Light: Waves or Particles?

Light is a wave, right? But, maybe you’ve heard that it can be a particle, too? And you swear that in school you were always doing experiments with light rays! So which is it!? All of them? None of them? Has everybody been lying to you!? And how can maps help us figure it out?

---

Maps are great.

They’re so great, in fact, that if you want to know the first thing about a place, you’ll probably look at a map of it.

This map tells me where buried treasure is hidden:

We say that it’s a good map if it makes correct predictions about the place it’s modelling. If I really do find treasure where the map tells me it is, then I’d say it’s a good map!

A map that always gives incorrect predictions is a bad map, and you’d probably stop using it quite quickly.

Some of the earliest maps were made by the ancient Greeks, but they were interested in lots of other things, too, like architecture, literature, and banishing their politicians…

…But also in light.

Around 360BC, a Greek guy named Plato modelled light as ‘streams of vision’, or rays, that travelled in straight lines between our eyes and the things we see^[1].

Plato’s version of the model had a few problems (for example, he thought these rays originated in the eyes rather than light sources like the sun), but this model was later refined by a Greek guy named Euclid, in his Book of Optics^[2].

Then again by a Greek guy named Ptolemy, in his Book of Optics^[3].

Then again by an Arab guy named Ibn al-Haytham, in his Kitāb al-Manāẓir.

…or Book of Optics^[4].

The refined version of the ray model was very good. Why? Because it allowed us to make correct predictions about the behaviour of light. It could explain reflection, refraction, and even things like why pinhole cameras produced images upside down!

But, in the 1600s, it became clear that it couldn’t explain everything.

---

In 1665, a man named Francesco Grimaldi noticed that light did something strange when passing around very thin objects, specifically that it seemed to create a series of light and dark areas, rather than a single shadow like the ray model would predict^[5]:

He called this phenomenon diffracte, from the Latin verb diffringere, meaning ‘to shatter’, and which in English we now call diffraction^[6].

Additionally, just 4 years later, a man named Rasmus Bartholin observed an unusual type of refraction through what he called Iceland crystal, where everything you looked at through it appeared to be doubled, as though there was a kind of double refraction^[7]:

These were things that the ray model just couldn’t explain.

---

This map is a pretty good model to use for finding buried treasure:

but there are still a lot of things that it doesn’t give me any information about! Is the treasure buried at ground level, or up on top of a hill? Do I have to walk there, or is there a bus route I could use? If I want to know about these things, I’m going to need a new map.

In order to explain diffraction and double refraction, we needed a new model of light, and in the late 1600s, there were two contenders. One model proposed by Isaac Newton (yes, that Newton) thought of light as small ‘bodies’, or particles, which he introduced in his Book of Optics^[8].

Newton’s book is most famous for his work on colour, where he figured out that white light is made up of all the other colours combined. Passing white light through a prism allowed you to see all the colours, he wrote, because each one was bent, or refracted, at a slightly different angle, separating them out.

Newton explained this by saying that different colours of light were formed from particles of different sizes, and as a result were attracted to the glass of the prism by different amounts. He used this particle model to explain diffraction by proposing that it was simply a different kind of refraction, where the particles were bent around a nearby object rather than through it. He also explained double refraction by suggesting that light particles might have different ‘sides’ with properties which caused them to refract differently.

Newton’s particle model seemed reasonable, but a man named Christiaan Huygens disagreed – he said that light wasn’t made of particles, but waves.

In his book Treatise on Light, Huygens proposed that light was formed of waves travelling through a substance called ether^[9]. He thought that each point on this ether acted as a source of small wavelets, which added together to form a single wave-like motion, or wave front.

Huygens used these wavelets to explain double refraction, proposing that they could be either spherical or ellipsoidal, and that the structure of Iceland crystal caused it to refract these shapes differently. The principle could also be used to explain diffraction, as blocking the path of some of these wavelets would naturally cause the overall wave front to curve around the object.

Something Huygens’ wave model didn’t explain was colour. A few decades later, it was suggested that colour could be related to frequency, how quickly a wave vibrates, by a man named Leonhard Euler^[10], who we probably won’t be hearing from again (well, maybe a few more times).

But despite this, Newton’s particle model was the prevailing model of light for around a century, which had absolutely nothing to do with the fact that he was president of The Royal Society of London.

But it wouldn’t remain the main theory forever, because in 1801 a man named Thomas Young said that light was a wave, and he could explain how, using something he called interference^[11].

---

Young’s Double Slit experiment (the first one because there are two) involved passing a beam of light through two neighbouring slits and observing the pattern made on a far surface.

Similarly to Grimaldi’s earlier experiments, he observed a series of light and dark stripes, but Young had an explanation for this. He said that, if light were a wave, then the waves coming from each of the two slits would interfere with each other. This meant that when the high points, peaks, or the low points, troughs, of each wave met, they would add together, producing the bright areas, but when a peak of one wave met a trough of the other, they would cancel out, producing the dark spots:

Young’s wave model was controversial at the time, but a few years later a man named Augustin-Jean Fresnel also used the wave model to explain double refraction, by saying that the two refractions were due to different orientations, or polarizations, of the waves^[12]:

which we can see by using a polarising filter!

It was by observing these polarizations of light that, in 1845, a man named Michael Faraday showed that light could be affected by magnetism, when he found that he could rotate the direction in which light was polarised by applying a magnetic force to the material it was travelling through^[13,14]. But, by this time, people had already discovered that magnetic force didn’t exist by itself, but came unified with the electric force, as electromagnetism. Faraday thought that light might be vibrations, or waves, in what he called electromagnetic lines of force^[15]. We can see Faraday’s ‘force lines’ by looking at the pattern that iron filings make around a magnet:

Faraday didn’t have a lot of confidence in this theory, but someone else did. While working on a set of equations to describe Faraday’s electromagnetic work, a Scotsman named James Clerk Maxwell calculated that disturbances in what he called the electromagnetic field did travel at the speed of light, supporting Faraday’s idea. Light was a wave in this electromagnetic field!^[16]

Maxwell’s equations predicted an infinite range of frequencies of these electromagnetic waves, producing what we now call the electromagnetic spectrum, which includes not just visible light, but also things like infrared, ultraviolet and microwaves, too!

So, it was finally settled: light had to be a wave, and could never be a particle.

…Or could it?

---

At the beginning of the 20th century, scientists were interested in how objects absorb and emit light.

To do this, they considered objects they called black bodies, which they named because they absorb all light that hits them, including microwaves and the rest of the electromagnetic spectrum, effectively making them black in colour. Black bodies are an idealisation, meaning they don’t really exist, but there are a lot of things that get pretty close, like stars, which makes them useful to think about.

However, when it came to black bodies, there was a problem.

According to the second law of thermodynamics, the energy emitted by a black body should be spread equally across all possible modes, in this case frequencies of light. It’s like if you were buying drinks for your picky friends so you wanted to spend your money equally across all the different brands.

The trouble is that there isn’t a fixed number of frequencies of light – there’s infinitely many of them! If you wanted to spend your money equally across an infinite number of drinks, you’d either need an infinite amount of money…

…or you’d only be able to buy an infinitesimally small amount of each one!

This made it seem like black bodies would either have to hold an infinite amount of energy, or emit no energy at all! But, in 1901, a man named Max Planck had a solution.

When we buy drinks, we can’t just get any amount we feel like: they’re sold in set quantities, like the can, or the pint, or the tankard:

Planck thought the same might be true of energy. He assumed that light could only carry energy in specific quantities, or quanta, and that the size of these quanta was dependent on the light’s frequency. Specifically, he calculated that they should be equal to the frequency multiplied by this number, which we now call Planck’s constant^[17]:

This solved the black body problem, as it meant that energy couldn’t be split equally over all modes! For example, very high frequency light could only carry energy in huge quanta, so was only ever emitted by extremely energetic objects. It would be like if all the most expensive drinks were only sold in huge bottles which you could never hope to afford!

When Planck first came up with energy quanta, he didn’t think they were real, just a clever trick to make the maths work out.

But somebody disagreed.

In 1905, Albert Einstein (yes, that Einstein) thought that, if these quanta were real, then they could explain a different physics puzzle.

In a metal, negatively charged electrons are stuck, or bound, to the surface by the positive charges within the metal.

By shining light on the metal’s surface, it can transfer enough energy to the electrons that they break free from this bond and escape the metal entirely, and this is what we call the photoelectric effect.

The energy of the incoming light could be increased in two ways: by increasing its intensity, or by increasing its frequency. The strange thing was that, in experiment, it was only the light’s frequency that seemed to have any effect on the energy of the electrons.

There was also a problem with how long it took for the electrons to start escaping, namely that it didn’t take very long at all. If light were a wave, then it should take a while for it to transfer all the energy that the electrons needed, but, in experiment, the metal started giving off electrons almost immediately!

In his paper on light, Einstein suggested that these results could be explained if light really did travel in quanta^[18]. It would explain why there was no time delay on electron emission, as the energy would be transferred all at once when the light quanta interacted with the electron. It also explained why increasing the intensity of the light didn’t change the electron energy – if the individual quanta didn’t carry enough energy to free the electron, then it wouldn’t matter how many of them you used, and Planck had found that quantum energy was proportional to frequency, not intensity.

But, hang on, all this talk of energy coming in quanta made it sound like light was a particle again! But we knew that it had to be a wave, because of the interference pattern we get from Young’s double slit experiment:

If we did a similar experiment with something we’re sure is a particle, like, say, an electron, then we wouldn’t see an interference—

…Oh.

In the early 1920s, Clinton Davisson and Lester Germer performed a Double Slit experiment (the second one) using electrons. And they did it by accident!

The experiment was originally to fire electrons at a nickel crystal to study its properties, but the regualar structure of atoms within the crystal created lots of very small gaps, unknowingly setting up the conditions for a double slit experiment^[19]. When they observed the scattering of the electrons, they were surprised to find that, collectively, the pattern formed was an interference pattern, just like the one made by light!

So now something we thought could be modelled entirely as a particle was also sometimes better modelled as a wave! Nobody could have predicted it!

Except they did.

In his PhD thesis, a man named Louis de Broglie postulated that matter, such as electrons, could have a wavelength, too^[20]. He even gave an equation to calculate what this wavelength would be, saying it should be equal to Planck’s constant (that tiny number from earlier) divided by momentum, and he was right!

$$\mathrm{wavelength}=\frac{\mathrm{Planck\text{'}s\; constant}}{\mathrm{momentum}}$$

So, now it seemed that there were all kinds of things which we sometimes had to treat like particles, and other times had to treat like waves. So, which are they?

---

This is a map of London:

This is also a map of London:

They look very different from each other, even though they’re describing the same place!

If I need to go somewhere on the underground, then the first map’s rubbish, but the underground map works great. Similarly, if I want to know the true distance between two places, the first map works well, but the underground map would be terrible.

You could say that London sometimes behaves like the first map, and sometimes behaves like the second map, but it’s a bit of a misleading way of explaining it, because nothing about the city is actually changing! It keeps behaving as it’s always behaved – as a city – the only thing that’s actually changed is what I’m interested in knowing about.

But what if we wanted to know about the public transport routes and know the true distances between places? Could we make a new map that tells us about both? Yes!

---

In 1925, a man named Werner Heisenberg created a new model to describe how quanta behaved, marking the birth of quantum mechanics^[21]. Unfortunately for him, people thought his paper was confusing, partly because he used these strange things called matrices^[22], which were things that mathematicians used, not physicists^[23i,24].

Fortunately for everyone else, later that year a man named Erwin Schrödinger (yes that Schrödinger) thought that, if things like electrons had wave-like properties, their motion must be able to be described by some sort of wave equation^[25], which everyone was much more comfortable with, and which we now know to be equivalent to Heisenberg’s approach^[26,27i].

The equation he came up with looks like this:

$$-\frac{ħ}{\mathrm{2m}}{\nabla }^2\psi +V\psi =E\psi$$

which seems complicated but is actually just describing conservation of energy using de Broglie’s wavelength-momentum relation.

An important symbol in the Schrödinger equation is $\psi$, the Greek letter psi, which represents the state of a quantum object. It’s these quantum states that we use to describe things like electrons instead of classical waves or particles. They’re also called wave functions because they’re solutions to a wave equation.

But if they aren’t classical waves, what are they? We thought about classical light waves as being waves in the electromagnetic field, so what about quantum wave functions? What’s waving?

Schrödinger thought that his electron wave functions might represent the spread of charge of the electron^[28], but a man named Max Born had a different idea. He thought that quantum wave functions weren’t describing charge, but probability^[23ii,29].

---

This wave function, $\psi$, is spread out over space, $x$:

If I asked you to tell me its position, it would be pretty difficult, because the question doesn’t make a lot of sense.

But in experiments we can measure the position of a quantum object, so how can it be described by this wave function!?

Born’s idea was that the wave function describes the probability that you would find a quantum object at any particular point when you measured it.

(…Hang on, this bit’s negative, we can’t have negative probability.)

Born’s idea was that the wave function squared describes the probability that you would find a quantum object at any particular point when you measured it:

So, if we measured the position of the quantum object described by this wave function, what would we find?

It’s important to remember that we have to interact with a quantum object in order to make measurements, and this interaction changes the wave function. This means that, after measurement, this wave function would have:

This reflects the fact that we’re now certain of the object’s position (that’s why the new function reaches all the way up to 100%!). This changing of the wave function due to measurement is known as collapse, and it’s why quantum objects sometimes look like they have a definite position, like a particle!

---

Lots of people hated this probability idea – they thought objects should always have definite properties, not some set of possible ones determined by random chance. Einstein wrote a letter to Born where he said the theory must be incomplete, famously saying that [God] does not play dice^[30]. Schrödinger similarly disliked the idea, because the whole thing seemed ridiculous once you tried to apply it to a large object, like a cat. If you made the state of a cat dependent on the state of one of these quantum objects, then wouldn’t the cat also have to have undetermined properties^[31]? And at what point would we know the state of the cat for certain? When does the ‘collapse’ happen?

Heisenberg, however, agreed with Born – he thought that it made complete sense for the state of a quantum object to be modelled as a probability distribution^[27ii].

In classical physics, we can predict the future state of an object based on information about its current state. If I want to know what the position of an object is going to be, all I need is its current position and its current velocity:

But what if I didn’t know exactly what its current position or velocity was? What if there was uncertainty in these values? Then I could only calculate what its future position would probably be:

Heisenberg said that the reason quantum states are probability distributions is because we can’t be certain of all an object’s properties at the same time. He called this the uncertainty principle^[32]. Heisenberg’s colleague Niels Bohr thought that the uncertainty principle could help explain why we sometimes had to treat quanta like waves, and sometimes like particles – if we knew one property for certain, then it would have definite, particle-like value, but, by the uncertainty principle, that would mean that another, complementary property would have an infinite range of values, like a wave!^[27iii,33]

So, the square of the wave function represents probability, but what does the wave function itself represent? Does it really exist? And why does it do this weird collapse thing? These are all interesting questions, which have equally interesting answers like Don’t know, Not sure, and Haven’t a clue.

But if we don’t even know what a wave function is, then how is it useful?

---

A good map makes correct predictions about the place it’s modelling. So long as a model gives us correct predictions, then it’s still useful to us!

This is why we use the quantum model, but also why the wave and particle models have stuck around: they’re not wrong, just useful in a more limited number of situations. Ultimately, quantum objects behave like quantum objects, not like waves or particles, but sometimes we can treat them as one or the other to simplify a problem!

Why use a complicated map like this…

If all we need to know is which tube station to get off at?

---

References

1. Plato, Timaeus, section 45b – c (English translation)
2. Euclid, Optika (English translation)
3. A. M. Smith, Ptolemy's Theory Of Visual Perception
4. A. I. Sabra, The Optics of Ibn al-Haytham: Books I – III (translated from the original Arabic)
5. F. M. Grimaldi, Physico-mathesis de lumine, coloribus et iride aliisque adnexis, pg. 1 – 11
6. R. Cecchini & G. Pelosi, Diffraction: the first recorded observation, IEEE Antennas and Propagation Magazine, Vol. 32, No. 2, pg. 27 – 30
7. E. Bartholin, Experimenta crystalli islandici disdiaclastici quibus mira & insolita refractio detegitur
8. I. Newton, Opticks, pg. 370 – 374
9. C. Huygens, Traité de la lumière (English translation)
10. C. Hakfoort, Optics in the age of Euler, pg. 97 – 100
11. T. Young, I. The Bakerian Lecture. Experiments and calculations relative to physical optics
12. A. Fresnel, Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées, Annales de chimie et de physique, Vol. 17, pg. 102 – 111 (English translation)
13. The Royal Institution, Michael Faraday’s magneto-optical apparatus
14. M. Faraday, Faraday's Diary, Vol. IV, pg. 263 – 267
15. M. Faraday, Thoughts on Ray-vibrations, Experimental Researches in Electricity, pg. 447 – 452
16. J. C. Maxwell, A Dynamical Theory of the Electromagnetic Field
17. M. Planck, Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum, Verhandlungen der Deutschen physikalischen Gesellschaft (1900), pg. 237 – 245 (English translation)
18. A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt, Annalen der Physik, Vol. 322, pg. 132 – 148 (English translation)
19. Physical Science Study Committee (PSSC), Matter Waves
20. L. de Broglie, Recherches sur la théorie des quanta, Annales de Physique, Vol. 10, No. 3, pg. 22 – 128 (English translation)
21. W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Zeitschrift für Physik, Vol. 33, pg. 879 – 893 (English translation)
22. M. Born & P. Jordan, Zur Quantenmechanik, Zeitschrift für Physik, Vol. 34, pg. 858 – 888 (English translation)
23. A. Pais, Inward Bound
    1. pg. 255
    2. pg. 256 – 261
24. M. Jammer, The Conceptual Development of Quantum Mechanics, pg. 206 – 207
25. E. Schrödinger, Quantisierung als Eigenwertproblem (Erste Mitteilung), Annalen der Physik, Vol. 384, pg. 361 – 376 (English translation)
26. C. M. Madrid Casado, A brief history of the mathematical equivalence between the two quantum mechanics, Latin-American Journal of Physics Education, Vol. 2, No. 2, pg. 152 – 155
27. A. Pais, Niels Bohr's Times, In Physics, Philosophy, and Polity
    1. pg. 284
    2. pg. 304 – 306
    3. pg. 314 – 315
28. E. Schrödinger, An Undulatory Theory of the Mechanics of Atoms and Molecules, Physical Review, Series II, Vol. 28, No. 6, pg. 1049 – 1070
29. M. Born, Zur Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik, Vol. 37, pg. 863 – 867 (English translation)
30. A. Einstein, To Max Born, The Collected Papers of Albert Einstein, Vol. 15 (English translation)
31. E. Schrödinger, Die gegenwärtige Situation in der Quantenmechanik, Naturwissenschaften, Vol. 23, Issue 48, pg. 807 – 812 (English translation)
32. W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, Vol. 43, pg. 172 – 198 (English translation)
33. N. Bohr, The Quantum Postulate and the Recent Development of Atomic Theory, Nature, Vol. 121, pg. 580 – 590