# Development of Quantum Physics IV – How We Found the Wavy Particles

Light can only appear as tiny packets of energy called quanta, as mentioned previously. The fact that there are smallest energy packets (i. e., photons) clearly limits the classical wave model of light. Let’s venture another revolutionary step today!
In 1924, the French physicist Louis de Broglie proposed that former “particles” could be described in terms of the wave-particle dualism as well as light. To be honest, there’s no actual reason why it couldn’t be that way, right?!
But what does this mean? Actually, what is light then? And what exactly are electrons?

You can find the other articles of this series here.

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Well, what is light? How can someone figure it?
These questions are difficult to explain indeed!
Anyhow, light consists of small and indivisible energy packets (“quanta”) which are called photons. Therefore, photons are particles in some sense, but they also have wave characteristics. It depends on the experiment’s setup whether light manifests as waves or as particles.
As we’ve already seen, photons are defined by their energy $E=h\cdot\nu$, where $h$ is Planck’s constant (a very tiny natural constant) and $\nu$ the photon’s frequency. You can additionally assign a formal mass $m=E/c^2=h\nu / c^2$ to a single photon with $c$ being the propagation speed, which is the speed of light in this case. (Photons don’t really have any mass in most of our models (click here for converse speculations) because they are travelling at $c$. Everything massive can’t move at the speed of light. This is the reason why the photon’s mass is only formal!) Another characteristic of a photon is its momentum $p=\hbar\cdot k$, where $\hbar$ is the Planck constant divided by $2\pi$. (We write $\hbar$ instead of $h/(2\pi)$ for economical reasons only.) $k$ is the so-called wavenumber. $k=2\pi\cdot\nu / c$.

If you look carefully, you’ll realize that all three properties of photons are merely defined by the photon’s frequency!

I think you would agree if I’d say that momentum, mass, etc. are rather particle characteristics than wave properties. But here we have described them in terms of the frequency $\nu$ and the wavelength $\lambda$, respectively. (Frequency and wavelength are connected by $\lambda\cdot\nu = c = \text{constant}$.)
At this point we can already guess a deeper correlation between the wave and the particle model of electromagnetic radiation.

Now, let’s assume that this wave model also applies to massive “particles” like electrons, neutrons, atoms, and molecules in the broadest sense. This is exactly what Louis de Broglie did.

Louis de Broglie, 1929

Suddenly, former typical particles are equipped with a wavelength – namely, the de Broglie wavelength -, because from now on, particles should have wave properties like the wave-like light reversely had particle properties back then.

For the sake of completeness, the de Broglie wavelength can be calculated as follows:

$\lambda = \frac{h}{p} = \frac{h}{m\cdot v} = \frac{h}{\sqrt{2m\cdot E_\text{kin}}}$

($h$…Planck’s constant, $p$…momentum, $m$…the particle’s mass, $v$…speed of the particle, $E_\text{kin}$…kinetic energy)

Here’s an amusing example of the de Broglie wavelength:
Imagine a ball (e.g. a soccer ball) of mass of 430 grams (= a little less than 1 pound) moving at a speed of 120 kilometers per hour (= 75 miles per hour). Then, its de Broglie wavenlength is 46·10-36 meters. This is a damn small wavelength! (The tininess of this wavelength is the reason why we don’t observe any interference effects etc. on the macroscopic objects in our everyday lifes. But the de Broglie wavelength of an electron can be in the range of nanometers (10-9 meteres), which of course is much larger than a soccer ball’s wavelength. Photons can have a still larger wavelength and therefore can see wave phenomena a lot easier.)

What are the consequences of these “wavy particles” for our understanding of the world?
If massive particles were defined by the same characteristics as photons, then they would have to behave in the same way under equal conditions, wouldn’t they?

Results of a double-slit-experiment performed by Dr. Tonomura showing the build-up of an interference pattern of single electrons. Numbers of electrons are 11 (a), 200 (b), 6000 (c), 40000 (d), 140000 (e)

In Young’s interference experiment photons are fired against a double-slit. The light beams travel through both slits and interfere as they hit a screen behind. Intensity maxima are formed if the incoming light beams are in phase, i. e. a wave peak encounters another peak. Vice versa, there are places on the screen where a peak and a trough come together and intensity minima are formed. But in quantum mechanics strange things can happen: If the light’s intensity is dimmed so much so that there’s only a single photon in the experimental setup at one time, there’s an interference pattern even in this case!

As you can see in the picture on the left, the points of impact of the single photons seem randomly distributed at first – and they really are. As more and more photons come in, a distinct interference pattern forms.
How is that possible? Photons hit the screen on a random place and are not able to interfere with other photons (because they are “alone in the experiment”) – nevertheless, we can clearly see interference effects. How? Why?
Quantum mechanics offer a theoretical interpretation of these observations. I’ll write about it later on!

Let’s assume we are not yet interested in the mechanism behind these observations. Then we can nevertheless claim: If we define electrons by the same characteristics as photons, then they must behave like photons when fired through a double slit.
As a matter of fact, Clinton Davisson and Lester Germer were able to show in 1926 that electrons after passing through a thin film of crystalline material (that’s a variant form of a double slit) generate a interference pattern in the form of diffraction rings. It’s really remarkable that electron diffraction works analogous to the diffraction of X-rays (= photons)! You can see the similarities by comparing the following images:

X-ray diffraction*

Electron diffraction*

Further experiments showed that not only electrons have wave properties, but also other particles like neutrons etc. By now, it has been demonstrated that certain molecules, which are objects consisting of a multitude of atoms, cause interference patterns behind a double slit.

De Broglie’s ambitious assumptions turned out to be true and naturally realized!
They shine new light on our view of nature and they have sweeping consequences!

But there still remains a question: What’s the underlying mechanism behind the diffraction of light/particles in Young’s double slit experiment? I’ll give a answer to that next time. As you will see, we have to introduce new concepts and ideas again.
We move forward in the development of quantum physics by giant strides!

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* (Image source: chemgapedia.at)

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