Wave Particle Duality

The idea of wave particle duality has been around for a long time, and is often used to explain phenomenon like the double slit experiment.  The idea goes like this: the particle (electron, photon, whatever) behaves like a wave some of the time - like when it is passing through the slits - and like a particle at other times - such as when it hits the screen and produces a flash in a single location.  This explains how it can behave as if it went through both slits at the same time despite being in just one location whenever we check - e.g. by making it collide with a surface.  The change from wave like behaviour to particle like behaviour is called the collapse of the wavefunction.

This idea raises many questions, chiefly: what does and what does not collapse the wavefunction?  For example, if you replace the screen with a mirror then a photon continues to exhibit wave behaviour after bouncing off of it.  The proponents of the wave particle duality theory never answered this question, but the theory has stuck and continues to be taught despite many, if not most, physicists having rejected it by now.  One reason for this seems to be that the early proponents such as Bohr were so influential, another is that it took many decades for alternatives to be proposed, and another still that when they came they were so very counter intuitive, and - for some - disquieting.

In this post I will attempt to explain where I think the misunderstanding about wave particle duality comes from.  Because it must work to some extent to have been so persistent an idea.

First we need to introduce configuration space....

Imagine a bathtub full of marbles.  You reach in and pull one out, look into it, and you see a complete universe inside, frozen in a moment of time ($\dagger$).   You put it back, and pick out one that was immediately next to it.  Looking into this new marble you see an almost identical universe, but differing in one tiny aspect - maybe one particle is in a very slightly different position.  In fact, radiating out in all directions from the original marble there are marbles depicting different universes, and these become more and more different the further you get from the original marble, and they become different in ways determined by the direction you take through the bathtub.

Each of the marbles can be addressed, for example you could say that the marble I am interested in (perhaps because it represents the moment I am experiencing) is 20 centimetres from the wall, 103 centimetres from the tap, and 30 centimetres down.  But the bathtub is just a model - in reality you need gazillions of numbers to address a single universe, a single point in configuration space ($\dagger^2$).

Suppose you perform an experiment in which you set up some apparatus in configuration A, and the system evolves to an end state B.  A and B are marbles in the bathtub, and in the classical model of physics there is a string of marbles - a path through configuration space - joining A and B.  This is a path of privileged points because they represent states of the universe which actually occurred.  Every marble off that path is just a mathematical construct - unless as the universe evolves it happens to snake back and pass through it.

In classical physics each little marble is associated with an arrow which tells you exactly in which direction the universe would evolve were it to pass through that point ($\dagger^3$).  So you can start at A and join all the dots by following the arrows to find out how we get to B.  This is what is meant when we say that classical physics is deterministic - it is what was meant by the Enlightenment thinkers who described the universe as clockwork. 

In Quantum mechanics the recipe looks very different.  You have to imagine shaking the marble at A and setting up a wave in the bathtub.  As a result - eventually - all the marbles in the bathtub will end up shaking, but to different degrees.  You cannot say that the experiment will definitely result in B, but you can calculate the degree to which B ends up shaking - relative to the other marbles - and give that as the likelihood of that result ($\dagger^{4,5}$).  This is different in two particularly important respects: firstly there is no privileged path of marbles from A to B; secondly if you remove any of the marbles anywhere in the bathtub as if they were irrelevant then the result of the calculation - the probability of getting result B - ends up different, and WRONG! 

As already pointed out, the bathtub is just a model - a three dimensional model - for configuration space.  So in reality the wave I've described is not a wave that you can easily picture in just three dimensions, but a harder-to-visualize wave in a gazillion dimensions.  But imagine for a moment that we are discussing a really really simple universe containing just one particle, say an electron.  This electron has just 3 attributes, namely its $x$, $y$, and $z$ coordinates ($\dagger^6$).  So the configuration space for this simple system really is three dimensional, and the wave needed to predict the probabilities of different outcomes really is a wave in three dimensional space.  This is why wave particle duality does actually work as a theory - but only for trivially simple systems.

When studying QM it takes a long time to move on from the trivially simple system of a single particle to more complicated systems, and it also took several decades for the pioneers of the field to do so.  I think that this explains why the useful-but-misleading model of wave particle duality stuck, and why it continues to be taught.

Notes

• ($\dagger$)  I'm using the position basis for describing the state of the system.  It makes no difference if you use the momentum basis, or any other.
• ($\dagger^2$) One dimension for each property of each particle, e.g. position coordinates, spin, &c.  Actually it's more complicated than that because the number of particles can vary from one point in configuration space to the next, and some dimensions are of non-continuous variables.
• ($\dagger^3$) And phase space - which includes momenta dimensions - is used instead of configuration space. 
• ($\dagger^4$) Displacements in the wave are complex numbers rather than the usual position vectors.  You could imagine each marble being given a colour with an intensity representing magnitude and hue representing phase.   The actual probability is the square of the amplitude - always a non-negative real number!
• ($\dagger^5$) This also raises the question of what we mean by probability.  My own view is that The Universal Wavefunction never collapses, the frozen moments represented by these marbles all exist in parallel, and that a high probability of B resulting from A simply means that B-like marbles contain more embedded evidence of A than do non B-like marbles.
• ($\dagger^6$) Yes, electrons also have spin!  But you can ignore that for some experiments.  For the rest you just need to model two 3D waves, not one 4D wave, so the argument still holds.