I realized that my last posts were just some crude and rude soundbites, so I thought it would be good to briefly summarize them into something more coherent. Please let me know what you think of it.
The Uncertainty Principle: epistemology versus physics
Anyone who has read anything about quantum physics will know that its concepts and principles are very non-intuitive. Several interpretations have therefore emerged. The mainstream interpretation of quantum mechanics is referred to as the Copenhagen interpretation. It mainly distinguishes itself from more frivolous interpretations (such as the many-worlds and the pilot-wave interpretations) because it is… Well… Less frivolous. Unfortunately, the Copenhagen interpretation itself seems to be subject to interpretation.
One such interpretation may be referred to as radical skepticism – or radical empiricism: we can only say something meaningful about Schrödinger’s cat if we open the box and observe its state. According to this rather particular viewpoint, we cannot be sure of its reality if we don’t make the observation. All we can do is describe its reality by a superposition of the two possible states: dead or alive. That’s Hilbert’s logic: the two states (dead or alive) are mutually exclusive but we add them anyway. If a tree falls in the wood and no one hears it, then it is both standing and not standing. Richard Feynman – who may well be the most eminent representative of mainstream physics – thinks this epistemological position is nonsensical, and I fully agree with him:
“A real tree falling in a real forest makes a sound, of course, even if nobody is there. Even if no one is present to hear it, there are other traces left. The sound will shake some leaves, and if we were careful enough we might find somewhere that some thorn had rubbed against a leaf and made a tiny scratch that could not be explained unless we assumed the leaf were vibrating.” (Feynman’s Lectures, III-2-6)
So what is the mainstream physicist’s interpretation of the Copenhagen interpretation of quantum mechanics then? To fully answer that question, I should encourage the reader to read all of Feynman’s Lectures on quantum mechanics. But then you are reading this because you don’t want to do that, so let me quote from his introductory Lecture on the Uncertainty Principle: “Making an observation affects the phenomenon. The point is that the effect cannot be disregarded or minimized or decreased arbitrarily by rearranging the apparatus. When we look for a certain phenomenon we cannot help but disturb it in a certain minimum way.” (ibidem)
It has nothing to do with consciousness. Reality and consciousness are two very different things. After having concluded the tree did make a noise, even if no one was there to hear it, he wraps up the philosophical discussion as follows: “We might ask: was there a sensation of sound? No, sensations have to do, presumably, with consciousness. And whether ants are conscious and whether there were ants in the forest, or whether the tree was conscious, we do not know. Let us leave the problem in that form.” In short, I think we can all agree that the cat is dead or alive, or that the tree is standing or not standing¾regardless of the observer. It’s a binary situation. Not something in-between. The box obscures our view. That’s all. There is nothing more to it.
Of course, in quantum physics, we don’t study cats but look at the behavior of photons and electrons (we limit our analysis to quantum electrodynamics – so we won’t discuss quarks or other sectors of the so-called Standard Model of particle physics). The question then becomes: what can we reasonably say about the electron – or the photon – before we observe it, or before we make any measurement. Think of the Stein-Gerlach experiment, which tells us that we’ll always measure the angular momentum of an electron – along any axis we choose – as either +ħ/2 or, else, as -ħ/2. So what’s its state before it enters the apparatus? Do we have to assume it has some definite angular momentum, and that its value is as binary as the state of our cat (dead or alive, up or down)?
We should probably explain what we mean by a definite angular momentum. It’s a concept from classical physics, and it assumes a precise value (or magnitude) along some precise direction. We may challenge these assumptions. The direction of the angular momentum may be changing all the time, for example. If we think of the electron as a pointlike charge – whizzing around in its own space – then the concept of a precise direction of its angular momentum becomes quite fuzzy, because it changes all the time. And if its direction is fuzzy, then its value will be fuzzy as well. In classical physics, such fuzziness is not allowed, because angular momentum is conserved: it takes an outside force – or torque – to change it. But in quantum physics, we have the Uncertainty Principle: some energy (force over a distance, remember) can be borrowed – so to speak – as long as it’s swiftly being returned – within the quantitative limits set by the Uncertainty Principle: ΔE·Δt = ħ/2.
Mainstream physicists – including Feynman – do not try to think about this. For them, the Stern-Gerlach apparatus is just like Schrödinger’s box: it obscures the view. The cat is dead or alive, and each of the two states has some probability – but they must add up to one – and so they will write the state of the electron before it enters the apparatus as the superposition of the up and down states. I must assume you’ve seen this before:
|ψ〉 = Cup|up〉 + Cdown|down〉
It’s the so-called Dirac or bra-ket notation. Cup is the amplitude for the electron spin to be equal to +ħ/2 along the chosen direction – which we refer to as the z-direction because we will choose our reference frame such that the z-axis coincides with this chosen direction – and, likewise, Cup is the amplitude for the electron spin to be equal to -ħ/2 (along the same direction, obviously). Cup and Cup will be functions, and the associated probabilities will vary sinusoidally – with a phase difference so as to make sure both add up to one.
The model is consistent, but it feels like a mathematical trick. This description of reality – if that’s what it is – does not feel like a model of a real electron. It’s like reducing the cat in our box to the mentioned fuzzy state of being alive and dead at the same time. Let’s try to come up with something more exciting. 😊
 Academics will immediately note that radical empiricism and radical skepticism are very different epistemological positions but we are discussing some basic principles in physics here rather than epistemological theories.
 The reference to Hilbert’s logic refers to Hilbert spaces: a Hilbert space is an abstract vector space. Its properties allow us to work with quantum-mechanical states, which become state vectors. You should not confuse them with the real or complex vectors you’re used to. The only thing state vectors have in common with real or complex vectors is that (1) we also need a base (aka as a representation in quantum mechanics) to define them and (2) that we can make linear combinations.
The ‘flywheel’ electron model
Physicists describe the reality of electrons by a wavefunction. If you are reading this article, you know how a wavefunction looks like: it is a superposition of elementary wavefunctions. These elementary wavefunctions are written as Ai·exp(-iθi), so they have an amplitude Ai and an argument θi = (Ei/ħ)·t – (pi/ħ)·x. Let’s forget about uncertainty, so we can drop the index (i) and think of a geometric interpretation of A·exp(-iθ) = A·e–iθ.
Here we have a weird thing: physicists think the minus sign in the exponent (-iθ) should always be there: the convention is that we get the imaginary unit (i) by a 90° rotation of the real unit (1) – but the rotation is counterclockwise rotation. I like to think a rotation in the clockwise direction must also describe something real. Hence, if we are seeking a geometric interpretation, then we should explore the two mathematical possibilities: A·e–iθ and A·e+iθ. I like to think these two wavefunctions describe the same electron but with opposite spin. How should we visualize this? I like to think of A·e–iθ and A·e+iθ as two-dimensional harmonic oscillators:
A·e–iθ = cos(-θ) + i·sin(-θ) = cosθ – i·sinθ
A·e+iθ = cosθ + i·sinθ
So we may want to imagine our electron as a pointlike electric charge (see the green dot in the illustration below) to spin around some center in either of the two possible directions. The cosine keeps track of the oscillation in one dimension, while the sine (plus or minus) keeps track of the oscillation in a direction that is perpendicular to the first one.
Figure 1: A pointlike charge in orbit
So we have a weird oscillator in two dimensions here, and we may calculate the energy in this oscillation. To calculate such energy, we need a mass concept. We only have a charge here, but a (moving) charge has an electromagnetic mass. Now, the electromagnetic mass of the electron’s charge may or may not explain all the mass of the electron (most physicists think it doesn’t) but let’s assume it does for the sake of the model that we’re trying to build up here. The point is: the theory of electromagnetic mass gives us a very simple explanation for the concept of mass here, and so we’ll use it for the time being. So we have some mass oscillating in two directions simultaneously: we basically assume space is, somehow, elastic. We have worked out the V-2 engine metaphor before, so we won’t repeat ourselves here.
Figure 2: A perpetuum mobile?
Previously unrelated but structurally similar formulas may be related here:
- The energy of an oscillator: E = (1/2)·m·a2ω2
- Kinetic energy: E = (1/2)·m·v2
- The rotational (kinetic) energy that’s stored in a flywheel: E = (1/2)·I·ω2 = (1/2)·m·r2·ω2
- Einstein’s energy-mass equivalence relation: E = m·c2
Of course, we are mixing relativistic and non-relativistic formulas here, and there’s the 1/2 factor – but these are minor issues. For example, we were talking not one but two oscillators, so we should add their energies: (1/2)·m·a2·ω2 + (1/2)·m·a2·ω2 = m·a2·ω2. Also, one can show that the classical formula for kinetic energy (i.e. E = (1/2)·m·v2) morphs into E = m·c2 when we use the relativistically correct force equation for an oscillator. So, yes, our metaphor – or our suggested physical interpretation of the wavefunction, I should say – makes sense.
If you know something about physics, then you know the concept of the electromagnetic mass – its mathematical derivation, that is – gives us the classical electron radius, aka as the Thomson radius. It’s the smallest of a trio of radii that are relevant when discussing electrons: the other two radii are the Bohr radius and the Compton scattering radius respectively. The Thomson radius is used in the context of elastic scattering: the frequency of the incident particle (usually a photon), and the energy of the electron itself, do not change. In contrast, Compton scattering does change the frequency of the photon that is being scattered, and also impacts the energy of our electron. [As for the Bohr radius, you know that’s the radius of an electron orbital, roughly speaking – or the size of a hydrogen atom, I should say.]
Now, if we combine the E = m·a2·ω2 and E = m·c2 equations, then a·ω must be equal to c, right? Can we show this? Maybe. It is easy to see that we get the desired equality by substituting the amplitude of the oscillation (a) for the Compton scattering radius r = ħ/(m·c), and ω (the (angular) frequency of the oscillation) by using the Planck relation (ω = E/ħ):
a·ω = [ħ/(m·c)]·[E/ħ] = E/(m·c) = m·c2/(m·c) = c
We get a wonderfully simple geometric model of an electron here: an electric charge that spins around in a plane. Its radius is the Compton electron radius – which makes sense – and the radial velocity of our spinning charge is the speed of light – which may or may not make sense. Of course, we need an explanation of why this spinning charge doesn’t radiate its energy away – but then we don’t have such explanation anyway. All we can say is that the electron charge seems to be spinning in its own space – that it’s racing along a geodesic. It’s just like mass creates its own space here: according to Einstein’s general relativity theory, gravity becomes a pseudo-force—literally: no real force. How? I am not sure: the model here assumes the medium – empty space – is, somehow, perfectly elastic: the electron constantly borrows energy from one direction and then returns it to the other – so to speak. A crazy model, yes – but is there anything better? We only want to present a metaphor here: a possible visualization of quantum-mechanical models.
However, if this model is to represent anything real, then many more questions need to be answered. For starters, let’s think about an interpretation of the results of the Stern-Gerlach experiment.
A spinning charge is a tiny magnet – and so it’s got a magnetic moment, which we need to explain the Stern-Gerlach experiment. But it doesn’t explain the discrete nature of the electron’s angular momentum: it’s either +ħ/2 or -ħ/2, nothing in-between, and that’s the case along any direction we choose. How can we explain this? Also, space is three-dimensional. Why would electrons spin in a perfect plane? The answer is: they don’t.
Indeed, the corollary of the above-mentioned binary value of the angular momentum is that the angular momentum – or the electron’s spin – is never completely along any direction. This may or may not be explained by the precession of a spinning charge in a field, which is illustrated below (illustration taken from Feynman’s Lectures, II-35-3).
Figure 3: Precession of an electron in a magnetic field
So we do have an oscillation in three dimensions here, really – even if our wavefunction is a two-dimensional mathematical object. Note that the measurement (or the Stein-Gerlach apparatus in this case) establishes a line of sight and, therefore, a reference frame, so ‘up’ and ‘down’, ‘left’ and ‘right’, and ‘in front’ and ‘behind’ get meaning. In other words, we establish a real space. The question then becomes: how and why does an electron sort of snap into place?
The geometry of the situation suggests the logical angle of the angular momentum vector should be 45°. Now, if the value of its z-component (i.e. its projection on the z-axis) is to be equal to ħ/2, then the magnitude of J itself should be larger. To be precise, it should be equal to ħ/√2 ≈ 0.7·ħ (just apply Pythagoras’ Theorem). Is that value compatible with our flywheel model?
Maybe. Let’s see. The classical formula for the magnetic moment is μ = I·A, with I the (effective) current and A the (surface) area. The notation is confusing because I is also used for the moment of inertia, or rotational mass, but… Well… Let’s do the calculation. The effective current is the electron charge (qe) divided by the period (T) of the orbital revolution: : I = qe/T. The period of the orbit is the time that is needed for the electron to complete one loop. That time (T) is equal to the circumference of the loop (2π·a) divided by the tangential velocity (vt). Now, we suggest vt = r·ω = a·ω = c, and the circumference of the loop is 2π·a. For a, we still use the Compton radius a = ħ/(m·c). Now, the formula for the area is A = π·a2, so we get:
μ = I·A = [qe/T]·π·a2 = [qe·c/(2π·a)]·[π·a2] = [(qe·c)/2]·a = [(qe·c)/2]·[ħ/(m·c)] = [qe/(2m)]·ħ
In a classical analysis, we have the following relation between angular momentum and magnetic moment:
μ = (qe/2m)·J
Hence, we find that the angular momentum J is equal to ħ, so that’s twice the measured value. We’ve got a problem. We would have hoped to find ħ/2 or ħ/√2. Perhaps it’s because a = ħ/(m·c) is the so-called reduced Compton scattering radius…
Maybe we’ll find the solution one day. I think it’s already quite nice we have a model that’s accurate up to a factor of 1/2 or 1/√2. 😊
Post scriptum: I’ve turned this into a small article which may or may not be more readable. You can link to it here. Comments are more than welcome.