right? Well… No. It can’t be. Feynman shows why that can’t be the case by adding a third apparatus once again, as shown below.

third apparatusLet me quote him here, as I can’t explain it any better:

“Suppose T is rotated by 360°; then, clearly, it is right back at zero degrees, and we should have C’+ = C+ and C’C or, what is the same thing, ei·m·2π = 1. We get m = 1. [But no!] This argument is wrong! To see that it is, consider that is rotated by 180°. If m were equal to 1, we would have C’+ei·πC+ = −C+ and C’ei·πC = −C. [Feynman works with states here, instead of the wavefunction of the particle as a whole. I’ll come back to this.] However, this is just the original state all over again. Both amplitudes are just multiplied by 1 which gives back the original physical system. (It is again a case of a common phase change.) This means that if the angle between and is increased to 180°, the system would be indistinguishable from the zero-degree situation, and the particles would again go through the (+) state of the apparatus. At 180°, though, the (+) state of the apparatus is the (xstate of the original S apparatus. So a (+x) state would become a (x) state. But we have done nothing to change the original state; the answer is wrong. We cannot have m = 1. We must have the situation that a rotation by 360°, and no smaller angle reproduces the same physical state. This will happen if m = 1/2.”

The result, of course, is this weird 720° symmetry. While we get the same physics after a 360° rotation of the T apparatus, we do not get the same amplitudes. We get the opposite (complex) number: C’+ei·2π/2C+ = −C+ and C’ei·2π/2C = −C. That’s OK, because… Well… It’s a common phase shift, so it’s just like changing the origin of time. Nothing more. Nothing less. Same physics. Same reality. But… Well… C’+ ≠ −C+ and C’ ≠ −C, right? We only get our original amplitudes back if we rotate the T apparatus two times, so that’s by a full 720 degrees—as opposed to the 360° we’d expect.

Now, space is isotropic, right? So this 720° business doesn’t make sense, right?

Well… It does and it doesn’t. We shouldn’t dramatize the situation. What’s the actual difference between a complex number and its opposite? It’s like x or −x, or t and −t. I’ve said this a couple of times already again, and I’ll keep saying it many times more: Nature surely can’t be bothered by how we measure stuff, right? In the positive or the negative direction—that’s just our choice, right? Our convention. So… Well… It’s just like that −eiθ function we got when looking at the same experimental set-up from the other side: our eiθ and −eiθ functions did not describe a different reality. We just changed our perspective. The reference frame. As such, the reference frame isn’t real. The experimental set-up is. And—I know I will anger mainstream physicists with this—the representation is. Yes. Let me say it loud and clear here:

A different representation describes a different reality.

In contrast, a different perspective—or a different reference frame—does not.

Conventions

While you might have had a lot of trouble going through all of the weird stuff above, the point is: it is not all that weird. We can understand quantum mechanics. And in a fairly intuitive way, really. It’s just that… Well… I think some of the conventions in physics hamper such understanding. Well… Let me be precise: one convention in particular, really. It’s that convention for measuring angles. Indeed, Mr. Leonhard Euler, back in the 18th century, might well be “the master of us all” (as Laplace is supposed to have said) but… Well… He couldn’t foresee how his omnipresent formula—eiθ = cosθ + i·sinθ—would, one day, be used to represent something real: an electron, or any elementary particle, really. If he would have known, I am sure he would have noted what I am noting here: Nature can’t be bothered by our conventions. Hence, if eiθ represents something real, then eiθ must also represent something real. [Coz I admire this genius so much, I can’t resist the temptation. Here’s his portrait. He looks kinda funny here, doesn’t he? :-)]

Leonhard_Euler

Frankly, he would probably have understood quantum-mechanical theory as easily and instinctively as Dirac, I think, and I am pretty sure he would have noted—and, if he would have known about circularly polarized waves, probably agreed to—that alternative convention for measuring angles: we could, effectively, measure angles clockwise or counterclockwise depending on the direction of our particle—as opposed to Euler’s ‘one-size-fits-all’ counterclockwise convention. But so we did not adopt that alternative convention because… Well… We want to keep honoring Euler, I guess. 🙂

So… Well… If we’re going to keep honoring Euler by sticking to that ‘one-size-fits-all’ counterclockwise convention, then I do believe that eiθ and eiθ represent two different realities: spin up versus spin down.

Yes. In our geometric interpretation of the wavefunction, these are, effectively, two different spin directions. And… Well… These are real directions: we see something different when they go through a Stern-Gerlach apparatus. So it’s not just some convention to count things like 0, 1, 2, etcetera versus 0, −1, −2 etcetera. It’s the same story again: different but related mathematical notions are (often) related to different but related physical possibilities. So… Well… I think that’s what we’ve got here. Think of it. Mainstream quantum math treats all wavefunctions as right-handed but… Well… A particle with up spin is a different particle than one with down spin, right? And, again, Nature surely cannot be bothered about our convention of measuring phase angles clockwise or counterclockwise, right? So… Well… Kinda obvious, right? 🙂

Let me spell out my conclusions here:

1. The angular momentum can be positive or, alternatively, negative: J = +ħ/2 or −ħ/2. [Let me note that this is not obvious. Or less obvious than it seems, at first. In classical theory, you would expect an electron, or an atomic magnet, to line up with the field. Well… The Stern-Gerlach experiment shows they don’t: they keep their original orientation. Well… If the field is weak enough.]

2. Therefore, we would probably like to think that an actual particle—think of an electron, or whatever other particle you’d think of—comes in two variants: right-handed and left-handed. They will, therefore, either consist of (elementary) right-handed waves or, else, (elementary) left-handed waves. An elementary right-handed wave would be written as: ψ(θi= eiθi = ai·(cosθi + i·sinθi). In contrast, an elementary left-handed wave would be written as: ψ(θie−iθi = ai·(cosθii·sinθi). So that’s the complex conjugate.

So… Well… Yes, I think complex conjugates are not just some mathematical notion: I believe they represent something real. It’s the usual thing: Nature has shown us that (most) mathematical possibilities correspond to real physical situations so… Well… Here you go. It is really just like the left- or right-handed circular polarization of an electromagnetic wave: we can have both for the matter-wave too! [As for the differences—different polarization plane and dimensions and what have you—I’ve already summed those up, so I won’t repeat myself here.] The point is: if we have two different physical situations, we’ll want to have two different functions to describe it. Think of it like this: why would we have two—yes, I admit, two related—amplitudes to describe the up or down state of the same system, but only one wavefunction for it? You tell me.

[…]

Authors like me are looked down upon by the so-called professional class of physicists. The few who bothered to react to my attempts to make sense of Einstein’s basic intuition in regard to the nature of the wavefunction all said pretty much the same thing: “Whatever your geometric (or physical) interpretation of the wavefunction might be, it won’t be compatible with the isotropy of space. You cannot imagine an object with a 720° symmetry. That’s geometrically impossible.”

Well… Almost three years ago, I wrote the following on this blog: “As strange as it sounds, a spin-1/2 particle needs two full rotations (2×360°=720°) until it is again in the same state. Now, in regard to that particularity, you’ll often read something like: “There is nothing in our macroscopic world which has a symmetry like that.” Or, worse, “Common sense tells us that something like that cannot exist, that it simply is impossible.” [I won’t quote the site from which I took this quotes, because it is, in fact, the site of a very respectable  research center!] Bollocks! The Wikipedia article on spin has this wonderful animation: look at how the spirals flip between clockwise and counterclockwise orientations, and note that it’s only after spinning a full 720 degrees that this ‘point’ returns to its original configuration after spinning a full 720 degrees.

720 degree symmetry

So… Well… I am still pursuing my original dream which is… Well… Let me re-phrase what I wrote back in January 2015:

Yes, we can actually imagine spin-1/2 particles, and we actually do not need all that much imagination!

In fact, I am tempted to think that I’ve found a pretty good representation or… Well… A pretty good image, I should say, because… Well… A representation is something real, remember? 🙂

Post scriptum (10 December 2017): Our flywheel model of an electron makes sense, but also leaves many unanswered questions. The most obvious one question, perhaps, is: why the up and down state only?

I am not so worried about that question, even if I can’t answer it right away because… Well… Our apparatus—the way we measure reality—is set up to measure the angular momentum (or the magnetic moment, to be precise) in one direction only. If our electron is captured by some harmonic (or non-harmonic?) oscillation in multiple dimensions, then it should not be all that difficult to show its magnetic moment is going to align, somehow, in the same or, alternatively, the opposite direction of the magnetic field it is forced to travel through.

Of course, the analysis for the spin up situation (magnetic moment down) is quite peculiar: if our electron is a mini-magnet, why would it not line up with the magnetic field? We understand the precession of a spinning top in a gravitational field, but… Hey… It’s actually not that different. Try to imagine some spinning top on the ceiling. 🙂 I am sure we can work out the math. 🙂 The electron must be some gyroscope, really: it won’t change direction. In other words, its magnetic moment won’t line up. It will precess, and it can do so in two directions, depending on its state. 🙂 […] At least, that’s why my instinct tells me. I admit I need to work out the math to convince you. 🙂

The second question is more important. If we just rotate the reference frame over 360°, we see the same thing: some rotating object which we, vaguely, describe by some e+i·θ function—to be precise, I should say: by some Fourier sum of such functions—or, if the rotation is in the other direction, by some ei·θ function (again, you should read: a Fourier sum of such functions). Now, the weird thing, as I tried to explain above is the following: if we rotate the object itself, over the same 360°, we get a different object: our ei·θ and ei·θ function (again: think of a Fourier sum, so that’s a wave packet, really) becomes a −e±i·θ thing. We get a minus sign in front of it. So what happened here? What’s the difference, really?

Well… I don’t know. It’s very deep. If I do nothing, and you keep watching me while turning around me, for a full 360°, then you’ll end up where you were when you started and, importantly, you’ll see the same thing. Exactly the same thing: if I was an e+i·θ wave packet, I am still an an e+i·θ wave packet now. Or if I was an ei·θ wave packet, then I am still an an ei·θ wave packet now. Easy. Logical. Obvious, right?

But so now we try something different: turn around, over a full 360° turn, and you stay where you are. When I am back where I was—looking at you again, so to speak—then… Well… I am not quite the same any more. Or… Well… Perhaps I am but you see me differently. If I was e+i·θ wave packet, then I’ve become a −e+i·θ wave packet now. Not hugely different but… Well… That minus sign matters, right? Or If I was wave packet built up from elementary a·ei·θ waves, then I’ve become a −ei·θ wave packet now. What happened?

It makes me think of the twin paradox in special relativity. We know it’s a paradox—so that’s an apparent contradiction only: we know which twin stayed on Earth and which one traveled because of the gravitational forces on the traveling twin. The one who stays on Earth does not experience any acceleration or deceleration. Is it the same here? I mean… The one who’s turning around must experience some force.

Can we relate this to the twin paradox? Maybe. Note that a minus sign in front of the e−±i·θ functions amounts a minus sign in front of both the sine and cosine components. So… Well… The negative of a sine and cosine is the sine and cosine but with a phase shift of 180°: −cosθ = cos(θ ± π) and −sinθ = sin(θ ± π). Now, adding or subtracting a common phase factor to/from the argument of the wavefunction amounts to changing the origin of time. So… Well… I do think the twin paradox and this rather weird business of 360° and 720° symmetries are, effectively, related. 🙂

Re-visiting the Uncertainty Principle

Let me, just like Feynman did in his last lecture on quantum electrodynamics for Alix Mautner, discuss some loose ends. Unlike Feynman, I will not be able to tie them up. However, just describing them might be interesting and perhaps you, my imaginary reader, could actually help me with tying them up ! Let’s first re-visit the wave function for a photon by way of introduction.

The wave function for a photon

Let’s not complicate things from the start and, hence, let’s first analyze a nice Gaussian wave packet, such as the right-hand graph below: Ψ(x, t). It could be a de Broglie wave representing an electron but here we’ll assume the wave packet might actually represent a photon. [Of course, do remember we should actually show both the real as well as the imaginary part of this complex-valued wave function but we don’t want to clutter the illustration and so it’s only one of the two (cosine or sine). The ‘other’ part (sine or cosine) is just the same but with a phase shift. Indeed, remember that a complex number reθ is equal to r(cosθ + isinθ), and the shape of the sine function is the same as the cosine function but it’s shifted to the left with π/2. So if we have one, we have the other. End of digression.] 

example of wave packet

The assumptions associated with this wonderful mathematical shape include the idea that the wave packet is a composite wave consisting of a large number of harmonic waves with wave numbers k1, k2k3,… all lying around some mean value μk. That is what is shown in the left-hand graph. The mean value is actually noted as k-bar in the illustration above but because I can’t find a k-bar symbol among the ‘special characters’ in the text editor tool bar here, I’ll use the statistical symbols μ and σ to represent a mean value (μ) and some spread around it (σ). In any case, we have a pretty normal shape here, resembling the Gaussian distribution illustrated below.

normal probability distributionThese Gaussian distributions (also known as a density function) have outliers, but you will catch 95,4% of the observations within the μ ± 2σ interval, and 99.7% within the μ ± 3σ interval (that’s the so-called two- and three-sigma rule). Now, the shape of the left-hand graph of the first illustration, mapping the relation between k and A(k), is the same as this Gaussian density function, and if you would take a little ruler and measure the spread of k on the horizontal axis, you would find that the values for k are effectively spread over an interval that’s somewhat bigger than k-bar plus or minus 2Δk. So let’s say 95,4% of the values of k lie in the interval [μ– 2Δk, μk + 2Δk]. Hence, for all practical purposes, we can write that μ– 2Δk  < k< μ+ 2Δk. In any case, we do not care too much about the rest because their contribution to the amplitude of the wave packet is minimal anyway, as we can see from that graph. Indeed, note that the A(k) values on the vertical axis of that graph do not represent the density of the k variable: there is only one wave number for each component wave, and so there’s no distribution or density function of k. These A(k) numbers represent the (maximum) amplitude of the component waves of our wave packet Ψ(x, t). In short, they are the values A(k) appearing in the summation formula for our composite wave, i.e. the wave packet:

Wave packet summation

I don’t want to dwell much more on the math here (I’ve done that in my other posts already): I just want you to get a general understanding of that ‘ideal’ wave packet possibly representing a photon above so you can follow the rest of my story. So we have a (theoretical) bunch of (component) waves with different wave numbers kn, and the spread in these wave numbers – i.e. 2Δk, or let’s take 4Δk to make sure we catch (almost) all of them – determines the length of the wave packet Ψ, which is written here as 2Δx, or 4Δx if we’d want to include (most of) the tail ends as well. What else can we say about Ψ? Well… Maybe something about velocities and all that? OK.

To calculate velocities, we need both ω and k. Indeed, the phase velocity of a wave (vp) is equal to v= ω/k. Now, the wave number k of the wave packet itself – i.e. the wave number of the oscillating ‘carrier wave’ so to say – should be equal to μaccording to the article I took this illustration from. I should check that but, looking at that relationship between A(k) and k, I would not be surprised if the math behind is right. So we have the k for the wave packet itself (as opposed to the k’s of its components). However, I also need the angular frequency ω.

So what is that ω? Well… That will depend on all the ω’s associated with all the k’s, isn’t it? It does. But, as I explained in a previous post, the component waves do not necessarily have to travel all at the same speed and so the relationship between ω and k may not be simple. We would love that, of course, but Nature does what it wants. The only reasonable constraint we can impose on all those ω’s is that they should be some linear function of k. Indeed, if we do not want our wave packet to dissipate (or disperse or, to put it even more plainly, to disappear), then the so-called dispersion relation ω = ω(k) should be linear, so ωn should be equal to ω= ak+ b. What a and b? We don’t know. Random constants. But if the relationship is not linear, then the wave packet will disperse and it cannot possibly represent a particle – be it an electron or a photon.

I won’t go through the math all over again but in my Re-visiting the Matter Wave (I), I used the other de Broglie relationship (E = ħω) to show that – for matter waves that do not disperse – we will find that the phase velocity will equal c/β, with β = v/c, i.e. the ratio of the speed of our particle (v) and the speed of light (c). But, of course, photons travel at the speed of light and, therefore, everything becomes very simple and the phase velocity of the wave packet of our photon would equal the group velocity. In short, we have:

v= ω/k = v= ∂ω/∂k = c

Of course, I should add that the angular frequency of all component waves will also be equal to ω = ck, so all component waves of the wave packet representing a photon are supposed to travel at the speed of light! What an amazingly simple result!

It is. In order to illustrate what we have here – especially the elegance and simplicity of that wave packet for a photon – I’ve uploaded two gif files (see below). The first one could represent our ‘ideal’ photon: group and phase velocity (represented by the speed of the green and red dot respectively) are the same. Of course, our ‘ideal’ photon would only be one wave packet – not a bunch of them like here – but then you may want to think that the ‘beam’ below might represent a number of photons following each other in a regular procession.

Wave - same group and phase velocity

The second animated gif below shows how phase and group velocity can differ. So that would be a (bunch of) wave packets representing a particle not traveling at the speed of light. The phase velocity here is faster than the group velocity (the red dot travels faster than the green dot). [One can actually also have a wave with positive group velocity and negative phase velocity – quite interesting ! – but so that would not represent a particle wave.] Again, a particle would be represented by one wave packet only (so that’s the space between two green dots only) but, again, you may want to think of this as representing electrons following each other in a very regular procession. 

Wave_groupThese illustrations (which I took, once again, from the online encyclopedia Wikipedia) are a wonderful pedagogic tool. I don’t know if it’s by coincidence but the group velocity of the second wave is actually somewhat slower than the first – so the photon versus electron comparison holds (electrons are supposed to move (much) slower). However, as for the phase velocities, they are the same for both waves and that would not reflect the results we found for matter waves. Indeed, you may or may not remember that we calculated superluminal speeds for the phase velocity of matter waves in that post I mentioned above (Re-visiting the Matter Wave): an electron traveling at a speed of 0.01c (1% of the speed of light) would be represented by a wave packet with a group velocity of 0.01c indeed, but its phase velocity would be 100 times the speed of light, i.e. 100c. [That being said, the second illustration may be interpreted as a little bit correct as the red dot does travel faster than the green dot, which – as I explained – is not necessarily always the case when looking at such composite waves (we can have slower or even negative speeds).]

Of course, I should once again repeat that we should not think that a photon or an electron is actually wriggling through space like this: the oscillation only represent the real or imaginary part of the complex-valued probability amplitude associated with our ‘ideal’ photon or our ‘ideal’ electron. That’s all. So this wave is an ‘oscillating complex number’, so to say, whose modulus we have to square to get the probability to actually find the photon (or electron) at some point x and some time t. However, the photon (or the electron) itself are just moving straight from left to right, with a speed matching the group velocity of their wave function.

Are they? 

Well… No. Or, to be more precise: maybe. WHAT? Yes, that’s surely one ‘loose end’ worth mentioning! According to QED, photons also have an amplitude to travel faster or slower than light, and they are not necessarily moving in a straight line either. WHAT? Yes. That’s the complicated business I discussed in my previous post. As for the amplitudes to travel faster or slower than light, Feynman dealt with them very summarily. Indeed, you’ll remember the illustration below, which shows that the contributions of the amplitudes associated with slower or faster speed than light tend to nil because (a) their magnitude (or modulus) is smaller and (b) they point in the ‘wrong’ direction, i.e. not the direction of travel.

Contribution interval

Still, these amplitudes are there and – Shock, horror ! – photons also have an amplitude to not travel in a straight line, especially when they are forced to travel through a narrow slit, or right next to some obstacle. That’s diffraction, described as “the apparent bending of waves around small obstacles and the spreading out of waves past small openings” in Wikipedia. 

Diffraction is one of the many phenomena that Feynman deals with in his 1985 Alix G. Mautner Memorial Lectures. His explanation is easy: “not enough arrows” – read: not enough amplitudes to add. With few arrows, there are also few that cancel out indeed, and so the final arrow for the event is quite random, as shown in the illustrations below.

Many arrows Few arrows

So… Not enough arrows… Feynman adds the following on this: “[For short distances] The nearby, nearly straight paths also make important contributions. So light doesn’t really travel only in a straight line; it “smells” the neighboring paths around it, and uses a small core of nearby space. In the same way, a mirror has to have enough size to reflect normally; if the mirror is too small for the core of neighboring paths, the light scatters in many directions, no matter where you put the mirror.” (QED, 1985, p. 54-56)

Not enough arrows… What does he mean by that? Not enough photons? No. Diffraction for photons works just the same as for electrons: even if the photons would go through the slit one by one, we would have diffraction (see my Revisiting the Matter Wave (II) post for a detailed discussion of the experiment). So even one photon is likely to take some random direction left or right after going through a slit, rather than to go straight. Not enough arrows means not enough amplitudes. But what amplitudes is he talking about?

These amplitudes have nothing to do with the wave function of our ideal photon we were discussing above: that’s the amplitude Ψ(x, t) of a photon to be at point x at point t. The amplitude Feynman is talking about is the amplitude of a photon to go from point A to B along one of the infinitely many possible paths it could take. As I explained in my previous post, we have to add all of these amplitudes to arrive at one big final arrow which, over longer distances, will usually be associated with a rather large probability that the photon will travel in a straight line and at the speed of light – which is why light seems to do at a macro-scale. 🙂

But back to that very succinct statement: not enough arrows. That’s obviously a very relative statement. Not enough as compared to what? What measurement scale are we talking about here? It’s obvious that the ‘scale’ of these arrows for electrons is different than for photons, because the 2012 diffraction experiment with electrons that I referred to used 50 nanometer slits (50×10−9 m), while one of the many experiments demonstrating light diffraction using pretty standard (red) laser light used slits of some 100 micrometer (that 100×10−6 m or – in units you are used to – 0.1 millimeter). 

The key to the ‘scale’ here is the wavelength of these de Broglie waves: the slit needs to be ‘small enough’ as compared to these de Broglie wavelengths. For example, the width of the slit in the laser experiment corresponded to (roughly) 100 times the wavelength of the laser light, and the (de Broglie) wavelength of the electrons in that 2012 diffraction experiment was 50 picometer – that was actually a thousand times the electron wavelength – but it was OK enough to demonstrate diffraction. Much larger slits would not have done the trick. So, when it comes to light, we have diffraction at scales that do not  involve nanotechnology, but when it comes to matter particles, we’re not talking micro but nano: that’s thousand times smaller.

The weird relation between energy and size

Let’s re-visit the Uncertainty Principle, even if Feynman says we don’t need that (we just need to do the amplitude math and we have it all). We wrote the uncertainty principle using the more scientific Kennard formulation: σxσ≥ ħ/2, in which the sigma symbol represents the standard deviation of position x and momentum p respectively. Now that’s confusing, you’ll say, because we were talking wave numbers, not momentum in the introduction above. Well… The wave number k of a de Broglie wave is, of course, related to the momentum p of the particle we’re looking at: p = ħk. Hence, a spread in the wave numbers amounts to a spread in the momentum really and, as I wanted to talk scales, let’s now check the dimensions.

The value for ħ is about 1×10–34 Joule·seconds (J·s) (it’s about 1.054571726(47)×10−34 but let’s go with the gross approximation as for now). One J·s is the same as one kg·m2/s because 1 Joule is a shorthand for km kg·m2/s2. It’s a rather large unit and you probably know that physicists prefer electronVolt·seconds (eV·s) because of that. However, even in expressed in eV·s the value for ħ comes out astronomically small6.58211928(15)×10−16 eV·s. In any case, because the J·s makes dimensions come out right, I’ll stick to it for a while. What does this incredible small factor of proportionality, both in the de Broglie relations as well in that Kennard formulation of the uncertainty principle, imply? How does it work out from a math point of view?

Well… It’s literally a quantum of measurement: even if Feynman says the uncertainty principle should just be seen “in its historical context”, and that “we don’t need it for adding arrows”, it is a consequence of the (related) position-space and momentum-space wave functions for a particle. In case you would doubt that, check it on Wikipedia: the author of the article on the uncertainty principle derives it from these two wave functions, which form a so-called Fourier transform pair. But so what does it say really?

Look at it. First, it says that we cannot know any of the two values exactly (exactly means 100%) because then we have a zero standard deviation for one or the other variable, and then the inequality makes no sense anymore: zero is obviously not greater or equal to 0.527286×10–34 J·s. However, the inequality with the value for  ħ plugged in shows how close to zero we can get with our measurements. Let’s check it out. 

Let’s use the assumption that two times the standard deviation (written as 2Δk or 2Δx on or above the two graphs in the very first illustration of this post) sort of captures the whole ‘range’ of the variable. It’s not a bad assumption: indeed, if Nature would follow normal distributions – and in our macro-world, that seems to be the case – then we’d capture 95.4 of them, so that’s good. Then we can re-write the uncertainty principle as:

Δx·σ≥ ħ or σx·Δp ≥ ħ

So that means we know x within some interval (or ‘range’ if you prefer that term) Δx or, else, we know p within some interval (or ‘range’ if you prefer that term) Δp. But we want to know both within some range, you’ll say. Of course. In that case, the uncertainty principle can be written as:

Δx·Δp ≥ 2ħ

Huh? Why the factor 2? Well… Each of the two Δ ranges corresponds to 2σ (hence, σ= Δx/2 and σ= Δp/2), and so we have (1/2)Δx·(1/2)Δp ≥ ħ/2. Note that if we would equate our Δ with 3σ to get 97.7% of the values, instead of 95.4% only, once again assuming that Nature distributes all relevant properties normally (not sure – especially in this case, because we are talking discrete quanta of action here – so Nature may want to cut off the ‘tail ends’!), then we’d get Δx·Δp ≥ 4.5×ħ: the cost of extra precision soars! Also note that, if we would equate Δ with σ (the one-sigma rule corresponds to 68.3% of a normally distributed range of values), then we get yet another ‘version’ of the uncertainty principle: Δx·Δp ≥ ħ/2. Pick and choose! And if we want to be purists, we should note that ħ is used when we express things in radians (such as the angular frequency for example: E = ħω), so we should actually use h when we are talking distance and (linear) momentum. The equation above then becomes Δx·Δp ≥ h/π.

It doesn’t matter all that much. The point to note is that, if we express x and p in regular distance and momentum units (m and kg·m/s), then the unit for ħ (or h) is 1×10–34. Now, we can sort of choose how to spread the uncertainty over x and p. If we spread it evenly, then we’ll measure both Δx and Δp  in units of 1×10–17  m and 1×10–17 kg·m/s. That’s small… but not that small. In fact, it is (more or less) imaginably small I’d say.

For example, a photon of a blue-violet light (let’s say a wavelength of around 660 nanometer) would have a momentum p = h/λ equal to some 1×10–22 kg·m/s (just work it out using the values for h and λ). You would usually see this value measured in a unit that’s more appropriate to the atomic scale: 6.25 eV/c. [Converting momentum into energy using E = pc, and using the Joule-electronvolt conversion (1 eV ≈ 1.6×10–19 J) will get you there.] Hence, units of 1×10–17  m for momentum are a hundred thousand times the rather average momentum of our light photon. We can’t have that so let’s reduce the uncertainty related to the momentum to that 1×10–22 kg·m/s scale. Then the uncertainty about position will be measured in units of 1×10–12 m. That’s the picometer scale in-between the nanometer (1×10–9 m) and the femtometer (1×10–9 m) scale. You’ll remember that this scale corresponds to the resolution of a (modern) electron microscope (50 pm). So can we see “uncertainty effects” ? Yes. I’ll come back to that.

However, before I discuss these, I need to make a little digression. Despite the sub-title I am using above, the uncertainties in distance and momentum we are discussing here are nowhere near to what is referred to as the Planck scale in physics: the Planck scale is at the other side of that Great Desert I mentioned: the Large Hadron Collider, which smashes particles with  (average) energies of 4 tera-electronvolt (i.e. 4 trillion eV – all packed into one particle !) is probing stuff measuring at a scale of a thousandth of a femtometer (0.001×10–12 m), but we’re obviously at the limits of what’s technically possible, and so that’s where the Great Desert starts. The ‘other side’ of that Great Desert is the Planck scale: 10–35 m. Now, why is that some kind of theoretical limit? Why can’t we just continue to further cut these scales down? Just like Dedekind did when defining irrational numbers? We can surely get infinitely close to zero, can we? Well… No. The reasoning is quite complex (and I am not sure if I actually understand it – the way I should) but it is quite relevant to the topic here (the relation between energy and size), and it goes something like this:

  1. In quantum mechanics, particles are considered to be point-like but they do take space, as evidenced from our discussion on slit widths: light will show diffraction at the micro-scale (10–6 m) but electrons will do that only at the nano-scale (10–9 m), so that’s a thousand times smaller. That’s related to their respective the de Broglie wavelength which, for electrons, is also a thousand times smaller than that of electrons. Now, the de Broglie wavelength is related to the energy and/or the momentum of these particles: E = hf and p = h/λ.
  2. Higher energies correspond to smaller de Broglie wavelengths and, hence, are associated with particles of smaller size. To continue the example, the energy formula to be used in the E = hf relation for an electron – or any particle with rest mass – is the (relativistic) mass-energy equivalence relation: E = γm0c2, with γ the Lorentz factor, which depends on the velocity v of the particle. For example, electrons moving at more or less normal speeds (like in the 2012 experiment, or those used in an electron microscope) have typical energy levels of some 600 eV, and don’t think that’s a lot: the electrons from that cathode ray tube in the back of an old-fashioned TV which lighted up the screen so you could watch it, had energies in the 20,000 eV range. So, for electrons, we are talking energy levels a thousand or a hundred thousand higher than for your typical 2 to 10 eV photon.
  3. Of course, I am not talking X or gamma rays here: hard X rays also have energies of 10 to 100 kilo-electronvolt, and gamma ray energies range from 1 million to 10 million eV (1-10 MeV). In any case, the point to note is ‘small’ particles must have high energies, and I am not only talking massless particles such as photons. Indeed, in my post End of the Road to Reality?, I discussed the scale of a proton and the scale of quarks: 1.7 and 0.7 femtometer respectively, which is smaller than the so-called classical electron radius. So we have (much) heavier particles here that are smaller? Indeed, the rest mass of the u and d quarks that make up a proton (uud) is 2.4 and 4.8 MeV/c2 respectively, while the (theoretical) rest mass of an electron is 0.511 Mev/conly, so that’s almost 20 times more: (2.4+2.4+4.8/0.5). Well… No. The rest mass of a proton is actually 1835 times the rest mass of an electron: the difference between the added rest masses of the quarks that make it up and the rest mass of the proton itself (938 MeV//c2) is the equivalent mass of the strong force that keeps the quarks together.
  4. But let me not complicate things. Just note that there seems to be a strange relationship between the energy and the size of a particle: high-energy particles are supposed to be smaller, and vice versa: smaller particles are associated with higher energy levels. If we accept this as some kind of ‘factual reality’, then we may understand what the Planck scale is all about: : the energy levels associated with theoretical ‘particles’ of the above-mentioned Planck scale (i.e. particles with a size in the 10–35 m range) would have energy levels in the 1019 GeV range. So what? Well… This amount of energy corresponds to an equivalent mass density of a black hole. So any ‘particle’ we’d associate with the Planck length would not make sense as a physical entity: it’s the scale where gravity takes over – everything.

Again: so what? Well… I don’t know. It’s just that this is entirely new territory, and it’s also not the topic of my post here. So let me just quote Wikipedia on this and then move on: “The fundamental limit for a photon’s energy is the Planck energy [that’s the 1019 GeV which I mentioned above: to be precise, that ‘limit energy’ is said to be 1.22 × 1019 GeV], for the reasons cited above [that ‘photon’ would not be ‘photon’ but a black hole, sucking up everything around it]. This makes the Planck scale a fascinating realm for speculation by theoretical physicists from various schools of thought. Is the Planck scale domain a seething mass of virtual black holes? Is it a fabric of unimaginably fine loops or a spin foam network? Is it interpenetrated by innumerable Calabi-Yau manifolds which connect our 3-dimensional universe with a higher-dimensional space? [That’s what’s string theory is about.] Perhaps our 3-D universe is ‘sitting’ on a ‘brane’ which separates it from a 2, 5, or 10-dimensional universe and this accounts for the apparent ‘weakness’ of gravity in ours. These approaches, among several others, are being considered to gain insight into Planck scale dynamics. This would allow physicists to create a unified description of all the fundamental forces. [That’s what’s these Grand Unification Theories (GUTs) are about.]

Hmm… I wish I could find some easy explanation of why higher energy means smaller size. I do note there’s an easy relationship between energy and momentum for massless particles traveling at the velocity of light (like photons): E = p(or p = E/c), but – from what I write above – it is obvious that it’s the spread in momentum (and, therefore, in wave numbers) which determines how short or how long our wave train is, not the energy level as such. I guess I’ll just have to do some more research here and, hopefully, get back to you when I understand things better. 

Re-visiting the Uncertainty Principle

You will probably have read countless accounts of the double-slit experiment, and so you will probably remember that these thought or actual experiments also try to watch the electrons as they pass the slits – with disastrous results: the interference pattern disappears. I copy Feynman’s own drawing from his 1965 Lecture on Quantum Behavior below: a light source is placed behind the ‘wall’, right between the two slits. Now, light (i.e. photons) gets scattered when it hits electrons and so now we should ‘see’ through which slit the electron is coming. Indeed, remember that we sent them through these slits one by one, and we still had interference – suggesting the ‘electron wave’ somehow goes through both slits at the same time, which can’t be true – because an electron is a particle.

Watching the electrons

However, let’s re-examine what happens exactly.

  1. We can only detect all electrons if the light is high intensity, and high intensity does not mean higher energy photons but more photons. Indeed, if the light source is deem, then electrons might get through without being seen. So a high-intensity light source allows us to see all electrons but – as demonstrated not only in thought experiments but also in the laboratory – it destroys the interference pattern.
  2. What if we use lower-energy photons, like infrared light with wavelengths of 10 to 100 microns instead of visible light? We can then use thermal imaging night vision goggles to ‘see’ the electrons. 🙂 And if that doesn’t work, we can use radiowaves (or perhaps radar!). The problem – as Feynman explains it – is that such low frequency light (associated with long wavelengths) only give a ‘big fuzzy flash’ when the light is scattered: “We can no longer tell which hole the electron went through! We just know it went somewhere!” At the same time, “the jolts given to the electron are now small enough so that we begin to see some interference effect again.” Indeed: “For wavelengths much longer than the separation between the two slits (when we have no chance at all of telling where the electron went), we find that the disturbance due to the light gets sufficiently small that we again get the interference curve P12.” [P12 is the curve describing the original interference effect.]

Now, that would suggest that, when push comes to shove, the Uncertainty Principle only describes some indeterminacy in the so-called Compton scattering of a photon by an electron. This Compton scattering is illustrated below: it’s a more or less elastic collision between a photon and electron, in which momentum gets exchanged (especially the direction of the momentum) and – quite important – the wavelength of the scattered light is different from the incident radiation. Hence, the photon loses some energy to the electron and, because it will still travel at speed c, that means its wavelength must increase as prescribed by the λ = h/p de Broglie relation (with p = E/c for a photon). The change in the wavelength is called the Compton shift. and its formula is given in the illustration: it depends on the (rest) mass of the electron obviously and on the change in the direction of the momentum (of the photon – but that change in direction will obviously also be related to the recoil direction of the electron).

Compton scattering

This is a very physical interpretation of the Uncertainty Principle, but it’s the one which the great Richard P. Feynman himself stuck to in 1965, i.e. when he wrote his famous Lectures on Physics at the height of his career. Let me quote his interpretation of the Uncertainty Principle in full indeed:

“It is impossible to design an apparatus to determine which hole the electron passes through, that will not at the same time disturb the electrons enough to destroy the interference pattern. If an apparatus is capable of determining which hole the electron goes through, it cannot be so delicate that it does not disturb the pattern in an essential way. No one has ever found (or even thought of) a way around this. So we must assume that it describes a basic characteristic of nature.”

That’s very mechanistic indeed, and it points to indeterminacy rather than ontological uncertainty. However, there’s weirder stuff than electrons being ‘disturbed’ in some kind of random way by the photons we use to detect them, with the randomness only being related to us not knowing at what time photons leave our light source, and what energy or momentum they have exactly. That’s just ‘indeterminacy’ indeed; not some fundamental ‘uncertainty’ about Nature.

We see such ‘weirder stuff’ in those mega- and now tera-electronvolt experiments in particle accelerators. In 1965, Feynman had access to the results of the high-energy positron-electron collisions being observed in the 3 km long Stanford Linear Accelerator (SLAC), which started working in 1961, but stuff like quarks and all that was discovered only in the late 1960s and early 1970s, so that’s after Feynman’s Lectures on Physics.So let me just mention a rather remarkable example of the Uncertainty Principle at work which Feynman quotes in his 1985 Alix G. Mautner Memorial Lectures on Quantum Electrodynamics.

In the Feynman diagram below, we see a photon disintegrating, at time t = T3into a positron and an electron. The positron (a positron is an electron with positive charge basically: it’s the electron’s anti-matter counterpart) meets another electron that ‘happens’ to be nearby and the annihilation results in (another) high-energy photon being emitted. While, as Feynman underlines, “this is a sequence of events which has been observed in the laboratory”, how is all this possible? We create matter – an electron and a positron both have considerable mass – out of nothing here ! [Well… OK – there’s a photon, so that’s some energy to work with…]

Photon disintegrationFeynman explains this weird observation without reference to the Uncertainty Principle. He just notes that “Every particle in Nature has an amplitude to move backwards in time, and therefore has an anti-particle.” And so that’s what this electron coming from the bottom-left corner does: it emits a photon and then the electron moves backwards in time. So, while we see a (very short-lived) positron moving forward, it’s actually an electron quickly traveling back in time according to Feynman! And, after a short while, it has had enough of going back in time, so then it absorbs a photon and continues in a slightly different direction. Hmm… If this does not sound fishy to you, it does to me.

The more standard explanation is in terms of the Uncertainty Principle applied to energy and time. Indeed, I mentioned that we have several pairs of conjugate variables in quantum mechanics: position and momentum are one such pair (related through the de Broglie relation p =ħk), but energy and time are another (related through the other de Broglie relation E = hf = ħω). While the ‘energy-time uncertainty principle’ – ΔE·Δt ≥ ħ/2 – resembles the position-momentum relationship above, it is apparently used for ‘very short-lived products’ produced in high-energy collisions in accelerators only. I must assume the short-lived positron in the Feynman diagram is such an example: there is some kind of borrowing of energy (remember mass is equivalent to energy) against time, and then normalcy soon gets restored. Now THAT is something else than indeterminacy I’d say.

uncertainty principle energy time

But so Feynman would say both interpretations are equivalent, because Nature doesn’t care about our interpretations.

What to say in conclusion? I don’t know. I obviously have some more work to do before I’ll be able to claim to understand the uncertainty principle – or quantum mechanics in general – somewhat. I think the next step is to solve my problem with the summary ‘not enough arrows’ explanation, which is – evidently – linked to the relation between energy and size of particles. That’s the one loose end I really need to tie up I feel ! I’ll keep you posted !