About three weeks ago, I brought my most substantial posts together in one document: it’s the Deep Blue page of this site. I also published it on Amazon/Kindle. It’s nice. It crowns many years of self-study, and many nights of short and bad sleep – as I was mulling over yet another paradox haunting me in my dreams. It’s been an extraordinary climb but, frankly, the view from the top is magnificent. 🙂

The offer is there: anyone who is willing to go through it and offer *constructive* and/or *substantial* comments will be included in the book’s acknowledgements section when I go for a second edition (which it needs, I think). First person to be acknowledged here is my wife though, Maria Elena Barron, as she has given me the spacetime and, more importantly, the freedom to take this bull by its horns.Below I just copy the foreword, just to give you a taste of it. 🙂

**Foreword**

Another introduction to quantum mechanics? Yep. I am not hoping to sell many copies, but I do hope my unusual background—I graduated as an economist, *not *as a physicist—will encourage you to take on the challenge and grind through this.

I’ve always wanted to thoroughly *understand*, rather than just vaguely *know*, those quintessential equations: the Lorentz transformations, the wavefunction and, above all, Schrödinger’s wave *equation*. In my bookcase, I’ve always had what is probably the most famous physics course in the history of physics: Richard Feynman’s *Lectures on Physics*, which have been used for decades, not only at Caltech but at many of the best universities in the world. Plus a few dozen other books. Popular books—which I now regret I ever read, because they were an utter waste of time: the language of physics is math and, hence, one should read physics in math—not in any other language.

But Feynman’s *Lectures on Physics*—three volumes of about fifty chapters each—are not easy to read. However, the experimental verification of the existence of the Higgs particle in CERN’s LHC accelerator a couple of years ago, and the award of the Nobel prize to the scientists who had predicted its existence (including Peter Higgs and François Englert), convinced me it was about time I take the bull by its horns. While, I consider myself to be of average intelligence only, I do feel there’s value in the ideal of the ‘Renaissance man’ and, hence, I think stuff like this is something we all should try to understand—*somehow*. So I started to read, and I also started a blog (www.readingfeynman.org) to externalize my frustration as I tried to cope with the difficulties involved. The site attracted hundreds of visitors every week and, hence, it encouraged me to publish this booklet.

So what is it about? What makes it special? In essence, it is a common-sense introduction to the key concepts in quantum physics. However, while common-sense, it does not shy away from the math, which is complicated, but not impossible. So this little book is surely *not* a Guide to the Universe for Dummies. I do hope it will guide some Not-So-Dummies. It basically recycles what I consider to be my more interesting posts, but combines them in a comprehensive structure.

It is a bit of a philosophical analysis of quantum mechanics as well, as I will – hopefully – do a better job than others in distinguishing the *mathematical *concepts from what they are supposed to *describe*, i.e. *physical *reality.

Last but not least, it does offer some new *didactic *perspectives. For those who know the subject already, let me briefly point these out:

**I.** Few, if any, of the popular writers seems to have noted that the argument of the wavefunction (θ = E·t – p·t) – using natural units (hence, the numerical value of ħ and *c *is one), and for an object moving at constant velocity (hence, x = *v*·t) – can be written as the product of the *proper time *of the object and its rest mass:

θ = E·t – p·x = E·t − p·x = m* _{v}*·t − m

*·*

_{v}*v*·x = m

*·(t −*

_{v}*v*·x)

⇔ θ = m_{0}·(t − *v*·x)/√(1 – *v*^{2}) = m_{0}·t’

Hence, the argument of the wavefunction is just the proper time of the object with the rest mass acting as a *scaling *factor for the time: the internal clock of the object ticks much faster if it’s heavier. This symmetry between the argument of the wavefunction of the object as measured in its own (inertial) reference frame, and its argument as measured by us, in our own reference frame, is remarkable, and allows to understand the nature of the wavefunction in a more intuitive way.

While this approach reflects Feynman’s idea of the photon stopwatch, the presentation in this booklet generalizes the concept for all wavefunctions, first and foremost the wavefunction of the *matter*-particles that we’re used to (e.g. electrons).

**II.** Few, if any, have thought of looking at Schrödinger’s wave *equation* as an energy propagation mechanism. In fact, when helping my daughter out as she was trying to understand non-linear regression (logit and Poisson regressions), it suddenly realized we can analyze the wavefunction as a *link *function that connects two *physical *spaces: the *physical *space of our moving object, and a *physical *energy space.

Re-inserting Planck’s quantum of action in the argument of the wavefunction – so we write θ as θ = (E/ħ)·t – (p/ħ)·x = [E·t – p·x]/ħ – we may assign a physical dimension to it: when interpreting ħ as a scaling factor only (and, hence, when we only consider its *numerical *value, not its physical dimension), θ becomes a quantity expressed in newton·meter·second, i.e. the (physical) dimension of action. It is only natural, then, that we would associate the real and imaginary part of the wavefunction with some physical dimension too, and a dimensional analysis of Schrödinger’s equation tells us this dimension must be energy.

This perspective allows us to look at the wavefunction as an energy propagation mechanism, with the real and imaginary part of the probability amplitude interacting in very much the same way as the electric and magnetic field vectors **E** and **B**. This leads me to the next point, which I make rather emphatically in this booklet: the propagation mechanism for electromagnetic energy – as described by Maxwell’s equations – is *mathematically *equivalent to the propagation mechanism that’s implicit in the Schrödinger equation.

I am, therefore, able to present the Schrödinger equation in a much more coherent way, describing not only how this famous equation works for electrons, or matter-particles in general (i.e. *fermions* or spin-1/2 particles), which is probably the only use of the Schrödinger equation you are familiar with, but also how it works for bosons, including the photon, of course, but also the theoretical zero-spin boson!

In fact, I am personally rather proud of this. Not because I am doing something that hasn’t been done before (I am sure many have come to the same conclusions before me), but because one always has to trust one’s intuition. So let me say something about that third innovation: the photon wavefunction.

**III.** Let me tell you the little story behind my photon wavefunction. One of my acquaintances is a retired nuclear scientist. While he knew I was delving into it all, I knew he had little time to answer any of my queries. However, when I asked him about the wavefunction for *photons*, he bluntly told me photons didn’t have a wavefunction. I should just study Maxwell’s equations and that’s it: there’s no wavefunction for photons: just this traveling electric and a magnetic field vector. Look at Feynman’s *Lectures*, or any textbook, he said. None of them talk about photon wavefunctions. That’s true, but I knew he had to be wrong. I mulled over it for several months, and then just sat down and started doing to fiddle with Maxwell’s equations, assuming the oscillations of the **E** and **B** vector could be described by regular sinusoids. And – *Lo and behold!* – I derived a wavefunction for the photon. It’s fully equivalent to the classical description, but the new expression solves the Schrödinger equation, if we modify it in a rather logical way: we have to double the diffusion constant, which makes sense, because **E** and **B** give you two waves for the price of one!

[…]

In any case, I am getting ahead of myself here, and so I should wrap up this rather long introduction. Let me just say that, through my rather long journey in search of *understanding* – rather than *knowledge* alone – I have learned there are so many wrong answers out there: wrong answers that *hamper *rather than *promote *a better understanding. Moreover, I was most shocked to find out that such wrong answers are not the preserve of amateurs alone! This emboldened me to write what I write here, and to publish it. Quantum mechanics is a logical and coherent framework, and it is* not* all that difficult to understand. One just needs good pointers, and that’s what I want to provide here.

As of now, it focuses on the *mechanics *in particular, i.e. the concept of the wavefunction and wave equation (better known as Schrödinger’s equation). The other aspect of quantum mechanics – i.e. the idea of *uncertainty* as implied by the quantum idea – will receive more attention in a later version of this document. I should also say I will limit myself to quantum electrodynamics (QED) only, so I won’t discuss quarks (i.e. quantum chromodynamics, which is an entirely different realm), nor will I delve into any of the other more recent advances of physics.

In the end, you’ll still be left with lots of unanswered questions. However, that’s quite OK, as Richard Feynman himself was of the opinion that he himself did not *understand* the topic the way he would like to understand it. But then that’s exactly what draws all of us to quantum physics: a common search for a *deep* and full *understanding* of reality, rather than just some superficial *description* of it, i.e. *knowledge* alone.

So let’s get on with it. I am *not *saying this is going to be easy reading. In fact, I blogged about much easier stuff than this in my blog—treating only *aspects *of the whole theory. This is the *whole *thing, and it’s not easy to swallow. In fact, it may well too big to swallow as a whole. But please do give it a try. I wanted this to be an intuitive but formally correct introduction to quantum math. However, when everything is said and done, you are the only who can judge if I reached that goal.

Of course, I should not forget the acknowledgements but… Well… It was a rather lonely venture, so I am only going to acknowledge my wife here, Maria, who gave me all of the spacetime and all of the freedom I needed, as I would get up early, or work late after coming home from my regular job. I sacrificed weekends, which we could have spent together, and – when mulling over yet another paradox – the nights were often short and bad. Frankly, it’s been an extraordinary climb, but the view from the top is magnificent.

I just need to insert one caution, my site (www.readingfeynman.org) includes animations, which make it much easier to grasp some of the mathematical concepts that I will be explaining. Hence, I warmly recommend you also have a look at that site, and its Deep Blue page in particular – as that page has the same contents, more or less, but the animations make it a much easier read.

Have fun with it!

Jean Louis Van Belle, *BA*, *MA*,* BPhil*, *Drs.*