One of the most astounding and controversial outcomes of the early investigation of quantum physics was the discovery that light acts as both a particle and a wave, the so-called particle-wave duality of matter. Physicists went further though, discovering it was not just light that behaved as if it were a particle and a wave depending on the conditions under which it was observed. All matter, up to a certain limit, behaves this way. How could something be both a particle and a wave? The answer is that if it is neither of course. All of this poses a further question if solid-matter has wave-like properties, why don’t we see travelling objects diffract around stationary objects in their path?

To answer this question, we’ll first consider the conclusions that had been reached about light and matter at the turn of the twentieth-century and then examine the findings and the physicists that turned these conclusions and our very understanding of nature on its head.

Light as a wave?

As of 1905, physicists had settled on the idea that light was wave-like in nature. This was despite the fact that the father of physics, Newton, had strongly believed that light travelled in ‘corpuscles’. It was Huygens and others at the start of the 19th Century that clearly demonstrated optical interference and diffraction. Light did not pass through a narrow opening like a stream of bullets for example. But like water waves passing an obstruction, surely demonstrating that light was wave-like in nature. To consider what interference means, imagine a water-wave passing a post or other object and creating two small wavelets. In the areas where the crest of these waves meet a large peak is created; this is constructive interference.

The fringe pattern is known as interference and it can be explained easily using an analogy to water waves. When two waves meet at the point of their maxim amplitude the overall amplitude is increased. The waves are in phase.

However, where a peak meets a trough, both cancel the other out, thus destructive interference occurs.

When two waves meet with maximum amplitude and minimum amplitude, the waves cancel. Unsurprisingly, this is termed destructive interference. (http://www.phys.uconn.edu) The waves are said to be out of phase

This was demonstrated experimentally by Thomas Young in his celebrated “double-slit experiment” in 1801. Using a simple light source and a screen with two openings, Young showed light to behave in a similar way, creating alternating light and dark bands on a photo-receptive screen (below). The experiment is conducted in physics classrooms around the world to this day, commonly using a simple laser to demonstrate interference patterns.

The wave-like behaviour of light as demonstrated by the double-slit experiment

Building on this, the Maxwell wave equations of light established most factors regarding how light behaved allowing physicists to conclude that all forms of light are just electromagnetic waves travelling at a set speed in a vacuum, term c, the speed of light. In 1887, Henrich Hertz discovered the Photoelectric effect which was then fully investigated by his assistant Philipp Lenard, an investigation for which Lenard received the 1905 Nobel prize. Metals exposed to high-frequency electromagnetic radiation eject electrons, which Lenard found, did not all emerge with the same kinetic energy. The problem in this arose from the fact that Maxwell’s equations suggested that the number of ejected electrons should depend on the intensity of the incident electromagnetic radiation. The intensity of the wave is the energy it transfers per unit area per unit time. Maxwell’s model saw the waves of light impinging on the metal, continuously delivering energy. Think of this problem like this, the ejection of an electron would be almost like water gradually filling a bucket with a ball floating in it. As the water overflows the ball is ejected. This also suggests that there should be a delay between a light source being switched on and the metal beginning to eject electrons.

Incident light of the necessary frequency causes the surface of the metal to release electrons

But this wasn’t what was observed.

Not only did the metal begin to eject the electron immediately, akin in our bucket analogy to the first drop of water bouncing out the ball, but the bucket could be full, and the ball would still not spill out. Almost as if the water must be a certain temperature before the ball can spill out, no matter how full the bucket.

Light as a particle?

The number of ejected photons depended not on the intensity of the incident light, but on its frequency: the number of peaks of the wave delivered per second. All forms of electromagnetic radiation have a characteristic frequency. In fact, if the frequency of the incident electromagnetic radiation was lower than a certain threshold, varying from metal to metal, no electrons would be emitted, no matter how intense it was. Also, electrons bombarded with a low-intensity light emerged with the same kinetic energy as electrons bombarded with a high-intensity light of the same frequency.

Einstein saw the contradiction very clearly thanks to the work of Max Planck. Planck had suggested that energy was transmitted in discrete amounts or quanta which was proportional to the frequency of the wave. High-frequency waves carry larger quanta of energy. This led to the development of Planck’s law: The energy (E) of an electromagnetic wave of frequency (f) is the product of Planck’s constant (h) and the frequency (f) or E= hf. This was verified experimentally by Robert Millikan in 1915. If light is delivered to the metal as packets of energy (quanta) of the value hf, this suggests that light must exist as particles. These particles going on to be named photons. But how could this be reconciled with the wave-nature of light, a cornerstone of physics, which had also been verified experimentally?

In 1909 G.I Taylor studied diffraction patterns of light using the double slit experiment and a low-intensity light source. After a short interval of time, scattered spots appeared on Taylor’s photoreceptive plate. The impressions of single photons- particle-like behaviour! This gradually built over a three-month exposure in a wave-like interference pattern. Thus, simultaneously demonstrating both wave and particle-like properties.

The most sensible conclusion is that light is neither a wave or a particle but behaves as either under certain conditions. This leads to an interesting question: if light is neither a particle or a wave but something with properties of both, what about other fundamental particles? What about the matter around us?

Electrons: particles, waves or something else?

In a 1924 paper Louis de Broglie suggested that not only could light be described as particle-like and wave-like, but that all matter could be mathematically modelled using wave equations. Following this, Davison and Germer used a series of experiments involving electron diffraction to show that matter possessed the same duality of nature as light. Using a computer simulation, we can show the results of this refined double-slit experiment with electrons taken the place of photons and replicate what this tells us about the nature of matter.

We start with a hypothetical apparatus set up as shown below.

Figure 1: An electron gun fires electrons through two narrow slits onto a fluorescent screen. Where an electron hits the screen a white dot is left behind. We aren’t going to concern ourselves too much with the widths of the slits(1 nm) or the energy of the electrons(38 V).

Figure 2: As the electrons stream through the slits one by one they appear on the screen seemingly randomly. In fact, this underlines how the probabilistic nature of quantum physics can still be reconciled with the deterministic nature of classical physics. We cannot predict with any certainty where the next particle will strike the screen, but we can predict with absolute certainty the overall distribution of a large number of electron hits.

Figure 3:

The final distribution shows wide bands of electron hits punctuated with thin black bars showing virtually no elections strikes. These bars don’t line up with the blockages in the equipment, and the distribution clearly doesn’t resemble what we would get from firing hard projectiles such as bullets through scaled-up slits where we would expect a graph of the hits to be two “humps” corresponding to the openings, with some lying between.

Figure 4: the graph yielded by our electron firing simulation

Figure 5: This fringe pattern is destroyed if we run the experiment again, but this time with one of the slits closed.

These patterns are achieved every time this experiment is conducted. Even though the precise build is random and probabilistic, the final distribution is deterministic and fully predictable.

These wavefunctions, known as de Broglie waves, are a mathematical interpretation of how a particle propagates through space. They are composed of all the possible positions of the particle at any time and the probability of finding a particle at that particular point. In what follows, you’ll see why it’s necessary to describe the propagation of a particle through space as a wave function. To attempt to understand these effects in the above experiment, we reopen both slits and turn down the current of the electron gun to allow one electron at a time to hit the screen. Remarkably the fringe pattern returns, albeit slowly, defying the idea that it is the wave function of one electron is interfering with its neighbour. In fact, it’s clear that the electron interferes with itself.

Stop laughing at the back!

Following this startling demonstration of electrons exhibiting wave-like behaviour, physicists began to experiment with progressively larger particles. Perhaps the most well-known example of the display of wave-like behaviour in large conglomerates of matter involves Carbon-60 or “Bucky-balls” (named after architect Buckminster Fuller who was famed for his geodesic structures which Carbon-60’s soccerball-like shape resembles) consisting of 60 carbon atoms.

The consequence of this is that we are forced to abandon the classic idea of a particle possessing a single defined trajectory through space. The passage above demonstrates that the particle can be considered passing through each slit, with the contribution of each slit in the wave pattern causing constructive and destructive interference.

Why don’t we diffract around lamp-posts? The De Broglie wave is why.

If all matter can be described using wave equations, why don’t we see diffraction effects in the everyday world? After all, I’m sure we’d all like to see the odd football diffract around a goalkeeper or cricket ball around a bat occasionally.

Penalty shoot-outs just became infinitely more stressful.[/caption]

To answer this, we need to consider the mathematics of the matter-wave (or De Broglie wave) and one of the most important and ubiquitous elements of quantum physics: Planck’s constant. The parameters of the matter-wave are given by the relationship:

Where h is Planck’s constant and p is the magnitude of the particle’s momentum which, in turn, for a particle with mass, given by:

The key to wave-like behaviour at the quantum scale lies in the size of Planck’s constant in comparison to the particle’s momentum. Planck’s constant has a value of 6.626 x 10–34 m2 kg/s which is a decimal point followed by 33 zeroes. As you can see, it’s an extremely small constant and is found throughout physics in this or another form. Diffraction only occurs when the size of the wavelength of the travelling wave or particle is comparable in size to the gap through which it passes or the object around which it’s diffracting. Think about FM radio waves which are electromagnetic radiation with a wavelength of roughly 1km, these will diffract around large objects like buildings. Whereas we know from the fact we can’t see what is directly behind large buildings that visible light with a wavelength of between 400 -700 nm cannot diffract around these objects.

Now for an electron with a mass of 10–30 kg travelling at roughly six million meters per second, the De Broglie wavelength is:

Which is roughly about a tenth of a nanometre.

Peter Parker may have radioactive spider-blood but he’s not defracting around any solid objects anytime soon! (Columbia Pictures)

Let’s compare this to the De Broglie wavelength of the average human being running for a bus, running straight into a lamp-post. Let’s take the average global body mass of 62 kg and let’s say our running speed is 3 m/s when we hit the post. Which gives us:

As you can see the De Broglie wavelength for a macroscopic object is incredibly small in comparison to that of an atom. To demonstrate this isn’t just the case for a human being, let’s look at the reason we don’t see a tennis ball diffract around a tennis racket at Wimbledon. Let’s take the mass of the ball to be 0.06 kg and the magnitude of its velocity to be 60 m/s.

So we wouldn’t expect a diffraction from a lamp-post simply because of the massive difference in size, and that massive difference in size continues through tennis balls and rackets way down to particles of roughly 5000 protons, neutrons and electrons with the formula C284.H190.F320.N4.S12. The momentum aspect of the De Broglie equation is simply too large, and Planck’s constant too small, for even small slow-moving objects for the wave-like behaviour to be demonstrated at macroscopic levels.

So we can see that the tiny value of Planck’s constant (h) is the reason we don’t see particle-wave duality in the macroscopic world, but it turns out this is the mitigating factor in why we don’t see other quantum effects such as the Heisenberg uncertainty principle in classical physics.

Examining the ‘fuzziness’ of nature

The visualisation of the wave/particle duality of light

The fundamental takeaway from all this is that we can see that at a quantum level, it can be incredibly difficult to pin definitions on the fundamental building blocks of nature. This ‘fuzziness’ underpins the non-intuitive nature of quantum physics. But if this all seems nebulous and ill-defined, take heart in recent developments in the investigation of the properties of light. In March 2015, Nature published the work of a team of physicists from the Laboratory for Ultrafast Microscopy and Electron Scattering in Switzerland who created a standing wave of UV light which they then bombarded with electrons. This allowed the disturbed photons of the wave to be seen distinctly, thus demonstrating the first picturable demonstration of particle-wave duality, perhaps giving us our first glimpse of the true nature of matter.