Moving on…

This is it, folks ! I am moving on ! It was nice camping out here. 🙂

This has been a very interesting journey for me. I wrote my first post in October 2013, so that’s almost five years ago. As mentioned in the ‘About‘ page, I started writing this blog because — with all those breakthroughs in science (some kind of experimental verification of what is referred to as the Higgs field in July 2012 and, more recently, the confirmation of the reality of gravitational waves in 2016 by Caltech’s LIGO Lab) — I felt I should make an honest effort to try to understand what it was all about.

Despite all of my efforts (including enrolling in MIT’s edX QM course, which I warmly recommend as an experience, especially because it’s for free), I haven’t moved much beyond quantum electrodynamics (QED). Hence, that Higgs field is a still a bit of a mystery to me. In any case, the summaries I’ve read about it say it’s just some scalar field. So that’s not very exciting: mass is some number associated with some position in spacetime. That’s nothing new, right?

In contrast, I am very enthusiastic about the LIGO Lab discovery. Why? Because it confirms Einstein was right all along.

If you have read any of my posts, you will know I actually disagree with Feynman. I have to thank him for his Lectures — and I would, once again, like to thank Michael Gottlieb and Rudolf Pfeiffer, who have worked for decades to get those Lectures online — but my explorations did confirm that guts feeling I had deep inside when starting this journey: the complexity in the quantum-mechanical framework does not match the intuition that, if the theory has a simple circle group structure, one should not be calculating a zillion integrals all over space over 891 4-loop Feynman diagrams to explain the magnetic moment of an electron in a Penning trap. And the interference of a photon with itself in the Mach-Zehnder interference experiment has a classical explanation too. The ‘zero state’ of a photon – or its zero states (plural), I should say – are the linear components of the circular polarization. In fact, I really wish someone would have gently told me that an actual beam splitter changes the polarization of light. I could have solved the Mach-Zehnder puzzle with that information like a year ago.]

This will probably sound like Chinese to you, so let me translate it: there is no mystery. Not in the QED sector of the Standard Model, at least. All can be explained by simple geometry and the idea of a naked charge: something that has no other property but its electric charge and – importantly – some tiny radius, which is given by the fine-structure constant (the ratio becomes a distance if we think of the electron’s Compton radius as a natural (distance) unit). So the meaning of God’s Number is clear now: there is nothing miraculous about it either. Maxwell’s equations combined with the Planck-Einstein Law (E = h·f) are all we need to explain the whole QED sector. No hocus-pocus needed. The elementary wavefunction exp(±i·θ) = exp(±ω·t) = exp[±(E/ħ)·t] represents an equally elementary oscillation. Physicists should just think some more about the sign convention and, more generally, think some more about Occam’s Razor Principle when modeling their problems. 🙂

Am I a crackpot? Maybe. I must be one, because I think the academics have a problem, not me. So… Well… That’s the definition of a crackpot, isn’t it? 🙂 It feels weird. Almost all physicists I got in touch with – spare two or three (I won’t mention their names because they too don’t quite know what to do with me) – are all stuck in their Copenhagen interpretation of quantum mechanics: reality is some kind of black box and we’ll never understand it the way we would want to understand it. Almost none of them is willing to think outside of the box. I blame vested interests (we’re talking Nobel Prize stuff, unfortunately) and Ivory Tower culture.

In any case, I found the answers to the questions I started out with, and I don’t think the academics I crossed (s)words with have found that peace of mind yet. So if I am a crackpot, then I am a happy one. 😊

The Grand Conclusion is that the Emperor is not wearing any clothes. Not in the QED sector, at least. In fact, I think the situation is a lot worse. The Copenhagen interpretation of quantum mechanics feels like a Bright Shining Lie. [Yes, I know that’s an ugly reference.] But… Yes. Just mathematical gimmicks to entertain students – and academics ! Of course, I can appreciate the fact that Nobel Prizes have been awarded and that academic reputations have to be upheld — posthumously or… I would want to write ‘humously’ here but that word doesn’t exist so I should replace it by ‘humorously’. 🙂 […] OK. Poor joke. 🙂

Frankly, it is a sad situation. :-/ Physics has become the domain of hype and canonical nonsense. To the few readers who have been faithful followers (this blog attracted about 154,034 visitors so far which is — of course — close to nothing), I’d say: think for yourself. Honor Boltzman’s spirit: “Bring forward what is true. Write it so that it is clear. Defend it to your last breath.” I actually like another quote of him too: “If you are out to describe the truth, leave elegance to the tailor.” But that’s too rough, isn’t it? And then I am also not sure he really said that. 🙂

Of course, QCD is another matter altogether — because of the non-linearity of the force(s) involved, and the multiplication of ‘colors’ but my research over the past five years (longer than that, actually) have taught me that there is no ‘deep mystery’ in the QED sector. All is logical – including the meaning of the fine-structure constant: that’s just the radius of the naked charge expressed in natural units. All the rest can be derived. And 99% of what you’ll read or google about quantum mechanics is about QED: perturbation theory, propagators, the quantized field, etcetera to talk about photons and electrons, and their interactions. If you have a good idea about what an electron and a photon actually are, then you do not need anything of that to understand QED.

In short, quantum electrodynamics – as a theory, and in its current shape and form – is incomplete: it is all about electrons and photons – and the interactions between the two – but the theory lacks a good description of what electrons and photons actually are. All of the weirdness of Nature is, therefore, in this weird description of the fields: gauge theories, Feynman diagrams, quantum field theory, etcetera. And the common-sense is right there: right in front of us. It’s easy and elegant: a plain common-sense interpretation of quantum mechanics — which, I should remind the reader, is based on Erwin Schrödinger’s trivial solution for Dirac’s wave equation for an electron in free space.

So is no one picking this up? Let’s see. Truth cannot be hidden, right? Having said that, I must admit I have been very surprised by the rigidity of thought of academics (which I know all too well from my experience as a PhD student in economics) in this domain. If math is the queen of science, then physics is the king, right? Well… Maybe not. The brightest minds seem to have abandoned the field.

But I will stop my rant here. I want to examine the QCD sector now. What theories do we have for the non-linear force(s) that keep(s) protons together? What explains electron capture by a proton—turning it into a neutron in the process? What’s the nature of neutrinos? How should we think of all these intermediary particles—which are probably just temporary resonances rather than permanent fixtures?

My new readingeinstein.blog will be devoted to that. I think I’ll need some time to post my first posts (pun intended)—but… Well… We’ve started this adventure and so I want to get to the next destination. It’s a mind thing, right? 🙂

Bringing physics back to Quantum Physics: classical quantum theory

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God’s Number explained

My posts on the fine-structure constant – God’s Number as it is often referred to – have always attracted a fair amount of views. I think that’s because I have always tried to clarify this or that relation by showing how and why exactly it pops us in this or that formula (e.g. Rydberg’s energy formula, the ratio of the various radii of an electron (Thomson, Compton and Bohr radius), the coupling constant, the anomalous magnetic moment, etcetera), as opposed to what most seem to try to do, and that is to further mystify it. You will probably not want to search through all of my writing so I will just refer you to my summary of these efforts on the viXra.org site: “Layered Motions: the Meaning of the Fine-Structure Constant.

However, I must admit that – till now – I wasn’t quite able to answer this very simple question: what is that fine-structure constant? Why exactly does it appear as a scaling constant or a coupling constant in almost any equation you can think of but not in, say, Einstein’s mass-energy equivalence relation, or the de Broglie relations?

I finally have a final answer (pun intended) to the question, and it’s surprisingly easy: it is the radius of the naked charge in the electron expressed in terms of the natural distance unit that comes out of our realist interpretation of what an electron actually is. [For those who haven’t read me before, this realist interpretation is based on Schrödinger’s discovery of the Zitterbewegung of an electron.] That natural distance unit is the Compton radius of the electron: it is the effective radius of an electron as measured in inelastic collisions between high-energy photons and the electron. I like to think of it as a quantum of space in which interference happens but you will want to think that through for yourself. 

The point is: that’s it. That’s all. All the other calculations follow from it. Why? It would take me a while to explain that but, if you carefully look at the logic in my classical calculations of the anomalous magnetic moment, then you should be able to  understand why these calculations are somewhat more fundamental than the others and why we can, therefore, get everything else out of them. 🙂

Post scriptum: I quickly checked the downloads of my papers on Phil Gibbs’ site, and I am extremely surprised my very first paper (the quantum-mechanical wavefunction as a gravitational wave) of mine still gets downloads. To whomever is interested in this paper, I would say: the realist interpretation we have been pursuing – based on the Zitterbewegung model of an electron – is based on the idea of a naked charge (with zero rest mass) orbiting around some center. The energy in its motion – a perpetual current ring, really – gives the electron its (equivalent) mass. That’s just Wheeler’s idea of ‘mass without mass’. But the force is definitely not gravitational. It cannot be. The force has to grab onto something, and all it can grab onto here is that naked charge. The force is, therefore, electromagnetic. It must be. I now look at my very first paper as a first immature essay. It did help me to develop some basic intuitive ideas on what any realist interpretation of QM should look like, but the quantum-mechanical wavefunction has nothing to do with gravity. Quantum mechanics is electromagnetics: we just add the quantum. The idea of an elementary cycle. Gravity is dealt with by general relativity theory: energy – or its equivalent mass – bends spacetime. That’s very significant, but it doesn’t help you when analyzing the QED sector of physics. I should probably pull this paper of the site – but I won’t. Because I think it shows where I come from: very humble origins. 🙂

Philosophy and Science: Dirac’s Principles

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The anomalous magnetic moment: classical calculations

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Interpreting quantum mechanics

My book is moving forward. I just produced a very first promotional video. Have a look and let me know what you think of it ! 🙂

The not-so-anomalous magnetic moment of an electron

Note: Check the revised paper on this topic. The substance is the same, but it is a more coherent development. 

Jean Louis Van Belle, 23 December 2018

Original post:

I am going to expose a bright shining lie in (quantum) physics in this post: what is referred to as the electron’s anomalous magnetic moment is actually not a magnetic moment, and it is not anomalous. Let’s start with the first remark. The anomalous magnetic moment is not a magnetic moment. It is just some (real) number: it’s a ratio, to be precise. It does not have any physical dimension. If it would be an actual magnetic moment then we would measure it as we usually do in the context of quantum mechanics, and that is in terms of the Bohr magneton, which is equal to: μB = qeħ/2m ≈ 9.274×10−24 joule per tesla.[1]

So what is the electron’s anomalous magnetic moment – denoted by ae – then? It is defined as the (half-)difference between (1) some supposedly real gyromagnetic ratio (ge) and (2) Dirac’s theoretical value for the gyromagnetic ratio of a spin-only electron (g = 2)[2]:F1This immediately triggers an obvious question: why would we use the g-factor of a spin-only electron. This is a very weird thing to do, because the electron in the cyclotron (a Penning trap) is actually not a spin-only electron: it follows an orbital motion – as we will explain shortly.

So… Well… It is also routinely said (and written) that its measured value is equal to 0.00115965218085(76). The 76 (between brackets) is the uncertainty – which looks pretty certain, because it is equal to 0.00000000000076. Hence, the precision here is equivalent to 76 parts per trillion (ppt). It is measured as a standard deviation.[3] However, the problem is that these experiments actually do not directly measure ae. What is being measured in the so-called Penning traps that are used in these experiments (think of them as a sort of cyclotron) are two slightly different frequencies – an orbital frequency and a precession frequency, to be precise – and ae is then calculated as the fractional difference between the two:F2Let us go through the motions here – literally. The orbital frequency fc is the cyclotron frequency: a charged particle in a Penning trap will move in a circular orbit whose frequency depends on the charge, its mass and the strength of the magnetic field only. Let us give you the formula (we will derive it for you in an instant):F3The subscript c stands for cyclotron – or circular, if you want. We should not think of the speed of light here! In fact, the orbital velocity is a (relatively small) fraction of the speed of light and we can, therefore, use non-relativistic formulas. The derivation of the formula is quite straightforward – but we find it useful to recap it. It is based on a simple analysis of the Lorentz force, which is just the magnetic force here[4]: F = v(q×B). Note that the frequency does not depend on the velocity or the radius of the circular motion. This is actually the whole idea of the trap: the electron can be inserted into the trap with a precise kinetic energy and will follow a circular trajectory if the frequency of the alternating voltage is kept constant. This is why we italicized only when writing that the orbital frequency depends on the charge, the mass and the strength of the magnetic field only. So what is the derivation? The Lorentz force is equal to the centripetal force here. We can therefore write:F4The v2/r factor is the centripetal acceleration. Hence, the F = m·v2/r does effectively represent Newton’s force law. The equation above yields the following formula for v and the v/r ratio:vNow, the cyclotron frequency fc will respect the following equation:F7Re-arranging and substituting v for q·r·b/m yields:F8The associated current will be equal to:F9Hence, the magnetic moment is equal to:F10The angular momentum – which we will denote by – is equal to[5]:F11Hence, we can write the g-factor as:F12It is what we would expect it to be: it is the gyromagnetic ratio for the orbital moment of the electron. It is one, not 2. Because gc is 1, we can write something very obvious:F13We should also note another equality here:F14Let us now look at the other frequency fs. It is the Larmor or precession frequency. It is (also) a classical thing: if we think of the electron as a tiny magnet with a magnetic moment that is proportional to its angular momentum, then it should, effectively, precess in a magnetic field. The analysis of precession is quite straightforward. The geometry of the situation is shown below and we may refer to (almost) any standard physics textbook for the derivation.[6]

precession frequency

pIt is tempting to use the equality above and write this as:F16However, we should not do this. The precession causes the electron to wobble: its plane of rotation – and, hence, the axis of the angular momentum (and the magnetic moment) – is no longer fixed. This wobbling motion changes the orbital and, therefore, we can no longer trust the values we have used in our formulas for the angular momentum and the magnetic moment. There is, therefore, nothing anomalous about the anomalous magnetic moment. In fact, we should not wonder why it is not zero, but – as we will argue – we should wonder why it is so nearly zero.

Let us continue our analysis. It is, in fact, a bit weird to associate a gyromagnetic ratio with this motion, but that is what the physicists doing these experiments do. We will denote this g-factor by gp:F17Hence, we can write the following tautology:F18You can verify that this is nothing but a tautology by writing it all out:F19We can, of course, measure the frequency in cycles per second (as opposed to radians per second):F20Hence, we get the following expression for the so-called anomalous magnetic moment of an electron ae:F21Hence, the so-called anomalous magnetic moment of an electron is nothing but the ratio of two mathematical factors – definitions, basically – which we can express in terms of actual frequencies:F22Our formula for ae now becomes:F23Of course, if we use the μ/J = 2m/q equality, then the fp/fc ratio will be equal to 1/2, and ae will not be zero but −1/2:F24However, as mentioned above, we should not do that. The precession causes the magnetic moment and the angular momentum to wobble. Hence, there is nothing anomalous about the anomalous magnetic moment. We should not wonder why its value is not zero. We should wonder why it is so nearly zero.

[1] Needless to say, the tesla is the SI unit for the magnitude of a magnetic field. We can also write it as [B] = N/(m∙A), using the SI unit for current, i.e. the ampere (A). Now, 1 C = 1 A∙s and, hence, 1 N/(m∙A) = 1 (N/C)/(m/s). Hence, the physical dimension of the magnetic field is the physical dimension of the electric field (N/C) divided by m/s. We like the [E] = [B]·m/s expression because it reflects the geometry of the electric and magnetic field vectors.

[2] See: Physics Today, 1 August 2006, p. 15 (https://physicstoday.scitation.org/doi/10.1063/1.2349714). The article also explains the methodology of the experiment in terms of the frequency measurements, which we explain above.

[3] See: G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom, New Determination of the Fine Structure Constant from the Electron g Value and QED, Phys. Rev. Lett. 97, 030802 (2006). More recent theory and experiments may have come up with an even more precise number.

[4] Our derivation is based on the following reference: https://www.didaktik.physik.uni-muenchen.de/elektronenbahnen/en/b-feld/anwendung/zyklotron2.php.

[5] J is the symbol which Feynman uses. In many articles and textbooks, one will read L instead of J. Note that the symbols may be confusing: I is a current, but I is the moment of inertia. It is equal to m·r2 for a rotating mass.

[6] We like the intuitive – but precise – explanation in Feynman’s Lectures (II-34-3), from which we also copied the illustration.

An intuitive interpretation of Einstein’s mass-energy equivalence relation

My dear readers – I haven’t published much lately, because I try to summarize my ideas now in short articles that might be suitable for publication in a journal. I think the latest one (on Einstein’s mass-energy relation) should be of interest. Let me just insert the summary here:

The radial velocity formula and the Planck-Einstein relation give us the Zitterbewegung (zbw) frequency (E = ħω = E/ħ) and zbw radius (a = c/ω = cħ/mc2 = ħ/mc) of  the electron. We interpret this by noting that the c = aω identity gives us the E = mc2 = ma2ω2 equation, which suggests we should combine the total energy (kinetic and potential) of two harmonic oscillators to explain the electron mass. We do so by interpreting the elementary wavefunction as a two-dimensional (harmonic) electromagnetic oscillation in real space which drives the pointlike charge along the zbw current ring. This implies a dual view of the reality of the real and imaginary part of the wavefunction:

  1. The x = acos(ωt) and y = a·sin(ωt) equations describe the motion of the pointlike charge.
  2. As an electromagnetic oscillation, we write it as E0 = E0cos(ωt+π/2) + i·E0·sin(ωt+π/2).

The magnitudes of the oscillation a and E0 are expressed in distance (m) and force per unit charge (N/C) respectively and are related because the energy of both oscillations is one and the same. The model – which implies the energy of the oscillation and, therefore, the effective mass of the electron is spread over the zbw disk – offers an equally intuitive explanation for the angular momentum, magnetic moment and the g-factor of charged spin-1/2 particles. Most importantly, the model also offers us an intuitive interpretation of Einstein’s enigmatic mass-energy equivalence relation. Going from the stationary to the moving reference frame, we argue that the plane of the zbw oscillation should be parallel to the direction of motion so as to be consistent with the results of the Stern-Gerlach experiment.

So… Well… Have fun with it ! I think I am going to sign off. 🙂 Yours – JL

Polarization states as hidden variables?

This post explores the limits of the physical interpretation of the wavefunction we have been building up in previous posts. It does so by examining if it can be used to provide a hidden-variable theory for explaining quantum-mechanical interference. The hidden variable is the polarization state of the photon.

The outcome is as expected: the theory does not work. Hence, this paper clearly shows the limits of any physical or geometric interpretation of the wavefunction.

This post sounds somewhat academic because it is, in fact, a draft of a paper I might try to turn into an article for a journal. There is a useful addendum to the post below: it offers a more sophisticated analysis of linear and circular polarization states (see: Linear and Circular Polarization States in the Mach-Zehnder Experiment). Have fun with it !

A physical interpretation of the wavefunction

Duns Scotus wrote: pluralitas non est ponenda sine necessitate. Plurality is not to be posited without necessity.[1] And William of Ockham gave us the intuitive lex parsimoniae: the simplest solution tends to be the correct one.[2] But redundancy in the description does not seem to bother physicists. When explaining the basic axioms of quantum physics in his famous Lectures on quantum mechanics, Richard Feynman writes:

“We are not particularly interested in the mathematical problem of finding the minimum set of independent axioms that will give all the laws as consequences. Redundant truth does not bother us. We are satisfied if we have a set that is complete and not apparently inconsistent.”[3]

Also, most introductory courses on quantum mechanics will show that both ψ = exp(iθ) = exp[i(kx-ωt)] and ψ* = exp(-iθ) = exp[-i(kx-ωt)] = exp[i(ωt-kx)] = -ψ are acceptable waveforms for a particle that is propagating in the x-direction. Both have the required mathematical properties (as opposed to, say, some real-valued sinusoid). We would then think some proof should follow of why one would be better than the other or, preferably, one would expect as a discussion on what these two mathematical possibilities might represent¾but, no. That does not happen. The physicists conclude that “the choice is a matter of convention and, happily, most physicists use the same convention.”[4]

Instead of making a choice here, we could, perhaps, use the various mathematical possibilities to incorporate spin in the description, as real-life particles – think of electrons and photons here – have two spin states[5] (up or down), as shown below.

Table 1: Matching mathematical possibilities with physical realities?[6]

Spin and direction Spin up Spin down
Positive x-direction ψ = exp[i(kx-ωt)] ψ* = exp[i(ωt-kx)]
Negative x-direction χ = exp[i(ωt-kx)] χ* = exp[i(kx+ωt)]

That would make sense – for several reasons. First, theoretical spin-zero particles do not exist and we should therefore, perhaps, not use the wavefunction to describe them. More importantly, it is relatively easy to show that the weird 720-degree symmetry of spin-1/2 particles collapses into an ordinary 360-degree symmetry and that we, therefore, would have no need to describe them using spinors and other complicated mathematical objects.[7] Indeed, the 720-degree symmetry of the wavefunction for spin-1/2 particles is based on an assumption that the amplitudes C’up = -Cup and C’down = -Cdown represent the same state—the same physical reality. As Feynman puts it: “Both amplitudes are just multiplied by −1 which gives back the original physical system. It is a case of a common phase change.”[8]

In the physical interpretation given in Table 1, these amplitudes do not represent the same state: the minus sign effectively reverses the spin direction. Putting a minus sign in front of the wavefunction amounts to taking its complex conjugate: -ψ = ψ*. But what about the common phase change? There is no common phase change here: Feynman’s argument derives the C’up = -Cup and C’down = -Cdown identities from the following equations: C’up = eCup and C’down = eCdown. The two phase factors  are not the same: +π and -π are not the same. In a geometric interpretation of the wavefunction, +π is a counterclockwise rotation over 180 degrees, while -π is a clockwise rotation. We end up at the same point (-1), but it matters how we get there: -1 is a complex number with two different meanings.

We have written about this at length and, hence, we will not repeat ourselves here.[9] However, this realization – that one of the key propositions in quantum mechanics is basically flawed – led us to try to question an axiom in quantum math that is much more fundamental: the loss of determinism in the description of interference.

The reader should feel reassured: the attempt is, ultimately, not successful—but it is an interesting exercise.

The loss of determinism in quantum mechanics

The standard MIT course on quantum physics vaguely introduces Bell’s Theorem – labeled as a proof of what is referred to as the inevitable loss of determinism in quantum mechanics – early on. The argument is as follows. If we have a polarizer whose optical axis is aligned with, say, the x-direction, and we have light coming in that is polarized along some other direction, forming an angle α with the x-direction, then we know – from experiment – that the intensity of the light (or the fraction of the beam’s energy, to be precise) that goes through the polarizer will be equal to cos2α.

But, in quantum mechanics, we need to analyze this in terms of photons: a fraction cos2α of the photons must go through (because photons carry energy and that’s the fraction of the energy that is transmitted) and a fraction 1-cos2α must be absorbed. The mentioned MIT course then writes the following:

“If all the photons are identical, why is it that what happens to one photon does not happen to all of them? The answer in quantum mechanics is that there is indeed a loss of determinism. No one can predict if a photon will go through or will get absorbed. The best anyone can do is to predict probabilities. Two escape routes suggest themselves. Perhaps the polarizer is not really a homogeneous object and depending exactly on where the photon is it either gets absorbed or goes through. Experiments show this is not the case.

A more intriguing possibility was suggested by Einstein and others. A possible way out, they claimed, was the existence of hidden variables. The photons, while apparently identical, would have other hidden properties, not currently understood, that would determine with certainty which photon goes through and which photon gets absorbed. Hidden variable theories would seem to be untestable, but surprisingly they can be tested. Through the work of John Bell and others, physicists have devised clever experiments that rule out most versions of hidden variable theories. No one has figured out how to restore determinism to quantum mechanics. It seems to be an impossible task.”[10]

The student is left bewildered here. Are there only two escape routes? And is this the way how polarization works, really? Are all photons identical? The Uncertainty Principle tells us that their momentum, position, or energy will be somewhat random. Hence, we do not need to assume that the polarizer is nonhomogeneous, but we need to think of what might distinguish the individual photons.

Considering the nature of the problem – a photon goes through or it doesn’t – it would be nice if we could find a binary identifier. The most obvious candidate for a hidden variable would be the polarization direction. If we say that light is polarized along the x-direction, we should, perhaps, distinguish between a plus and a minus direction? Let us explore this idea.

Linear polarization states

The simple experiment above – linearly polarized light going through a polaroid – involves linearly polarized light. We can easily distinguish between left- and right-hand circular polarization, but if we have linearly polarized light, can we distinguish between a plus and a minus direction? Maybe. Maybe not. We can surely think about different relative phases and how that could potentially have an impact on the interaction with the molecules in the polarizer.

Suppose the light is polarized along the x-direction. We know the component of the electric field vector along the y-axis[11] will then be equal to Ey = 0, and the magnitude of the x-component of E will be given by a sinusoid. However, here we have two distinct possibilities: Ex = cos(ω·t) or, alternatively, Ex = sin(ω·t). These are the same functions but – crucially important – with a phase difference of 90°: sin(ω·t) = cos(ω·t + π/2).

  Figure 1: Two varieties of linearly polarized light?[12]

600px-Sine_cosine_one_period

Would this matter? Sure. We can easily come up with some classical explanations of why this would matter. Think, for example, of birefringent material being defined in terms of quarter-wave plates. In fact, the more obvious question is: why would this not make a difference?

Of course, this triggers another question: why would we have two possibilities only? What if we add an additional 90° shift to the phase? We know that cos(ω·t + π) = –cos(ω·t). We cannot reduce this to cos(ω·t) or sin(ω·t). Hence, if we think in terms of 90° phase differences, then –cos(ω·t) = cos(ω·t + π)  and –sin(ω·t) = sin(ω·t + π) are different waveforms too. In fact, why should we think in terms of 90° phase shifts only? Why shouldn’t we think of a continuum of linear polarization states?

We have no sensible answer to that question. We can only say: this is quantum mechanics. We think of a photon as a spin-one particle and, for that matter, as a rather particular one, because it misses the zero state: it is either up, or down. We may now also assume two (linear) polarization states for the molecules in our polarizer and suggest a basic theory of interaction that may or may not explain this very basic fact: a photon gets absorbed, or it gets transmitted. The theory is that if the photon and the molecule are in the same (linear) polarization state, then we will have constructive interference and, somehow, a photon gets through.[13] If the linear polarization states are opposite, then we will have destructive interference and, somehow, the photon is absorbed. Hence, our hidden variables theory for the simple situation that we discussed above (a photon does or does not go through a polarizer) can be summarized as follows:

Linear polarization state Incoming photon up (+) Incoming photon down (-)
Polarizer molecule up (+) Constructive interference: photon goes through Destructive interference: photon is absorbed
Polarizer molecule down (-) Destructive interference: photon is absorbed Constructive interference: photon goes through

Nice. No loss of determinism here. But does it work? The quantum-mechanical mathematical framework is not there to explain how a polarizer could possibly work. It is there to explain the interference of a particle with itself. In Feynman’s words, this is the phenomenon “which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics.”[14]

So, let us try our new theory of polarization states as a hidden variable on one of those interference experiments. Let us choose the standard one: the Mach-Zehnder interferometer experiment.

Polarization states as hidden variables in the Mach-Zehnder experiment

The setup of the Mach-Zehnder interferometer is well known and should, therefore, probably not require any explanation. We have two beam splitters (BS1 and BS2) and two perfect mirrors (M1 and M2). An incident beam coming from the left is split at BS1 and recombines at BS2, which sends two outgoing beams to the photon detectors D0 and D1. More importantly, the interferometer can be set up to produce a precise interference effect which ensures all the light goes into D0, as shown below. Alternatively, the setup may be altered to ensure all the light goes into D1.

Figure 2: The Mach-Zehnder interferometer[15]

Mach Zehnder

The classical explanation is easy enough. It is only when we think of the beam as consisting of individual photons that we get in trouble. Each photon must then, somehow, interfere with itself which, in turn, requires the photon to, somehow, go through both branches of the interferometer at the same time. This is solved by the magical concept of the probability amplitude: we think of two contributions a and b (see the illustration above) which, just like a wave, interfere to produce the desired result¾except that we are told that we should not try to think of these contributions as actual waves.

So that is the quantum-mechanical explanation and it sounds crazy and so we do not want to believe it. Our hidden variable theory should now show the photon does travel along one path only. If the apparatus is set up to get all photons in the D0 detector, then we might, perhaps, have a sequence of events like this:

Photon polarization At BS1 At BS2 Final result
Up (+) Photon is reflected Photon is reflected Photon goes to D0
Down () Photon is transmitted Photon is transmitted Photon goes to D0

 

Of course, we may also set up the apparatus to get all photons in the D1 detector, in which case the sequence of events might be this:

Photon polarization At BS1 At BS2 Final result
Up (+) Photon is reflected Photon is transmitted Photon goes to D1
Down () Photon is transmitted Photon is reflected Photon goes to D1

This is a nice symmetrical explanation that does not involve any quantum-mechanical weirdness. The problem is: it cannot work. Why not? What happens if we block one of the two paths? For example, let us block the lower path in the setup where all photons went to D0. We know – from experiment – that the outcome will be the following:

Final result Probability
Photon is absorbed at the block 0.50
Photon goes to D0 0.25
Photon goes to D1 0.25

How is this possible? Before blocking the lower path, no photon went to D1. They all went to D0. If our hidden variable theory was correct, the photons that do not get absorbed should also go to D0, as shown below.

Photon polarization At BS1 At BS2 Final result
Up (+) Photon is reflected Photon is reflected Photon goes to D0
Down () Photon is absorbed Photon was absorbed Photon was absorbed

Conclusion

Our hidden variable theory does not work. Physical or geometric interpretations of the wavefunction are nice, but they do not explain quantum-mechanical interference. Their value is, therefore, didactic only.

Jean Louis Van Belle, 2 November 2018

References

This paper discusses general principles in physics only. Hence, references were limited to references to general textbooks and courses and physics textbooks only. The two key references here are the MIT introductory course on quantum physics and Feynman’s Lectures – both of which can be consulted online. Additional references to other material are given in the text itself (see footnotes).

[1] Duns Scotus, Commentaria.

[2] See: https://en.wikipedia.org/wiki/Occam%27s_razor.

[3] Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 5, Section 5.

[4] See, for example, the MIT’s edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 4, Section 3.

[5] Photons are spin-one particles but they do not have a spin-zero state.

[6] Of course, the formulas only give the elementary wavefunction. The wave packet will be a Fourier sum of such functions.

[7] See, for example, https://warwick.ac.uk/fac/sci/physics/staff/academic/mhadley/explanation/spin/, accessed on 30 October 2018

[8] Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 6, Section 3.

[9] Jean Louis Van Belle, Euler’s wavefunction (http://vixra.org/abs/1810.0339, accessed on 30 October 2018)

[10] See: MIT edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 1, Section 3 (Loss of determinism).

[11] The z-direction is the direction of wave propagation in this example. In quantum mechanics, we often define the direction of wave propagation as the x-direction. This will, hopefully, not confuse the reader. The choice of axes is usually clear from the context.

[12] Source of the illustration: https://upload.wikimedia.org/wikipedia/commons/7/71/Sine_cosine_one_period.svg..

[13] Classical theory assumes an atomic or molecular system will absorb a photon and, therefore, be in an excited state (with higher energy). The atomic or molecular system then goes back into its ground state by emitting another photon with the same energy. Hence, we should probably not think in terms of a specific photon getting through.

[14] Feynman’s Lectures on Quantum Mechanics, Vol. III, Chapter 1, Section 1.

[15] Source of the illustration: MIT edX Course 8.04.1x (Quantum Physics), Lecture Notes, Chapter 1, Section 4 (Quantum Superpositions).

Surely You’re Joking, Mr Feynman !

I think I cracked the nut. Academics always throw two nasty arguments into the discussion on any geometric or physical interpretations of the wavefunction:

  1. The superposition of wavefunctions is done in the complex space and, hence, the assumption of a real-valued envelope for the wavefunction is, therefore, not acceptable.
  2. The wavefunction for spin-1/2 particles cannot represent any real object because of its 720-degree symmetry in space. Real objects have the same spatial symmetry as space itself, which is 360 degrees. Hence, physical interpretations of the wavefunction are nonsensical.

Well… I’ve finally managed to deconstruct those arguments – using, paradoxically, Feynman’s own arguments against him. Have a look: click the link to my latest paper ! Enjoy !

The metaphysics of physics

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[1]: 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[2]: 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. 😊

[1] 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.

[2] 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·eiθ.

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·eiθ 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·eiθ and A·e+iθ as two-dimensional harmonic oscillators:

eiθ = cos(-θ) + i·sin(-θ) = cosθ – i·sinθ

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

Circle_cos_sin

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?

V2

Previously unrelated but structurally similar formulas may be related here:

  1. The energy of an oscillator: E = (1/2)·m·a2ω2
  2. Kinetic energy: E = (1/2)·m·v2
  3. The rotational (kinetic) energy that’s stored in a flywheel: E = (1/2)·I·ω2 = (1/2)·m·r2·ω2
  4. 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.

Precession

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 fieldprecession

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…

Well… No.

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.

Certainty and uncertainty

A lot of the Uncertainty in quantum mechanics is suspiciously certain. For example, we know an electron will always have its spin up or down, in any direction along which we choose to measure it, and the value of the angular momentum will, accordingly, be measured as plus or minus ħ/2. That doesn’t sound uncertain to me. In fact, it sounds remarkably certain, doesn’t it? We know – we are sure, in fact, because of countless experiments – that the electron will be in either of those two states, and we also know that these two states are separated by ħ, Planck’s quantum of action, exactly.

Of course, the corollary of this is that the idea of the direction of the angular momentum is a rather fuzzy concept. As Feynman convincingly demonstrates, it is ‘never completely along any direction’. Why? Well… Perhaps it can be explained by the idea of precession?

In fact, the idea of precession might also explain the weird 720° degree symmetry of the wavefunction.

Hmm… Now that is an idea to look into ! 🙂

A Survivor’s Guide to Quantum Mechanics?

When modeling electromagnetic waves, the notion of left versus right circular polarization is quite clear and fully integrated in the mathematical treatment. In contrast, quantum math sticks to the very conventional idea that the imaginary unit (i) is – always! – a counter-clockwise rotation by 90 degrees. We all know that –i would do just as an imaginary unit as i, because the definition of the imaginary unit says the only requirement is that its square has to be equal to –1, and (–i)2 is also equal to –1.

So we actually have two imaginary units: i and –i. However, physicists stubbornly think there is only one direction for measuring angles, and that is counter-clockwise. That’s a mathematical convention, Professor: it’s something in your head only. It is not real. Nature doesn’t care about our conventions and, therefore, I feel the spin ‘up’ versus spin ‘down’ should correspond to the two mathematical possibilities: if the ‘up’ state is represented by some complex function, then the ‘down’ state should be represented by its complex conjugate.

This ‘additional’ rule wouldn’t change the basic quantum-mechanical rules – which are written in terms of state vectors in a Hilbert space (and, yes, a Hilbert space is an unreal as it sounds: its rules just say you should separate cats and dogs while adding them – which is very sensible advice, of course). However, they would, most probably (just my intuition – I need to prove it), avoid these crazy 720 degree symmetries which inspire the likes of Penrose to say there is no physical interpretation on the wavefunction.

Oh… As for the title of my post… I think it would be a great title for a book – because I’ll need some space to work it all out. 🙂

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.

Should we reinvent wavefunction math?

Preliminary note: This post may cause brain damage. 🙂 If you haven’t worked yourself through a good introduction to physics – including the math – you will probably not understand what this is about. So… Well… Sorry. 😦 But if you have… Then this should be very interesting. Let’s go. 🙂

If you know one or two things about quantum math – Schrödinger’s equation and all that – then you’ll agree the math is anything but straightforward. Personally, I find the most annoying thing about wavefunction math are those transformation matrices: every time we look at the same thing from a different direction, we need to transform the wavefunction using one or more rotation matrices – and that gets quite complicated !

Now, if you have read any of my posts on this or my other blog, then you know I firmly believe the wavefunction represents something real or… Well… Perhaps it’s just the next best thing to reality: we cannot know das Ding an sich, but the wavefunction gives us everything we would want to know about it (linear or angular momentum, energy, and whatever else we have an operator for). So what am I thinking of? Let me first quote Feynman’s summary interpretation of Schrödinger’s equation (Lectures, III-16-1):

“We can think of Schrödinger’s equation as describing the diffusion of the probability amplitude from one point to the next. […] But the imaginary coefficient in front of the derivative makes the behavior completely different from the ordinary diffusion such as you would have for a gas spreading out along a thin tube. Ordinary diffusion gives rise to real exponential solutions, whereas the solutions of Schrödinger’s equation are complex waves.”

Feynman further formalizes this in his Lecture on Superconductivity (Feynman, III-21-2), in which he refers to Schrödinger’s equation as the “equation for continuity of probabilities”. His analysis there is centered on the local conservation of energy, which makes me think Schrödinger’s equation might be an energy diffusion equation. I’ve written about this ad nauseam in the past, and so I’ll just refer you to one of my papers here for the details, and limit this post to the basics, which are as follows.

The wave equation (so that’s Schrödinger’s equation in its non-relativistic form, which is an approximation that is good enough) is written as:formula 1The resemblance with the standard diffusion equation (shown below) is, effectively, very obvious:formula 2As Feynman notes, it’s just that imaginary coefficient that makes the behavior quite different. How exactly? Well… You know we get all of those complicated electron orbitals (i.e. the various wave functions that satisfy the equation) out of Schrödinger’s differential equation. We can think of these solutions as (complex) standing waves. They basically represent some equilibrium situation, and the main characteristic of each is their energy level. I won’t dwell on this because – as mentioned above – I assume you master the math. Now, you know that – if we would want to interpret these wavefunctions as something real (which is surely what want to do!) – the real and imaginary component of a wavefunction will be perpendicular to each other. Let me copy the animation for the elementary wavefunction ψ(θ) = a·ei∙θ = a·ei∙(E/ħ)·t = a·cos[(E/ħ)∙t]  i·a·sin[(E/ħ)∙t] once more:

Circle_cos_sin

So… Well… That 90° angle makes me think of the similarity with the mathematical description of an electromagnetic wave. Let me quickly show you why. For a particle moving in free space – with no external force fields acting on it – there is no potential (U = 0) and, therefore, the Vψ term – which is just the equivalent of the the sink or source term S in the diffusion equation – disappears. Therefore, Schrödinger’s equation reduces to:

∂ψ(x, t)/∂t = i·(1/2)·(ħ/meff)·∇2ψ(x, t)

Now, the key difference with the diffusion equation – let me write it for you once again: ∂φ(x, t)/∂t = D·∇2φ(x, t) – is that Schrödinger’s equation gives us two equations for the price of one. Indeed, because ψ is a complex-valued function, with a real and an imaginary part, we get the following equations:

  1. Re(∂ψ/∂t) = −(1/2)·(ħ/meffIm(∇2ψ)
  2. Im(∂ψ/∂t) = (1/2)·(ħ/meffRe(∇2ψ)

Huh? Yes. These equations are easily derived from noting that two complex numbers a + i∙b and c + i∙d are equal if, and only if, their real and imaginary parts are the same. Now, the ∂ψ/∂t = i∙(ħ/meff)∙∇2ψ equation amounts to writing something like this: a + i∙b = i∙(c + i∙d). Now, remembering that i2 = −1, you can easily figure out that i∙(c + i∙d) = i∙c + i2∙d = − d + i∙c. [Now that we’re getting a bit technical, let me note that the meff is the effective mass of the particle, which depends on the medium. For example, an electron traveling in a solid (a transistor, for example) will have a different effective mass than in an atom. In free space, we can drop the subscript and just write meff = m.] 🙂 OK. Onwards ! 🙂

The equations above make me think of the equations for an electromagnetic wave in free space (no stationary charges or currents):

  1. B/∂t = –∇×E
  2. E/∂t = c2∇×B

Now, these equations – and, I must therefore assume, the other equations above as well – effectively describe a propagation mechanism in spacetime, as illustrated below:

propagation

You know how it works for the electromagnetic field: it’s the interplay between circulation and flux. Indeed, circulation around some axis of rotation creates a flux in a direction perpendicular to it, and that flux causes this, and then that, and it all goes round and round and round. 🙂 Something like that. 🙂 I will let you look up how it goes, exactly. The principle is clear enough. Somehow, in this beautiful interplay between linear and circular motion, energy is borrowed from one place and then returns to the other, cycle after cycle.

Now, we know the wavefunction consist of a sine and a cosine: the cosine is the real component, and the sine is the imaginary component. Could they be equally real? Could each represent half of the total energy of our particle? I firmly believe they do. The obvious question then is the following: why wouldn’t we represent them as vectors, just like E and B? I mean… Representing them as vectors (I mean real vectors here – something with a magnitude and a direction in a real space – as opposed to state vectors from the Hilbert space) would show they are real, and there would be no need for cumbersome transformations when going from one representational base to another. In fact, that’s why vector notation was invented (sort of): we don’t need to worry about the coordinate frame. It’s much easier to write physical laws in vector notation because… Well… They’re the real thing, aren’t they? 🙂

What about dimensions? Well… I am not sure. However, because we are – arguably – talking about some pointlike charge moving around in those oscillating fields, I would suspect the dimension of the real and imaginary component of the wavefunction will be the same as that of the electric and magnetic field vectors E and B. We may want to recall these:

  1. E is measured in newton per coulomb (N/C).
  2. B is measured in newton per coulomb divided by m/s, so that’s (N/C)/(m/s).

The weird dimension of B is because of the weird force law for the magnetic force. It involves a vector cross product, as shown by Lorentz’ formula:

F = qE + q(v×B)

Of course, it is only one force (one and the same physical reality), as evidenced by the fact that we can write B as the following vector cross-product: B = (1/c)∙ex×E, with ex the unit vector pointing in the x-direction (i.e. the direction of propagation of the wave). [Check it, because you may not have seen this expression before. Just take a piece of paper and think about the geometry of the situation.] Hence, we may associate the (1/c)∙ex× operator, which amounts to a rotation by 90 degrees, with the s/m dimension. Now, multiplication by i also amounts to a rotation by 90° degrees. Hence, if we can agree on a suitable convention for the direction of rotation here, we may boldly write:

B = (1/c)∙ex×E = (1/c)∙iE

This is, in fact, what triggered my geometric interpretation of Schrödinger’s equation about a year ago now. I have had little time to work on it, but think I am on the right track. Of course, you should note that, for an electromagnetic wave, the magnitudes of E and B reach their maximum, minimum and zero point simultaneously (as shown below). So their phase is the same.

E and B

In contrast, the phase of the real and imaginary component of the wavefunction is not the same, as shown below.wavefunction

In fact, because of the Stern-Gerlach experiment, I am actually more thinking of a motion like this:

Wavefunction 2But that shouldn’t distract you. 🙂 The question here is the following: could we possibly think of a new formulation of Schrödinger’s equation – using vectors (again, real vectors – not these weird state vectors) rather than complex algebra?

I think we can, but then I wonder why the inventors of the wavefunction – Heisenberg, Born, Dirac, and Schrödinger himself, of course – never thought of that. 🙂

Hmm… I need to do some research here. 🙂

Post scriptum: You will, of course, wonder how and why the matter-wave would be different from the electromagnetic wave if my suggestion that the dimension of the wavefunction component is the same is correct. The answer is: the difference lies in the phase difference and then, most probably, the different orientation of the angular momentum. Do we have any other possibilities? 🙂

P.S. 2: I also published this post on my new blog: https://readingeinstein.blog/. However, I thought the followers of this blog should get it first. 🙂

Superconductivity and flux quantization

This post continues my mini-series on Feynman’s Seminar on Superconductivity. Superconductivity is a state which produces many wondrous phenomena, but… Well… The flux quantization phenomenon may not be part of your regular YouTube feed but, as far as I am concerned, it may well be the most amazing manifestation of a quantum-mechanical phenomenon at a macroscopic scale. I mean… Super currents that keep going, with zero resistance, are weird—they explain how we can trap a magnetic flux in the first place—but the fact that such fluxes are quantized is even weirder.

The key idea is the following. When we cool a ring-shaped piece of superconducting material in a magnetic field, all the way down to the critical temperature that causes the electrons to condense into a superconducting fluid, then a super current will emerge—think of an eddy current, here, but with zero resistance—that will force the magnetic field out of the material, as shown below. This current will permanently trap some of the magnetic field, even when the external field is being removed. As said, that’s weird enough by itself but… Well… If we think of the super current as an eddy current encountering zero resistance, then the idea of a permanently trapped magnetic field makes sense, right? In case you’d doubt the effect… Well… Just watch one of the many videos on the effect on YouTube. 🙂 The amazing thing here is not the permanently trapped magnetic field, but the fact that it’s quantized.

trapped flux

To be precise, the trapped flux will always be an integer times 2πħ/q. In other words, the magnetic field which Feynman denotes by Φ (the capitalized Greek letter phi), will always be equal to:

Φ = 2πħ/q, with = 0, 1, 2, 3,…

Hence, the flux can be 0, 2πħ/q, 4πħ/q, 6πħ/q , and so on. The fact that it’s a multiple of 2π shows us it’s got to do with the fact that our piece of material is, effectively, a ring. The nice thing about this phenomenon is that the mathematical analysis is, in fact, fairly easy to follow—or… Well… Much easier than what we discussed before. 🙂 Let’s quickly go through it.

We have a formula for the magnetic flux. It must be equal to the line integral of the vector potential (A) around a closed loop Τ, so we write:

flux

Now, we can choose the loop Τ to be well inside the body of the ring, so that it never gets near the surface, as illustrated below. So we know that the current J is zero there. [In case you doubt this, see my previous post.]

curve

One of the equations we introduced in our previous post, ħθ = m·v + q·A, will then reduce to:

ħθ = q·A

Why? The v in the m·v term (the velocity of the superconducting fluid, really), is zero. Remember the analysis is for this particular loop (well inside the ring) only. So… Well… If we integrate the expression above, we get:

integral

Combining the two expressions with the integrals, we get:

integral 2Now, the line integral of a gradient from one point to another (say from point 1 to point 2) is the difference of the values of the function at the two points, so we can write:

integral 3

Now what constraints are there on the values of θ1 and θ2? Well… You might think that, if they’re associated with the same point (we’re talking a closed loop, right?), then the two values should be the same, but… Well… No. All we can say is that the wavefunction must have the same value. We wrote that wavefunction as:

ψ = ρ(r)1/2eθ(r)

The value of this function at some point r is the same if θ changes by 2π. Hence, when doing one complete turn around the ring, the ∫∇θ·ds integral in the integral formulas we wrote down must be equal to 2π. Therefore, the second integral expression above can be re-written as:

result

That’s the result we wanted to explain so… Well… We’re done. Let me wrap up by quoting Feynman’s account of the 1961 experiment which confirmed London’s prediction of the effect, which goes back to 1950! It’s interesting, because… Well… It shows how up to date Feynman’s Lectures really are—or were, back in 1963, at least!feynman overview of experiment

Feynman’s Seminar on Superconductivity (2)

We didn’t get very far in our first post on Feynman’s Seminar on Superconductivity, and then I shifted my attention to other subjects over the past few months. So… Well… Let me re-visit the topic here.

One of the difficulties one encounters when trying to read this so-called seminar—which, according to Feynman, is ‘for entertainment only’ and, therefore, not really part of the Lectures themselves—is that Feynman throws in a lot of stuff that is not all that relevant to the topic itself but… Well… He apparently didn’t manage to throw all that he wanted to throw into his (other) Lectures on Quantum Mechanics and so he inserted a lot of stuff which he could, perhaps, have discussed elsewhere. :-/ So let us try to re-construct the main lines of reasoning here.

The first equation is Schrödinger’s equation for some particle with charge q that is moving in an electromagnetic field that is characterized not only by the (scalar) potential Φ but also by a vector potential A:

schrodinger

This closely resembles Schrödinger’s equation for an electron that is moving in an electric field only, which we used to find the energy states of electrons in a hydrogen atom: i·ħ·∂ψ/∂t = −(1/2)·(ħ2/m)∇2ψ + V·ψ. We just need to note the following:

  1. On the left-hand side, we can, obviously, replace −1/i by i.
  2. On the right-hand side, we can replace V by q·Φ, because the potential of a charge in an electric field is the product of the charge (q) and the (electric) potential (Φ).
  3. As for the other term on the right-hand side—so that’s the −(1/2)·(ħ2/m)∇2ψ term—we can re-write −ħ2·∇2ψ as [(ħ/i)·∇]·[(ħ/i)·∇]ψ because (1/i)·(1/i) = 1/i2 = 1/(−1) = −1. 🙂
  4. So all that’s left now, is that additional −q·A term in the (ħ/i)∇ − q·A expression. In our post, we showed that’s easily explained because we’re talking magnetodynamics: we’ve got to allow for the possibility of changing magnetic fields, and so that’s what the −q·A term captures.

Now, the latter point is not so easy to grasp but… Well… I’ll refer you that first post of mine, in which I show that some charge in a changing magnetic field will effectively gather some extra momentum, whose magnitude will be equal to p = m·v = −q·A. So that’s why we need to introduce another momentum operator here, which we write as:

new-momentum-operator

OK. Next. But… Then… Well… All of what follows are either digressions—like the section on the local conservation of probabilities—or, else, quite intuitive arguments. Indeed, Feynman does not give us the nitty-gritty of the Bardeen-Cooper-Schrieffer theory, nor is the rest of the argument nearly as rigorous as the derivation of the electron orbitals from Schrödinger’s equation in an electrostatic field. So let us closely stick to what he does write, and try our best to follow the arguments.

Cooper pairs

The key assumption is that there is some attraction between electrons which, at low enough temperatures, can overcome the Coulomb repulsion. Where does this attraction come from? Feynman does not give us any clues here. He just makes a reference to the BCS theory but notes this theory is “not the subject of this seminar”, and that we should just “accept the idea that the electrons do, in some manner or other, work in pairs”, and that “we can think of thos−e pairs as behaving more or less like particles”, and that “we can, therefore, talk about the wavefunction for a pair.”

So we have a new particle, so to speak, which consists of two electrons who move through the conductor as one. To be precise, the electron pair behaves as a boson. Now, bosons have integer spin. According to the spin addition rule, we have four possibilities here but only three possible values:− 1/2 + 1/2 = 1; −1/2 + 1/2 = 0; +1/2 − 1/2 = 0; −1/2 − 1/2 = − 1. Of course, it is tempting to think these Cooper pairs are just like the electron pairs in the atomic orbitals, whose spin is always opposite because of the Fermi exclusion principle. Feynman doesn’t say anything about this, but the Wikipedia article on the BCS theory notes that the two electrons in a Cooper pair are, effectively, correlated because of their opposite spin. Hence, we must assume the Cooper pairs effectively behave like spin-zero particles.

Now, unlike fermions, bosons can collectively share the same energy state. In fact, they are likely to share the same state into what is referred to as a Bose-Einstein condensate. As Feynman puts it: “Since electron pairs are bosons, when there are a lot of them in a given state there is an especially large amplitude for other pairs to go to the same state. So nearly all of the pairs will be locked down at the lowest energy in exactly the same state—it won’t be easy to get one of them into another state. There’s more amplitude to go into the same state than into an unoccupied state by the famous factor √n, where n−1 is the occupancy of the lowest state. So we would expect all the pairs to be moving in the same state.”

Of course, this only happens at very low temperatures, because even if the thermal energy is very low, it will give the electrons sufficient energy to ensure the attractive force is overcome and all pairs are broken up. It is only at very low temperature that they will pair up and go into a Bose-Einstein condensate. Now, Feynman derives this √n factor in a rather abstruse introductory Lecture in the third volume, and I’d advise you to google other material on Bose-Einstein statistics because… Well… The mentioned Lecture is not among Feynman’s finest. OK. Next step.

Cooper pairs and wavefunctions

We know the probability of finding a Cooper pair is equal to the absolute square of its wavefunction. Now, it is very reasonable to assume that this probability will be proportional to the charge density (ρ), so we can write:

|ψ|= ψψ* ∼ ρ(r)

The argument here (r) is just the position vector. The next step, then, is to write ψ as the square root of ρ(r) times some phase factor θ. Abstracting away from time, this phase factor will also depend on r, of course. So this is what Feynman writes:

ψ = ρ(r)1/2eθ(r)

As Feynman notes, we can write any complex function of r like this but… Well… The charge density is, obviously, something real. Something we can measure, so we’re not writing the obvious here. The next step is even less obvious.

In our first post, we spent quite some time on Feynman’s digression on the local conservation of probability and… Well… I wrote above I didn’t think this digression was very useful. It now turns out it’s a central piece in the puzzle that Feynman is trying to solve for us here. The key formula here is the one for the so-called probability current, which—as Feynman shows—we write as:

probability-current-2

This current J can also be written as:

probability-current-1

Now, Feynman skips all of the math here (he notes “it’s just a change of variables” but so he doesn’t want to go through all of the algebra), and so I’ll just believe him when he says that, when substituting ψ for our wavefunction ψ = ρ(r)1/2eθ(r), then we can express this ‘current’ (J) in terms of ρ and θ. To be precise, he writes J as: current formulaSo what? Well… It’s really fascinating to see what happens next. While J was some rather abstract concept so far—what’s a probability current, really?—Feynman now suggests we may want to think of it as a very classical electric current—the charge density times the velocity of the fluid of electrons. Hence, we equate J to J =  ρ·v. Now, if the equation above holds true, but J is also equal to J = ρ·v, then the equation above is equivalent to:

moment

Now, that gives us a formula for ħθ. We write:

ħθ = m·v + q·A

Now, in my previous post on this Seminar, I noted that Feynman attaches a lot of importance to this m·v + q·A quantity because… Well… It’s actually an invariant quantity. The argument can be, very briefly, summarized as follows. During the build-up of (or a change in) a magnetic flux, a charge will pick up some (classical) momentum that is equal to p = m·v = −q·A. Hence, the m·v + q·A sum is zero, and so… Well… That’s it, really: it’s some quantity that… Well… It has a significance in quantum mechanics. What significance? Well… Think of what we’ve been writing here. The v and the A have a physical significance, obviously. Therefore, that phase factor θ(r) must also have a physical significance.

But the question remains: what physical significance, exactly? Well… Let me quote Feynman here:

“The phase is just as observable as the charge density ρ. It is a piece of the current density J. The absolute phase (θ) is not observable, but if the gradient of the phase (θ) is known everywhere, then the phase is known except for a constant. You can define the phase at one point, and then the phase everywhere is determined.”

That makes sense, doesn’t it? But it still doesn’t quite answer the question: what is the physical significance of θ(r). What is it, really? We may be able to answer that question after exploring the equations above a bit more, so let’s do that now.

Superconductivity

The phenomenon of superconductivity itself is easily explained by the mentioned condensation of the Cooper pairs: they all go into the same energy state. They form, effectively, a superconducting fluid. Feynman’s description of this is as follows:

“There is no electrical resistance. There’s no resistance because all the electrons are collectively in the same state. In the ordinary flow of current you knock one electron or the other out of the regular flow, gradually deteriorating the general momentum. But here to get one electron away from what all the others are doing is very hard because of the tendency of all Bose particles to go in the same state. A current once started, just keeps on going forever.”

Frankly, I’ve re-read this a couple of times, but I don’t think it’s the best description of what we think is going on here. I’d rather compare the situation to… Well… Electrons moving around in an electron orbital. That’s doesn’t involve any radiation or energy transfer either. There’s just movement. Flow. The kind of flow we have in the wavefunction itself. Here I think the video on Bose-Einstein condensates on the French Tout est quantique site is quite instructive: all of the Cooper pairs join to become one giant wavefunction—one superconducting fluid, really. 🙂

OK… Next.

The Meissner effect

Feynman describes the Meissner effect as follows:

“If you have a piece of metal in the superconducting state and turn on a magnetic field which isn’t too strong (we won’t go into the details of how strong), the magnetic field can’t penetrate the metal. If, as you build up the magnetic field, any of it were to build up inside the metal, there would be a rate of change of flux which would produce an electric field, and an electric field would immediately generate a current which, by Lenz’s law, would oppose the flux. Since all the electrons will move together, an infinitesimal electric field will generate enough current to oppose completely any applied magnetic field. So if you turn the field on after you’ve cooled a metal to the superconducting state, it will be excluded.

Even more interesting is a related phenomenon discovered experimentally by Meissner. If you have a piece of the metal at a high temperature (so that it is a normal conductor) and establish a magnetic field through it, and then you lower the temperature below the critical temperature (where the metal becomes a superconductor), the field is expelled. In other words, it starts up its own current—and in just the right amount to push the field out.”

The math here is interesting. Feynman first notes that, in any lump of superconducting metal, the divergence of the current must be zero, so we write:  ∇·J = 0. At any point? Yes. The current that goes in must go out. No point is a sink or a source. Now the divergence operator (∇·J) is a linear operator. Hence, that means that, when applying the divergence operator to the J = (ħ/m)·[θ − (q/ħ)·A]·ρ equation, we’ll need to figure out what ∇·θ =   = ∇2θ and ∇·A are. Now, as explained in my post on gauges, we can choose to make ∇·A equal to zero so… Well… We’ll make that choice and, hence, the term with ∇·A in it vanishes. So… Well… If ∇·J equals zero, then the term with ∇2θ has to be zero as well, so ∇2θ has to be zero. That, in turn, implies θ has to be some constant (vector).

Now, there is a pretty big error in Feynman’s Lecture here, as it notes: “Now the only way that ∇2θ can be zero everywhere inside the lump of metal is for θ to be a constant.” It should read: ∇2θ can only be zero everywhere if θ is a constant (vector). So now we need to remind ourselves of the reality of θ, as described by Feynman (quoted above): “The absolute phase (θ) is not observable, but if the gradient of the phase (θ) is known everywhere, then the phase is known except for a constant. You can define the phase at one point, and then the phase everywhere is determined.” So we can define, or choose, our constant (vector) θ to be 0.

Hmm… We re-set not one but two gauges here: A and θ. Tricky business, but let’s go along with it. [If we want to understand Feynman’s argument, then we actually have no choice than to go long with his argument, right?] The point is: the (ħ/m)·θ term in the J = (ħ/m)·[θ − (q/ħ)·A]·ρ vanishes, so the equation we’re left with tells us the current—so that’s an actual as well as a probability current!—is proportional to the vector potential:

currentNow, we’ve neglected any possible variation in the charge density ρ so far because… Well… The charge density in a superconducting fluid must be uniform, right? Why? When the metal is superconducting, an accumulation of electrons in one region would be immediately neutralized by a current, right? [Note that Feynman’s language is more careful here. He writes: the charge density is almost perfectly uniform.]

So what’s next? Well… We have a more general equation from the equations of electromagnetism:

A and J

[In case you’d want to know how we get this equation out of Maxwell’s equations, you can look it up online in one of the many standard textbooks on electromagnetism.] You recognize this as a Poisson equation… Well… Three Poisson equations: one for each component of A and J. We can now combine the two equations above by substituting in that Poisson equation, so we get the following differential equation, which we need to solve for A:

A

The λ2 in this equation is, of course, a shorthand for the following constant:

lambda

Now, it’s very easy to see that both e−λr as well as e−λr are solutions for that Poisson equation. But what do they mean? In one dimension, r becomes the one-dimensional position variable x. You can check the shapes of these solutions with a graphing tool.

graph

Note that only one half of each graph counts: the vector potential must decrease when we go from the surface into the material, and there is a cut-off at the surface of the material itself, of course. So all depends on the size of λ, as compared to the size of our piece of superconducting metal (or whatever other substance our piece is made of). In fact, if we look at e−λx as as an exponential decay function, then τ = 1/λ is the so-called scaling constant (it’s the inverse of the decay constant, which is λ itself). [You can work this out yourself. Note that for = τ = 1/λ, the value of our function e−λx will be equal to e−λ(1/λ) = e−1 ≈ 0.368, so it means the value of our function is reduced to about 36.8% of its initial value. For all practical purposes, we may say—as Feynman notes—that the field will, effectively, only penetrate to a thin layer at the surface: a layer of about 1/1/λ in thickness. He illustrates this as follows:

illustration

Moreover, he calculates the 1/λ distance for lead. Let me copy him here:

calculation

Well… That says it all, right? We’re talking two millionths of a centimeter here… 🙂

So what’s left? A lot, like flux quantization, or the equations of motion for the superconducting electron fluid. But we’ll leave that for the next posts. 🙂