2017 World Championship 24-Hour Run, Belfast, UK

### Running and Quantum Mechanics

In my day job, I'm a physicist. So I thought it would fun to draw an analogy between running and quantum mechanics. Even if you haven't officially studied physics in school (or if you did study it, but hated it), your real-life experience makes you an excellent experimental physicist. So bear with me, as I try to explain quantum mechanics in terms of running.

Mechanics is the study of how macroscopic (big, tangible) objects behave as they move, or as forces are applied to them. From your own life as an experimental physicist, you have a pretty good intuition for "classical mechanics" – i.e. the trajectory of a baseball when you throw it, or the effect on a car when it smashes into a wall. You know that throwing the ball harder gives it more "momentum" ($p$), which makes it harder to hit or catch (i.e. harder to change the direction of motion). You know that driving the car faster gives it more "energy" ($E$), which makes it crumple more when it hits the wall (i.e. the extra energy can deform the metal more). Quantum mechanics describes the motion of microscopic objects, like the atoms and electrons that are the basic quantized building blocks of the world around us. The catch is that when particles are really tiny, they act like waves as well as particles, which is outside the realm of our daily experience. But I'll give a simple running example to show how this works.

In a classical mechanics, you could write down some equations for the motion of objects, such as:

$$\label{eq:velocity} x = vt\qquad \mathsf{(distance = velocity}\times\mathsf{time)}$$ $$\label{eq:momentum} p = mv\qquad \mathsf{(momentum = mass}\times\mathsf{velocity)}$$ $$\label{eq:energy} E = \tfrac{1}{2}mv^2\qquad \mathsf{(energy = }\,\tfrac{1}{2}\mathsf{mass}\times\mathsf{velocity}^2\mathsf{)}$$

From your own life as an experimental physicist, you also have some intuition about light. If you shine light on a solid object, the object will cast a shadow, i.e. the light behaves as little particles (called "photons") that hit the solid object and can't get through. But if you look closely, you'll see that some light can bend, or "diffract" around the edge of the object.

Another familiar example of diffraction is a rainbow, where the different colors of sunlight diffract through water droplets in the sky at different angles, depending on their wavelength $\lambda$.

How can we reconcile your two experiences with light (rainbows vs. shadows)? There is no intuitive answer, but in physics we call it the "wave-particle duality", and we relate the particle-like properties of light (the energy $E$ and momentum $p$ of a single photon) to the wave-like properties of light (the frequency $f$ and the wavelength $\lambda$) using Planck's constant, $h = 6.6 \times 10^{-34}$ J$\cdot$s.

$$\label{eq:wavelength} p = \frac{h}{\lambda}\qquad \left(\mathsf{momentum = }\,\frac{h}{\mathsf{wavelength}}\right)$$ $$\label{eq:frequency} E = hf\qquad \mathsf{(energy = }\,h\times\mathsf{frequency)}$$

Now here comes the crazy part: even tangible objects – the everyday stuff of "classical mechanics" – also have a wave-particle duality! The wavelength of an object is given by the de Broglie formula, which combines equations (\ref{eq:momentum}) and (\ref{eq:wavelength}).

$$\label{eq:deBroglie} \lambda = \frac{h}{mv}$$

Here's an example of the de Broglie formula that brings us back to running. In 2016, I won the USA Track & Field National Championships, by running 142.07 miles (228.64 km) in 24 hours.

My average speed was: $$v = \frac{\mathsf{distance}}{\mathsf{time}} = \frac{228,640\,\mathsf{meters}}{24\,\mathsf{hrs}\times 60\,\mathsf{min/hr}\times 60\,\mathsf{sec/min}} = 2.64\,\mathsf{m/s}$$

My mass was: $m = 130$ lbs (59 kg).

So my wavelength was: $$\lambda = \frac{h}{mv} = \frac{6.6\times10^{-34}\,\mathsf{J}\cdot\mathsf{s}}{59\,\mathsf{kg}\times2.64\,\mathsf{m/s}} = 4\times 10^{-36}\,\mathsf{meters}$$

That’s really small – so small that you wouldn't notice it in your daily life. That explains why you don't have intuition about quantum mechanics as personal experimental physicist (and frankly, neither do I, even as a professional experimental physicist). But let's play with this a little more, to demonstrate some real quantum mechanics.

Another property of waves is that they can "interfere" with each other. If two waves are "in phase", i.e. their crests are lined up, then they add constructively, i.e. the light gets brighter or the sound gets louder. If two waves are "out of phase", i.e. the crest of one wave is lined up with the trough of another, then they add destructively, i.e. the light or sound is cancelled out. A well-designed theater will place the speakers and chairs such that the aisles correspond to the locations of maximum destructive interference (sound cancellation) at typical audible frequencies.

The course for the USATF National Championship 24-Hour Run is a 1.45 kilometer loop around Edgewater Park on the shore of Lake Erie in Cleveland, Ohio.

So if I'm running around this loop over and over again (158 times!) then I have a problem: if I go the wrong speed, my wave on lap 1 might interfere destructively with my wave on lap 2, so I would cancel myself out and disappear! In order to keep running all day long without disappearing, my wavelength must be constrained to interfere constructively on each lap. The only way for my wave to interfere constructively with itself, is if an "integer" (whole number) of my wavelengths fit into a 1.45 km loop.

So my velocity is constrained, or "quantized" to a set of values that satisfy the following equation for integer value of $N$.

$$d = 1.45\,\mathsf{km} = N \lambda = \frac{N h}{m v}$$ $$\rightarrow v = \frac{Nh}{md} = N(7.5 \times 10^{−39}\,\mathsf{m⁄s})$$

So I can travel only in integer increments of $7.5 \times 10^{-39}$ m/s. Again, this is ridiculously small, and there's no way that a personal experimental physicist would notice it, so you can be forgiven for overlooking this evidence of quantum mechanics. But now let's consider an electron, with mass $9\times 10^{-31}$ kg, and a typical velocity $10^6$ m/s. So a typical wavelength of an electron is $7\times 10^{-10}$ meters, or 0.7 nanometers, which is large enough to be observed with the kinds of microscopes that I build in my day job.

The bottom line is that all light, and all objects, big or small, experience a particle-wave duality, which leads to quantization. The quantization exists for all objects, but it's not relevant when we're dealing with macroscopic objects – which is why you don't notice it on a daily basis. But when we're dealing with light or electrons, it can lead to some pretty cool properties that affect your enjoyment of a rainbow, as well as the technological device on which you are reading this blog.