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1 {electron microscope image of a diatoms}
1 Caption: Smaller than the eye can see
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A sport in which balls are not hit close to the speed of light. 
In that sense it's like special relativity. If humans made a habit of moving around and playing sports in which we hit balls at close to the speed of light, our senses and reflexes would be adapted to the odd changes in time and space that occur at those speeds, and we would think they were perfectly natural and intuitive. But we don't. We live in a world where we move and interact with things going much, much slower than the speed of light, and relativistic effects are negligible. So our senses and reflexes and intuition about moving objects are adapted to that, and the effects of relativity seem weird.
It's the same thing with the wave/particle nature of light. If we were of the right scale and time perception to routinely interact with things that display wave/particle duality, we would think it was perfectly natural and our intuitions would deal with it. We'd probably think that objects that don't display both wave and particle properties were strange and unintuitive.
I've hinted there that wave/particle duality is found in more places than just the behaviour of light. For indeed it is. In previous discussions of photons I've given the mathematical relationship between the frequency of light ν and the energy of a photon E:
Waves made of particles (of sand). 
where h is Planck's constant.
Now, having recently discussed Einstein's discovery of massenergy equivalence:
E = mc^{2},
where m is mass and c is the speed of light, we have two equations in which the left hand side represent the same thing: energy. If we set these expressions equal to each other and divide through by the value h we get:
ν = mc^{2}/h.
This expression gives a relationship between the frequency of a wave ν on the left hand side and the mass of an object m on the right hand side. So far this is mathematical manipulation; what does it actually mean, if anything?
Result of a crash test. Creative Commons AttributionNonCommercialShareAlike image by Andrew Bossi. 
For everyday objects moving slowly, momentum is a quantity equal to mass times velocity. It's sometimes referred to as the "quantity of motion" in an object. The effect of it is most evident in collisions. Imagine a small car moving at some speed, say 30 km/h. Imagine a large, heavily laden semitrailer truck moving at the same speed, in the opposite direction. Imagine a collision the likes of which would have the MythBusters drooling over their slowmo replays. Head on. (With no people inside the vehicles.) Despite the fact that both vehicles are moving at the same speed before the collision, your intuition is probably saying that when they collide, the tangled mess of wreckage will end up sliding in the direction the truck was moving, as its enormous weight dwarfs that of the car. And you'd be right. It's not the speed that determines the motion after the collision, it's the momentum.
This effect of momentum—that it can transfer motion from one object to another during a collision—occurs in reverse when objects fly apart, such as in an explosion. Imagine firing a cannon. The explosion in the chamber propels the cannonball out the front at high speed. This speed, times the mass of the cannonball, is its momentum. But that momentum of its motion has to come from somewhere. Prior to the shot neither the cannon nor cannonball was moving. To balance the forward momentum of the cannonball, the cannon itself also acquires exactly the same amount of momentum, but in the reverse direction.^{[1]} We call this backwards motion of the cannon recoil. But the cannon weighs a lot more than the cannonball! So to have the same amount of momentum (mass times velocity), the more massive cannon has to have a smaller velocity than the cannonball. And so it is, by the correct amount to satisfy equality of momentum: If the cannon has 100 times the mass of the ball, it recoils at 1/100 of the speed of the ball.
Cannon recoil. Creative Commons AttributionNonCommercial image by Michael Kappel. 
Anyway, back to photons, which also have momentum. What does this mean? It means that if you hit an object with photons, you are adding momentum to the object. The photons will push it back. Also, if an object emits photons, it's like firing them from a gun  the emitting object will recoil. But light is made of photons. What this implies is that if you shine a torch (a.k.a. flashlight) on someone, firstly you should feel a recoil from the torch firing photons, and secondly that the person being hit by the light beam will feel a force pushing them backwards, as if you fired a canonball at them. As it happens, the momentum carried by the photons is very small  so small as to be imperceptible to our human senses.^{[2]} But it is indeed there, and can be measured by sensitive equipment.
In places where interactions with the gravity of the Earth and the pressure of its atmosphere are negligible, such as in deep space, the momentum of photons can potentially be harnessed. Imagine a space ship, with great sails billowing around it, angled to catch the light of the sun. All the photons of sunlight could supply enough momentum to the sails to push the ship around. Such solar sails have been proposed. In fact, even the planned trajectories of the relatively small bodies of interplanetary probes we have sent to other planets need to take into account the added momentum of sunlight along their journey, lest they end up slightly off course.
Einstein's complete formulation of the relativistic relation between energy, mass, and momentum, which is valid for all objects (both massive objects and photons), is:
Solar sail concept art. Creative Commons AttributionNonCommercial image by NASA. 
where p is momentum. For photons, which have no mass, this reduces to E = pc. (For objects with mass, it is essentially E = mc^{2} (where m is the rest mass), plus a term related to the relativistic momentum, which can either be interpreted as relativistic mass increase, or left alone as an additional term.)^{[3]}
If we use this photon formulation for energy, and set it equal to equation A above, we get:
hν = pc.
Then, using the wave relationship that frequency ν equals speed (c for light) divided by the wavelength λ, and rearranging the equation slightly, gives us the following:
λ = h/p. (equation B)
In other words, the momentum of a photon is directly related to its wavelength. The constants h and c are such that the momentum of a photon of typical wavelengths (such as visible light) is too small to be noticed by humans, but can be confirmed with delicate instruments. So that all works. Right, so much for photons. We now have equation B, which tells us something useful about the relationship between wavelength and momentum, can we interpret it for objects other than photons? Objects that do have a mass?
Louis de Broglie. Public domain image from Wikimedia Commons. 
For starters, the Planck constant h has such a tiny value that for anything with momentum of a scale perceivable by humans, the resulting wavelength would be incredibly tiny  much, much smaller than the size of an atom. If you throw a ball to someone, it has quite a bit of momentum. The wavelength according to de Broglie is too small to notice or have any sort of visible effect. But there are objects which have much smaller momenta, by virtue of having much smaller masses. The smallest mass object we can deal with relatively easily is an electron.
In 1927, the American physicists Clinton Davisson and Lester Germer were interested in the detailed surface structure of a sample of nickel. Working together at Bell Labs in New Jersey, they devised an experiment to probe their nickel sample by firing a beam of electrons at it, and measuring at what angles the electrons were reflected. The sample was polished to a mirrorlike finish in visible light. But they theorised that at scales too small to be visible to the eye, the surface structure of the crystals making up the metal would be rough. They expected the electrons would therefore bounce off at essentially random angles and end up scattered all over the place.
This didn't happen. Instead, they observed a pattern of scattered electrons, with many being detected at some angles, and few being detected at intermediate angles. In fact, the pattern looked pretty much like the diffraction patterns you get when you fire xrays (which are electromagnetic radiation, like light) at crystal structures. Those diffraction patterns are caused by the fact that xrays are waves, and the scattered waves off the different layers of atoms within the crystal interfere with one another, forming interference patterns (which I've discussed a while ago). What's more, Davisson and Germer generated their electron beam by accelerating the electrons through an electric voltage difference. By adjusting the voltage of their electron gun, they could change the speed (and hence momentum) of the electrons. When they did this, the pattern of reflected electrons shifted, as though they were changing the wavelength of the waves that were generating the pattern.
And when they plugged in the numbers to the known formula for diffraction of waves from crystals, the "wavelength" of the electrons they calculated matched the wavelength that de Broglie predicted, at every different electron momentum. This provided experimental evidence that de Broglie's hypothesis was not only correct, but that the mysterious wavelength he assigned to physical objects actually produced measurable consequences that could only be explained by a wavelike behaviour. Beams of electrons could not be treated simply as a stream of particles. They also had a wave property that resulted in wavelike diffraction effects.
An electron microscope. Creative Commons AttributionNonCommercialShareAlike image by Pacific Northwest National Laboratory. 
Another classic demonstration of the wave nature of light is Young's double slit experiment, in which a light beam is passed through two narrow slits to produce a distinctive interference pattern. It wasn't until 1961 that German physicist Claus Jönsson managed to perform a successful double slit experiment with an electron beam. Electrons fired through a pair of narrow slits produce exactly the sort of interference pattern you'd expect for waves of the wavelength given by de Broglie's formula.
However, well before this, in the 1930s, several people realised that one consequence of Davisson, Germer, and Thomson's discovery was that you could potentially use electrons as a sort of light beam. What's more, the wavelength of electrons moving at easily achievable speeds was much smaller than the wavelength of visible light, 1000 to 10,000 times smaller. This prompted them to think about a way around one of the fundamental limitations of microscopes.
Many of the objects we might wish to study carefully are actually smaller than the wavelength of visible light. Many plant and animal cells, for example, are not much bigger than a wavelength of light. So although they can be seen under a microscope, the light waves themselves are too big to get into all the details of the structure and make that detail visible to us. And some structures have details that are way smaller than that. If you could instead fire a beam of electrons at a small object, the wavelength of the electron beam would allow you to probe much smaller structures, and by reconstructing those details from the patterns of reflected electrons (in the same way as your eye constructs and image from patterns of reflected photons), we can build an extremely detailed 3dimensional model of what the electrons are hitting.
And in the 1930s, several people built machines that could do exactly this. They are called electron microscopes. They allow us to see incredible detail on the tiniest objects, at far better resolution than is possible with light waves. And it's all because objects at the scale of electrons don't behave just like tiny particles. They behave like both waves and particles.
[1] In reality things are slightly more complicated than this, because the explosion of powder also produces rapidly expanding hot gases, which are confined by the cannon barrel to travel in a certain direction (the same as the cannonball). To fully calculate things, you need to account for the momentum of this gas as well, though it is very small compared to the momenta of the cannon and the ball, since the gas has very little mass.
[2] Thankfully. Otherwise we'd be knocked over by the sunlight every time we went outside.
[3] I took a rather roundabout route over three separate annotations to get to this equation, probing some deadend alleys along the way. This may have seemed like the wrong approach and potentially headscratching to those already familiar with Einstein's relativistic mass/energy/momentum result, and wondering why I didn't mention this equation earlier. Sorry about that.
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