## When it comes to light, evidence presents us with a paradox. It propagates through space as a continuous wave, but somehow exchanges its energy in the form of discrete particles. So, “wave” or “particle”—what is light?

### The Paradox of Light

So, there are two famous experiments that look to explain the fundamental nature of light. On the one hand, Thomas Young’s two-slit experiment demonstrates that light is a wave—a periodic disturbance, such as sound that may either be traveling or standing. We can measure the wavelength of light, which is less than a millionth of a meter.

On the other hand, Albert Einstein’s analysis of the photoelectric effect demonstrates that light is composed of discrete particles called photons.

Learn More: Two Revolutionaries—Planck and Einstein

The number of photons, the light intensity, determines the number of the electrons produced in the experiment. The energy of the photons—the light frequency—determines the energy of the electrons that are produced.

We can’t just dismiss either of these experimental results. Our understanding of light must somehow encompass both the wave and the particle ideas. The quantum view can be summed up in a single phrase: Wave-particle duality. What does that mean?

### De Broglie’s Radical New Idea

The true nature of light cannot be described in simple terms. Our language just isn’t equipped to describe the way light really is. We need both wave and particle pictures to explain the behavior of light, but that might make us uneasy. How can we use both wave pictures and particle pictures? Isn’t that a logical contradiction? That’s an immensely tricky question. It’s maybe the trickiest question in all of science.

The first rule of thumb is that light travels in the form of waves with frequency and wavelength exhibiting constructive and destructive interference and so on. Light travels as a wave.

The first rule of thumb is that light travels in the form of waves with frequency and wavelength exhibiting constructive and destructive interference and so on. Light travels as a wave.

The other rule of thumb is that light interacts; it’s emitted or absorbed in the form of discrete particles, discrete photons, lumps of discrete energy. The answer for light appears to be that light is both discrete and continuous. It has properties of both kinds, which is a very strange new answer to an old question. That picture of wave-particle duality is a bit oversimplified, but it’s not too bad for a start. Light travels as a wave, interacts as a particle.

Even in the early days of quantum theory, it was clear that quantum theory was not just about the nature of light.

In 1924, along comes Louis de Broglie. Louis de Broglie is a French aristocrat as well as a physics doctoral student. In his doctoral thesis, he proposes a radical new idea. He proposes that quantum wave-particle duality applies not only to light, but also to matter. He says that particles like electrons must also have wave characteristics.

Electrons have frequency. They have wavelength. This sounds very strange. We think of electrons as particles. We think of electrons as little baseballs. How can a baseball have a wavelength?

Even though the idea is very strange, de Broglie’s idea is very soon spectacularly confirmed in the laboratory. In an experiment that was done in several different places, scientists shot electrons at a crystal. In a crystal, the atoms are arranged in very orderly ranks and rows. The electrons go through the crystal and they come out the other side, but because of this regular arrangement of atoms, they only come out in certain directions.

Why is that? The waves of the electron constructively interfere in some directions and destructively interfere in others. The electrons only come out in certain directions and, in fact, the scientists can measure the wavelength of an electron. De Broglie was right!

Learn More: Particles of Light, Waves of Matter

### Building on Planck’s Constant

Let’s suppose we have a particle of mass, m, moving at a speed, v. For example, take a pitched baseball. The baseball has two important characteristics. First of all, it has energy. The energy of the baseball is E = ½ mv2; this is sometimes called kinetic energy, energy due to its motion.

What does the energy tell us? The energy basically tells us how much work the pitcher has to do to throw it. A closely related characteristic of the baseball is the momentum of the baseball, which is just the mass times the velocity—how fast it’s going. Energy and momentum are characteristics of a particle.

Waves are characterized by their frequency, f, and their wavelength, which is denoted by the Greek λ. The frequency of a musical note is a few hundred waves per second. That’s the number of waves that pass by a fixed point in space, like your ear, each second. The wavelength of a musical note is something like a meter. It’s a pretty good-sized wave.

De Broglie’s idea was to somehow connect the particle properties—energy and momentum—to the wave properties of frequency and wavelength. The connection between them is going to involve Planck’s constant—the constant that German physicist Max Planck found that relates energy and frequency—specifically, Planck’s formula that the energy of the particle is equal to Planck’s constant times the frequency of the wave.

De Broglie adds to this. He says that momentum and the wavelength are related. The momentum of the particle, p, is equal to Planck’s constant divided by the wavelength of the wave λ: p = h/ λ. That’s the relation between momentum and wavelength.

Keep in mind that Planck’s constant is a very tiny number. That means that the typical wavelength that we’re talking about for electrons and atoms is going to be really tiny. The wavelength for an electron and an atom is less than 1 billionth of a meter in size. These are very short wavelengths and already you can see that the more massive the particle is, the more its momentum will be at a given speed.

If its momentum is larger, then de Broglie’s formula tells us that its wavelength will be shorter. The electron wavelengths are already really tiny. The wavelengths for more massive particles are even smaller and, because the wavelengths are so small, it’s extremely difficult to do interference experiments and see the interference effects for large particles.

### Born’s Important Rule

The Planck–de Broglie relations connect particle properties, energy and momentum, to wave properties, wavelength and frequency. The link between the particle properties and the wave properties is Planck’s constant, h, but that still leaves things very mysterious. When we talk about electron waves, what do we mean? What are we talking about?

A sound wave is a periodic disturbance in air pressure that travels through the room. A light wave is a periodic disturbance in the electromagnetic field that travels through a room. But an electron wave is a periodic disturbance in what? What’s waving?

That brings us to the second connection between particles and waves. That’s called the “Born Rule,” named after Max Born, the great German physicist and one of the inventors of quantum mechanics.

Here’s the basic problem that Born was considering. A particle is something that has a definite position in space. A wave, on the other hand, is spread out all over space. How do we reconcile these two pictures in quantum theory? Here’s Born’s great discovery: The intensity of the quantum wave at a point tells us the probability of finding the particle at that point.

### Fixing the Odds

Wave intensity turns out to be proportional to the square of the wave amplitude.

Imagine waves; the wave amplitude is the height of the waves. How high the waves are and the intensity of those waves is how much energy those waves carry. It turns out that the intensity is given by the square of the amplitude.

Waves that are twice as high actually carry four times as much energy; a wave that’s twice as high is really four times as powerful. That’s wave intensity. It’s the square of the amplitude and it measures the amount of energy carried by the wave. Wave intensity gives particle probability.

Here’s Born’s insight. Quantum theory does not tell us where a particle is. Quantum theory can’t tell us where a particle is. Quantum mechanics tells us the probability that the particle may be found here or found there. Quantum mechanics only tells us the probabilities. Where does the particle wind up? That’s random. It’s a gamble. Quantum mechanics fixes the odds.

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