Introductions…

As mentioned in my previous post, I thought it might be a good idea to release bits and pieces of my book on the go. It gives me the feeling I get something done, at least. 🙂

Introduction

If you are reading this book, then you are like me. You want to understand. Something inside of you tells you that the idea that we will never be able to understand quantum mechanics “the way we would like to understand it” does not make any sense. The quote is from Richard Feynman – probably the most eminent of all post-World War II physicists – and, yes, this is an aggressive opening. But you are right: our mind is flexible. We can imagine weird shapes and hybrid models. Hence, we should be able to understand quantum physics in some kind of intuitive way.

But what is intuitive? A lot of the formulas in this book feel intuitive to me, but that is only because I have been working with them for quite a while now. They may not feel very intuitive to you. However, I have confidence you will also sort of understand what they represent – intuitively, that is – because all of the formulas I use represent something we can imagine in space and in time – and I mean 3D space here: our Universe. Not the Universe of strings and hidden dimensions. An intuitive understanding of things means an understanding in terms of their geometry and their physicality.

You bought the right book. No hocus-pocus here. All physicists – and popular writers on physics – will tell you it is not possible. You see, the wavefunction of a particle – say, an electron – has this weird 720-degree symmetry, which we cannot really imagine. Of course, we have these professors doing the Dirac belt trick on YouTube – and wonderful animations (Jason Hise – whom I’ve been in touch with – makes the best ones) but, still, these visualizations all assume some weird relation between the object and the subject. In short, we cannot really imagine an object with a 720-degree symmetry.

The good news is: you don’t have to. The early theorists made a small mistake: they did not fully exploit the power of Euler’s ubiquitous a·eiθ function. The mistake is illustrated below – but don’t worry if this looks like you won’t understand: we’ll come back to it. It is a very subtle thing. Quantum physicists will tell you they don’t really think of the elementary wavefunction as representing anything real but, in fact, they do. And they will tell you, rather reluctantly because they are not so sure about what is what, that it might represent some theoretical spin-zero particle. Now, we all know spin-zero particles do not exist. All real particles – electrons, photons, anything – have spin, and spin (a shorthand for angular momentum) is always in one direction or the other: it is just the magnitude of the spin that differs. So it’s completely odd that the plus (+) or the minus (-) sign of the imaginary unit (i) in the a·e±iθ function is not being used to include the spin direction in the mathematical description.

clock

Figure 1: The meaning of +i and –i

Indeed, most introductory courses in quantum mechanics will show that both a·ei·θ = a·ei·(wtkx) and a·e+i·θ = a·e+i·(wtkx) are acceptable waveforms for a particle that is propagating in a given direction (as opposed to, say, some real-valued sinusoid). We would think physicists would then proceed to provide some argument showing why one would be better than the other, or some discussion on why they might be different, but that is not the case. The professors usually conclude that “the choice is a matter of convention” and, that “happily, most physicists use the same convention.” In case you wonder, this is a quote from the MIT’s edX course on quantum mechanics (8.01.1x).

That leads to the false argument that the wavefunction of spin-½ particles have a 720-degree symmetry. Again, you should not worry if you don’t get anything of what I write here – because I will come back to it – but the gist of the matter is the following: because they think the elementary wavefunction describes some theoretical zero-spin particle, physicists treat -1 as a common phase factor: they think we can just multiply a set of amplitudes – let’s say two amplitudes, to focus our mind (think of a beam splitter or alternative paths here) – with -1 and we’re going to get the same states. We find it rather obvious that that is not necessarily the case: -1 is not necessarily a common phase factor. We should think of -1 as a complex number itself: the phase factor may be +π or, alternatively, -π. To put it simply, when going from +1 to -1, it matters how you get there – and vice versa – as illustrated below.

rotation

Figure 2: e+iπ ¹ eiπ

I know this sounds like a bad start for a book that promises to be easy – but I just thought it would be good to be upfront about why this book is very different than anything you’ve ever read about quantum physics. If we exploit the full descriptive power of Euler’s function, then all weird symmetries disappear – and we just talk standard 360-degree symmetries in space. Also, weird mathematical conditions – such as the Hermiticity of quantum-mechanical operators – can easily be explained as embodying some common-sense physical law. In this particular case (Hermitian operators), we are talking physical reversibility: when we see something happening at the elementary particle level, then we need to be able to play the movie backwards. Physicists refer to it as CPT-symmetry, but that’s what it is really: physical reversibility.

The argument above involved geometry, and this brings me to a second mistake of the early quantum physicists: a total neglect of what I refer to as the form factor in physics. Why would an electron be some perfect sphere, or some perfect disk? In fact, we will argue it is not. It is a regular geometric shape – the Dirac-Kerr-Newman model suggests it’s an oblate spheroid – but so that’s not a perfect sphere. Once you acknowledge that, the so-called anomalous magnetic moment is not-so-anomalous anymore.

The mistake is actually more general than what I wrote above. We are thinking of the key constants in Nature as some number. Most notably, we think of Planck’s quantum of action (h ≈ 6.626×10−34 N·m·s) as some (scalar) number. Why would it be? It is – obviously – some vector quantity or – let me be precise – some matrix quantity: h is the product of a force (some vector in three-dimensional space), a distance (another three-dimensional concept) and time (one direction only). Somehow, those dimensions disappeared in the analysis. Vector equations became flat: vector quantities became magnitudes. Schrödinger’s equation should be rewritten as a vector or matrix equation. We do think of Planck’s quantum of action as some vector. We, therefore, think that the uncertainty – or the probabilistic nature of Nature, so to speak[1] – is not in its magnitude: it’s in its direction. But we are getting ahead of ourselves here – as usual. We should go step by step. Let us first acknowledge where we came from.

Before doing so, I would like to make yet another remark – one that is actually not so relevant for what we are going to try to do this in this book – and that is to understand the QED sector of the Standard Model geometrically – or physically, I should say. The innate nature of man to generalize did not contribute to greater clarity – in my humble opinion, that is. Feynman’s weird Lecture (Volume III, Chapter 4) on the key difference between bosons and fermions does not have any practical value: it just confuses the picture.

Likewise, it makes perfect sense to me to think that each sector of the Standard Model requires its own mathematical approach. I will briefly summarize this idea in totally non-scientific language. We may say that mass comes in one ‘color’ only: it is just some scalar number. Hence, Einstein’s geometric approach to gravity makes total sense. In contrast, the electromagnetic force is based on the idea of an electric charge, which can come in two ‘colors’ (+ or -), so to speak. Maxwell’s equation seemed to cover it all until it was discovered the nature of Nature – sorry for the wordplay – might be discrete and probabilistic. However, that’s fine. We should be able to modify the classical theory to take that into account. There is no need to invent an entirely new mathematical framework (I am talking quantum field and gauge theories here). Now, the strong force comes in three colors, and the rules for mixing them, so to speak, are very particular. It is, therefore, only natural that its analysis requires a wholly different approach. Hence, I would think the new mathematical framework should be reserved for that sector. I don’t like the reference of Aitchison and Hey to gauge theories as ‘the electron-figure’. The electron figure is a pretty classical idea to me. Hence, I do hope one day some alien will show us that the application of the Dyson-Feynman-Schwinger-Tomonaga ‘electron-figure’ to what goes on inside of the nucleus of an atom was, perhaps, not all that useful. A simple exponential series should not be explained by calculating a zillion integrals over 891 Feynman diagrams. It should be explained by a simple set of equations. If I have not lost you by now, please follow me to the acknowledgments section.

[1] A fair amount of so-called thought experiments in quantum mechanics – and we are not (only) talking the more popular accounts on what quantum mechanics is supposed to be all about – do not model the uncertainty in Nature, but on our uncertainty on what might actually be going on. Einstein was not worried about the conclusion that Nature was probabilistic (he fully agreed we cannot know everything): a quick analysis of the full transcriptions of his oft-quoted remarks reveal that he just wanted to see a theory that explains the probabilities. A theory that just describes them didn’t satisfy him.

PS: The book might take a while – as I am probably going to co-publish with another author (it’s perhaps a bit too big for me alone). In the meanwhile, you can just read the articles that are going to make up its contents.

Advertisements

My new book project

Dear readers of this blog – As you may or may not know, I had already published two or three books on amazon.com with some of the ideas on the geometric of physical interpretation of the wavefunction that I have been promoting on this blog. These books sold some copies but – all in all – were not a huge success. That’s fine – because I just wanted to try things out.

I will soon come up with an entirely new book. Its working title is what is mentioned in the current draft of the acknowledgments – copied below. The e-book will be published in a few weeks from now. It may – by some magic 🙂 – coincide with the publication of a convincing classical explanation of the anomalous magnetic moment of an electron – not written by me, of course, but by one of the foremost experts on quantum gravity (and QED in general). 🙂 It would upset the orthodox/mainstream/Copenhagen interpretation of quantum electrodynamics, and that will be a good thing: it will bring more reality to the interpretation (read: just a much easier way to truly understand everything).

If so, my book should sell – if only because it will document a history of scientific discovery. 🙂

The Emperor has no clothes:

The sorry state of Quantum Physics.

Acknowledgements

Although Dr. Alex Burinskii, Dr. Giorgio Vassallo and Dr. Christoph Schiller would probably prefer not to be associated with anything we write, they gave us the benefit of the doubt in their occasional, terse, but consistent communications and, hence, we would like to thank them here – not for believing in anything we write but for encouraging us for at least trying to understand.

More importantly, they made me realize that QED, as a theory, is probably 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. Hence, all of the weirdness of Nature is now, somehow, in this weird description of the fields: perturbation theory, gauge theories, Feynman diagrams, quantum field theory, etcetera. This complexity in the mathematical framework does not match the intuition that, if the theory has a simple circle group structure[1], 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.[2] We feel validated because, in his latest communication, Dr. Burinskii wrote he takes our idea of trying to corroborate his Dirac-Kerr-Newman electron model by inserting it into models that involve some kind of slow orbital motion of the electron – as it does in the Penning trap – seriously.[3]

There are some more professors who may or may not want to be mentioned but who have, somehow, been responsive and, therefore, encouraging. I fondly recall that, back in 2015, Dr. Lloyd N. Trefethen from the Oxford Math Institute reacted to a blog article on mine[4] – in which I pointed out a potential flaw in one of Richard Feynman’s arguments. It was on a totally unrelated topic – the rather mundane topic of shielding, to be precise – but his acknowledgement that Feynman’s argument was, effectively, flawed and that he and his colleagues had solved the issue in 2014 only (Chapman, Hewett and Trefethen, The Mathematics of the Faraday Cage) was an eye-opener for me. Trefethen concluded his email as follows: “Most texts on physics and electromagnetism, weirdly, don’t treat shielding at all, neither correctly nor incorrectly. This seems a real oddity of history given how important shielding is to technology.” When I read this, it made me think: how is it possible that engineers, technicians, physicists just took these equations for granted? How is it possible that scientists, for almost 200 years,[5], worked with a correct formula based on the wrong argument? This, too, resulted in a firm determination to not take any formula for granted but re-visit its origin instead.[6]

We have also been in touch with Dr. John P. Ralston, who wrote one of a very rare number of texts that address the honest questions of amateur physicists and philosophers upfront. I love the self-criticism of the profession: “Quantum mechanics is the only subject in physics where teachers traditionally present haywire axioms they don’t really believe, and regularly violate in research.”[7] We both concluded that our respective interpretations of the wavefunction are very different and, hence, that we should not  waste any electrons on trying to convince each other. However, the discussions were interesting.

I am grateful to my brother, Dr. Jean Paul Van Belle, for totally unrelated discussions on his key topic of research (which is information systems and artificial intelligence), which included discussions on Roger Penrose’s books – mainly The Emperor’s New Mind and The Road to Reality. These books made me think of the working title for this book: The Emperor has no clothes: the sorry state of Quantum Physics. We should go for another mountainbike or mountain-climbing adventure when this project is over.

Among other academics, I would like to single out Dr. Ines Urdaneta who – benefiting from more academic freedom than other researchers, perhaps – has just been plain sympathetic and, as such, provided great moral support. I also warmly thank Jason Hise, whose wonderful animations of 720-degree symmetries did not convince me that electrons (or spin-1/2 particles in general) actually have such symmetries – but whose communications stimulated my thinking on the subject-object relation in quantum mechanics.

Finally, I would like to thank all my friends and my family for keeping me sane. I would like to thank, in particular, my children – Hannah and Vincent – and my wife, Maria, for having given me the emotional, intellectual and financial space to pursue this intellectual adventure.

[1] QED is an Abelian gauge theory with the symmetry group U(1). This sounds extremely complicated – and it is. However, it can be translated as: its mathematical structure is basically the same as that of classical electromagnetics.

[2] We refer to the latest theoretical explanation of the anomalous magnetic moment here: Stefano Laporta, High-precision calculation of the 4-loop contribution to the electron g-2 in QED, 10 July 2017, https://arxiv.org/abs/1704.06996.

[3] Prof. Dr. Burinskii, email communication, 29 December 2018 2.13 pm (Brussels time). To be precise, he just wrote me to say he is ‘working on the magnetic moment’. I interpret this as saying he is looking at his model again to calculate the magnetic moment of the Dirac-Kerr-Newman electron so we will be in a position to show how the Kerr-Newman geometry – which I refer to as the (neglected) form factor in QED – might affect it. To be fully transparent, Dr. Burinskii made it clear his terse reactions do not amount to any endorsement or association of the ideas expressed in this and other papers. It only amounts to an admission our logic may have flaws but no fatal errors – not at first reading, at least.

[4] Jean Louis Van Belle, The field from a grid, 31 August 2015, https://readingfeynman.org/2015/08/.

[5] We should not be misunderstood here: the formulas – the conclusions – are fully correct, but the argument behind was, somehow, misconstrued. As Faraday performed his experiment with a metal mesh (instead of a metal shell) in 1836, we may say it took mankind 2014 – 1836 = 178 years to figure this out. In fact, the original experiments on Faraday’s cage were done by Benjamin Franklin – back in 1755, so that is 263 years ago!

[6] We reached out to Dr. Trefethen and some of his colleagues again to solicit comments on our more recent papers, but we received no reply. Only Dr. André Weideman wrote us back saying that this was completely out of his field and that he would, therefore, not invest in it.

[7] John P. Ralston, How to understand quantum mechanics (2017), p. 1-10.

The Emperor has no clothes

Hi guys (and ladies) – I should copy the paper into this post but… Well… That’s rather tedious. :-/ The topic is one that is of interest of you. You’re looking for a classical explanation of the anomalous magnetic moment, right? Well… We don’t have one – but we’re pretty sure this paper has all the right ingredients for one. We also designed a test for it. Also check out my other paper on the fine-structure constant. It explains everything.

Everything? Well… Almost everything. 🙂 The Revolution has started. The (quantum-mechanical) Emperor seems to have no clothes. 🙂

I am damn serious. This is what I wrote on my FB page today:

The only thing I can be proud of this year is a series of papers on quantum math. I will probably turn them into a popular book on physics. Its working title is “The Emperor Has No Clothes !” Indeed – if anything – these papers show that a lot of the highbrow stuff is just unnecessary complexity or deliberate hyping up of models that can be simplified significantly.

Worse, through my interactions with some physicists, I found some serious research into the nature of matter and energy is being neglected or ignored just because it challenges the Copenhagen interpretation of quantum physics. Most papers of Alexander Burinskii, for example, a brilliant physicist who developed a very plausible model of an electron, have been re-classified from ‘quantum physics’ to ‘general physics’ – which means no one will read them. Worse, he has had trouble just getting stuff published over the last four years! It’s plain censorship! 

I now summarize the Copenhagen interpretation as: “Calculate, don’t think !”  It’s a Diktatur, really! And I now also understand why the founding fathers of quantum mechanics (Dirac, Heisenberg, Pauli, Schroedinger,…) thought the theory they helped to create didn’t quite make the cut. It’s going to be a sad story to tell. In fact, I think Burinskii is in trouble because his model may show that a lot of the research on the anomalous magnetic moment is plain humbug – but so that got some people a Nobel Prize in 1955 and it’s popularly referred to as the ‘high-precision test’ of QED, so… Well… I looked at it too, and for quite a long time, and I’ve come to the conclusion that it’s plain nonsense – but so that cannot be said.

Hmm… If the state of physics is so poor, then we should not be surprised that we are constantly being misled in other fields as well. Let us remember Boltzmann:

“Bring forth what is true. Write it so it it’s clear. Defend it to your last breath.”

Oh – and I have a sort of classical explanation for what happens in the one-photon Mach-Zehnder experiment too. Check it out here. Quantum mechanics is not a mystery. Mr. Bell has got it all wrong. 🙂

Kind regards – JL

Call to Arms

Sent: Thursday, December 20, 2018 12:59 PM
To: All Rebels
Subject: A Manifesto for the Revolution?

Dear All – Thanks for the bilateral exchanges. Perhaps it is time to bring all spacetime rebels together here. 😊

I think we are all agreed on the fact that the Big Ship is not moving anymore. It feels like there has been a sort of academic brain freeze ever since Heisenberg imposed his Diktatur on how we should think about quantum mechanics. Orthodox quantum mechanics is broken beyond repair. Hence, we need to build our own spaceship to venture out to the New Universe. It should be small and nimble. The Seeds of the Revolution are the following:

1. The + or – sign in front of the argument of the wavefunction has a meaning. It’s a degree of freedom in the mathematical description that has not been exploited by physicists. If we want to give it a meaning, then it’s probably the spin direction. It is plain weird that we need the concept of spin in all of our discussions and models on quantum physics but that the Founding Fathers of QM chose to limit the power of Euler’s function to describe a spin-zero particle only.

Once we acknowledge that, all these weird symmetries (720-degree symmetry for spin-1/2) disappear, so there is no ‘excuse’ anymore to not think about a geometric/physical interpretation of the wavefunction. That should trigger a new burst of creative thinking. For starters, we’ll have a different interpretation of Schroedinger’s wave equation. In fact, I would dare to say that, for the first time, we will actually have a (geometric) interpretation of Schroedinger’s wave equation (and its solutions – the orbitals – of course).

2. The difference between the g-factor for spin versus orbital momentum (2 versus 1) can easily be explained by a form factor. If we think of the (free) electron as some disk-like structure (a two-dimensional oscillation, that is), then we’ll have a ½ factor in the formula for its angular momentum and the ‘mystery’ is solved. The anomalous magnetic moment is then not anomalous anymore: it’s just a coupling between the spin and orbital angular momentum that occurs because of the Larmor precession.

Schwinger’s α/2π factor says it all here: if the fine-structure constant is just a dimensional scaling factor explaining the disk-like shape of the (free) electron, then we would expect to see it pop up in some form in the final equations for the motion of real-life electron, which combines orbital motion, Larmor precession (just the effect of magnetism) and spin. I’ve re-written my paper on the anomalous magnetic moment in this sense (it’s on the Los Alamos site for rebels – yes, sorry, I don’t bother to even try to get stuff published) – but I need to do so more work on it. These motions are complicated and to get the coupling factor, we can – unfortunately – not just superpose motions: there is only one value for the magnetic field vector, and the magnetic moment/angular momentum of the whole thing (i.e. the real-life (disk-like) electron moving in this complicated orbital).

3. Interference and diffraction – stuff like the Mach-Zehner experiment – should be explained the way one would usually explain diffraction and interference: if we are going to force a wave through a slit or an aperture, the wave shape is going to change. We need to distinguish between linear and circular polarization ‘states’ – which become real states here! And we should think about how plane waves become spherical waves when they go through an aperture. I think a photon is a circularly polarized wave, but when it goes through the beam splitter, it might be broken up in two linearly polarized waves – each going in a different direction (to the top or, alternatively, to the bottom mirror). If one of them finds its way blocked, it will – somehow – rejoin the other direction (it might just bounce back, right?). Weak measurement shows there is something there. Weak measurement shows the idea of an amplitude is real. It’s not just a mathematical thing. We just need to do some hard thinking on wave shapes and form factors.

We’re not challenging any basic results of quantum mechanics here. We’re just challenging the standard Copenhagen interpretation, which is – basically – that we should not even try to understand what’s going on.

I have a lot of crazy followers on my physics blog (https://readingfeynman.org/). I am going to re-direct them another site – which I really wanted to reserve for the truly crazy ideas (https://readingeinstein.blog/).

On-on ! Let’s honor the Spirit of Ludwig Boltzmann: “Bring forth what is true. Write it so it’s clear. Defend it to your last breath.”

I would add: Please enjoy while doing so! 😊

Dōgen

Enter the void…

I woke up this morning with an inspiring idea. I wrote it on my Facebook page. It is this:

Turbulence… This animation (https://www.youtube.com/watch?v=C680Zz7MyP) has been made by another passenger on this spaceship. We’re traveling into a new Universe. We’re not re-writing the laws of physics. We’re re-writing the way they are written. We have a new language. It’s been a very lonely trip. We’re just a small band of travelers, and the big guns out there don’t like us. All those professors who tell us our ideas look good but that we should not try to challenge the current academic brain freeze… We have only one answer to the skepticism: our ship is small, but it moves. Your ships are big – but they’ve been frozen in time ever since the imposition of the Heisenberg Diktatur. Onwards ! I have a set of papers on the Los Alamos safe haven for rebels that encompasses virtually everything that needs to be explained: the anomalous magnetic moment, Mach-Zehnder interference, Einstein’s E = mc2 equation itself, the spin form factor,… The seeds of the Revolution have been sown.  

Because this site has become very crowded with posts and pages, I will use my other site to relaunch. The other site was supposed to be a site on Einstein – but I hope He is with us in Spirit. 🙂 

Jean Louis Van Belle, 20 December 2018

PS: For some reason, WordPress blocks the YouTube link. Just cut and paste the address in your browser. It’s worth it. 🙂

Looking back…

Well… I think this is it, folks ! With my last posts on superconductivity, I think I am done. I’ve gone through all of the Lectures and it’s been a amazing adventure.

Looking back at it, I’d say: there is really no substitute for buying these Lectures yourself, and just grind through it. The only thing this blog really does is, perhaps, raise a question here and there – or help with figuring something out. But then… Well… If I can do it, you can do it. Don’t go for other sources if you can go for the original writings ! Read a classic rather than yet another second-hand or half-cooked thing !

I should also note that I started off using the print copy of Feynman’s Lectures but, at this point, I realize I should really acknowledge the incredible effort of two extraordinary people: Michael Gottlieb and Rudolf Pfeiffer, who have worked for decades to get those Lectures online. I borrowed a lot of stuff from it. In fact, in the coming weeks and months, I want to make sure I duly acknowledge that for all of the illustrations and quotes I’ve used, and if I haven’t been paraphrasing a bit too much, but… Well… That will be quite an effort. These two extraordinary guys also created a website for these Lectures which offers many more resources. That makes it accessible to all and everyone.

However, let me repeat: there is no substitute for buying the Lectures yourself, and grinding through it yourself. I wish you all the best on this journey. It’s been a nice journey for me, and I am therefore pretty sure you’ll enjoy it at least as much as I did.

Jean Louis Van Belle, 26 February 2018

Post scriptum: The material I have copied and republished from this wonderful online edition of Gottlieb and Pfeiffer is under copyright. The site mentions that, without explicit permission, only some limited copying is permitted under Fair Use laws, for non-commercial publications (which this blog surely is), and with proper attribution. I realize that, despite my best efforts to provide hyperlinks to the Lectures themselves whenever I’d borrow from them, I should probably go through it all to make sure that’s effectively the case. If I have been lacking in this regard, it was surely not intentional.

Potential energy and amplitudes: energy conservation and tunneling effects

This post is intended to help you think about, and work with, those mysterious amplitudes. More in particular, I’ll explore how potential differences change amplitudes. But let’s first recapitulate the basics.

In my previous post, I explained why the young French Comte Louis de Broglie, when writing his PhD thesis back in 1924, i.e. before Schrödinger, Born, Heisenberg and others had published their work, boldly proposed to the ω·t − k·x argument in the wavefunction of a particle with the relativistic invariant product of the momentum and position four-vectors pμ = (E, p) = (E, px, py, pz,) and xμ = (t, x) = (t, x, y, z), provided the energy and momentum are re-scaled in terms of ħ. Hence, he wrote:

θ = ω·t − k·x = (pμxμ)/ħ = (E∙t − px)/ħ = (E/ħ)∙t − (p/ħ)∙x

As it’s usually instructive to do a quick dimensional analysis, let’s do one here too. Energy is expressed in joule, and dividing it by the quantum of action, which is expressed in joule·seconds (J·s) gives us the dimension of an (angular) frequency indeed, which, in turns, yields a pure number. Likewise, linear momentum can be expressed in newton·seconds which, when divided by joule·seconds (J·s), yields a quantity expressed per meter. Hence, the dimension of p/ħ is m–1, which again yields a pure number when multiplied with the dimension of the coordinates x, y or z.

In the mentioned post, I also gave an unambiguous answer to the question as to what energy concept should be used in the equation: it is the total energy of the particle we are trying to describe, so that includes its kinetic energy, its rest mass energy and, finally, its potential energy in whatever force field it may find itself, such as a gravitational and/or electromagnetic force field. Now, while we know that, when talking potential energy, we have some liberty in choosing the zero point of our energy scale, this issue is easily overcome by noting that we are always talking about the amplitude to go from one state to another, or to go from one point in spacetime to another. Hence, what matters is the potential difference, really.

Feynman, in his description of the conservation of energy in a quantum-mechanical context, distinguishes:

  1. The rest energy m0c2, which he describes as the rest energy ‘of the parts of the particle’. [One should remember he wrote this before the existence of quarks and the associated theory of matter was confirmed.]
  2. The energy ‘over and above’ the rest energy, which includes both the kinetic energy, i.e. m∙v2/2 = p2/(2m), as well as the ‘binding and/or excitation energy’, which he refers to as ‘internal energy’.
  3. Finally, there is the potential energy, which we’ll denote by U.

In my previous post, I also gave you the relativistically correct formula for the energy of a particle with momentum p:

Capture

However, we will follow Feynman in his description, who uses the non-relativistic formula E= Eint + p2/(2m) + U. This is quite OK if we assume that the classical velocity of our particle does not approach the speed of light, so that covers a rather large number of real-life situations. Also, to make the example more real, we will assume the potential energy is electrostatic, and given by the formula U = q·Φ, with Φ the electrostatic potential (so just think of a number expressed in volt). Of course, q·Φ will be negative if the signs of q (i.e. the electric charge of our particle) and Φ are opposite, and positive if both have the same sign, as opposites attract and like repel when it comes to electric charge.

The illustration below visualizes the situation for Φ< Φ1. For example, we may assume Φ1 is zero, that Φis negative, and that our particle is positively charged, so U= qΦ< 0. So it’s all rather simple really: we have two areas with a potential equal to U= qΦand U= qΦ< 0 respectively. Hence, we need to use E= Eint + p12/(2m) + U1 to substitute ωfor E1/ħ in the first area, and then E= Eint + p22/(2m) + Uto substitute ωfor E2/ħ in the second area, which U– U< 0.

potential

The corresponding amplitudes, or wavefunctions, are:

  1. Ψ11) = Ψ1(x, t) = a·eiθ1 = a·e−i[(Eint + p12/(2m) + U1)·t − p1∙x]/ħ 
  2. Ψ22) = Ψ2(x, t) = a·e−iθ2 = a·e−i[(Eint + p22/(2m) + U2)·t − p2∙x]/ħ 

Now how should we think about these two equations? We are definitely talking different wavefunctions. However, having said that, there is no reason to assume the different potentials would have an impact on the temporal frequency. Therefore, we can boldly equate ωand ωand, therefore, write that:

Eint + p12/(2m) + U=  Eint + p22/(2m) + U⇔ p12/(2m) − p22/(2m) = U– U< 0

⇒ p1− p2< 0 ⇔ p2 > p1

What this says is that the kinetic energy, and/or the momentum, of our particle is greater in the second area, which is what we would classically expect, as a positive charged particle will pick up speed – and, therefore, momentum and kinetic energy – as it moves from an area with zero potential to an area with negative potential. However, the λ = h/p relation then implies that λ2 = h/p2 is smaller than λ1 = h/p2, which is what is illustrated by the dashed lines in the illustration above – which represent surfaces of equal phase, or wavefronts – and also by the second diagram in the illustration, which shows the real part of the complex-valued amplitude and compares the wavelengths λ1 and λ2. [As you know, the imaginary part is just like the real part but with a phase shift equal to π/2. Ideally, we should show both, but you get the idea.]

To sum it all up, the classical statement energy conservation principle is equivalent to the quantum-mechanical statement that the temporal frequency f or ω, i.e. the time-rate of change of the phase of the wavefunction, does not change – as long as the conditions do not change with time, of course – but that the spatial frequency, i.e. the wave number k or the wavelength λ – changes as the potential energy and/or kinetic energy change.

Tunneling

The p12/(2m) − p22/(2m) = U– Uequation may be re-written to illustrate the quantum-mechanical effect of tunneling, i.e. the penetration of a potential barrier. Indeed, we can re-write p12/(2m) − p22/(2m) = U– Uas

p22 = 2m·[p12/(2m) − (U– U1)]

and, importantly, try to analyze what happens if U– U1 is larger than p12/(2m), so we get a negative value for p22. Just imagine that Φ1 is zero again, and that our particle is positively charged, but that Φis also positive (instead of negative, as in the example above), so our particle is being repelled. In practical terms, it means that our particle just doesn’t have enough energy to “climb the potential hill”. Quantum-mechanically, however, the amplitude is still given by that equation above, and we have a purely imaginary number for p2, as the square root of a negative number is a purely imaginary number, just like √−4 = 2i. So let’s denote p2 as i·p’ and let’s analyze what happens by breaking our a·eiθ2 function up in two separate parts by writing: a·e−iθ2 = a·e−i[(E2/ħ)∙t − (i·p’/ħ)x] = a·e−i(E2/ħ)∙t·ei2·p’·x/ħ = a·e−i(E2/ħ)∙t·e−p’·x/ħ.

Now, the e−p’·x/ħ factor in our formula for a·e−iθ2 is a real-valued exponential function, and it’s a decreasing function, with the same shape as the general e−x function, which I depict below.

graph

This e−p’·x/ħ basically ‘kills’ our wavefunction as we move in the positive x-direction, past the potential barrier, which is what is illustrated below.

potential barrier

However, the story doesn’t finish here. We may imagine that the region with the prohibitive potential is rather small—like a few wavelengths only—and that, past that region, we’ve got another region where p22 = 2m·[p12/(2m) − (U– U1)] is not negative. That’s the situation that’s depicted below, which also shows what might happen: the amplitude decays exponentially, but does not reach zero and, hence, there is a possibility that a particle might make it through the barrier, and that it will be found on the other side, with a real-valued and positive momentum and, hence, with a regular wavefunction.

potential barrier 2

Feynman gives a very interesting example of this: alpha-decay. Alpha decay is a type of radioactive decay in which an atomic nucleus emits an α-particle (so that’s a helium nucleus, really), thereby transforming or ‘decaying’ into an atom with a reduced mass and atomic number. The Wikipedia article on it hais not bad, but Feynman’s explanation is more to the point, especially when you’ve understood all of the above. The graph below illustrates the basic idea as it shows the potential energy U of an α-particle as a function of the distance from the center. As Feynman puts it: “If one tried to shoot an α-particle with the energy E into the nucleus, it would feel an electrostatic repulsion from the nucleus and would, classically, get no closer than the distance r1where its total energy is equal to U. Closer in, however, the potential energy is much lower because of the strong attraction of the short-range nuclear forces. How is it then that in radioactive decay we find α-particles which started out inside the nucleus coming out with the energy E? Because they start out with the energy E inside the nucleus and “leak” through the potential barrier.”

potential energy

As for the numbers involved, the mean life of an α-particle in the uranium nucleus is as long as 4.5 billion years, according to Feynman, whereas the oscillations inside the nucleus are in the range of 1022 cycles per second! So how can one get a number like 109 years from 10−22 seconds? The answer, as Feynman notes, is that that exponential gives a factor of about e−45. So that gives the very small but definite probability of leakage. Once the α-particle is in the nucleus, there is almost no amplitude at all for finding it outside. However, if you take many nuclei and wait long enough, you’ll find one. 🙂

Now, that should be it for today, but let me end this post with something I should have told you a while ago, but then I didn’t, because I thought it would distract you from the essentials. If you’ve read my previous post carefully, you’ll note that I wrote the wavefunction as Ψ(θ) = a·eiθ, rather as a·eiθ, with the minus sign in front of the complex exponent. So why is that?

There is a long and a short answer to that. I’ll give the short answer. You’ll remember that the phase of our wavefunction is like the hand of a stopwatch. Now we could imagine a stopwatch going counter-clockwise, and we could actually make one. Now, there is no arbitrariness here: it’s one way or the other, depending on our other conventions, and the phase of our complex-valued wavefunction does actually turn clockwise if we write things the way we’re writing them, rather than anti-clockwise. That’s a direction that’s actually not as per the usual mathematical convention: an angle in the unit circle is usually measured counter-clockwise. If you’d want it that way, we can fix easily by reversing the signs inside of the bracket, so we could write θ = k·x − ω·t, which is actually what you’ll often see. But so there’s only way to get it right: there’s a direction to it, and if we use the θ = ω·t − k·x, then we need the minus sign in the Ψ(θ) = a·e−iθ equation.

It’s just one of those things that is easy to state, but actually gives us a lot of food for thought. Hence, I’ll probably come back to this one day. As for now, however, I think you’ve had enough. Or I’ve had enough, at least. 🙂 I hope this was not too difficult, and that you enjoyed it.