The de Broglie hypothesis states that particles of matter can behave as both waves and particles, just like light. In this lesson, we’ll learn the basics of the de Broglie hypothesis and how it related to other theories released at the same time.

## De Broglie Hypothesis

In quantum mechanics, matter is believed to behave both like a particle and a wave at the sub-microscopic level.

The particle behavior of matter is obvious. When you look at a table, you think of it like a solid, stationary piece of matter with a fixed location. At this macroscopic scale, this holds true. But when we zoom into the subatomic level, things begin to get more complicated, and matter doesn’t always exhibit the particle behavior that we expect.This non-particle behavior of matter was first proposed in 1923, by **Louis de Broglie**, a French physicist. In his PhD thesis, he proposed that particles also have wave-like properties. Although he did not have the ability to test this hypothesis at the time, he derived an equation to prove it using Einstein’s famous mass-energy relation and the Planck equation.

## Deriving the de Broglie Equation

Albert Einstein was the first scientist to draw a relationship between mass and energy, culminating in his now-famous equation: *E* = *mc*^2. In this equation, *e* is energy, *m* is mass, and *c* is the speed of light.German physicist Max Planck created the equation now known as the **Planck equation** or **Einstein-Planck relation** to describe the energy in a photon wave. The equation is *E* = *h*nu*, where *e* is energy, *h* is the Planck constant, and *nu* is frequency of the wave. Now, Planck’s constant is a proportionality constant to describe the relation between the energy and the frequency. Constants are known values in science, and we can look up their value and directly plug them into equations.Louis de Broglie figured that if matter also behaved like waves, just like light, the Planck equation would also apply to matter.

So he combined the Einstein and Planck equations because Einstein’s equation solely dealt with the energy of matter, and Planck’s equation dealt with the energy of waves. As both equations had energy on one side of the equation, de Broglie made both sides equal to each other giving us:*mc*^2 = *h*nu*Unlike light waves, particles cannot travel at the speed of light, so he altered the equation to enter in a velocity, rather than the speed of light, giving us:*mv*^2 = *h*nu*It’s important to remember that although the *nu* from the Planck equation looks like a *v* from the Roman alphabet, it is actually the lowercase Greek letter *nu*. The Greek letter *nu* is often italicized to avoid some confusion (see video).Let’s recall the relationship between wavelength and frequency in a wave. **Wavelength** is the distance between two successive peaks in a wave. **Frequency** is the number of peaks that pass through a fixed point during a given time interval. They are related in that velocity = wavelength x frequency or:*v* = *nu***lambda*De Broglie substituted wavelength into his equation by:*mv*^2 = (*h*v*)/*lambda*.

This allows us to solve for lambda and simplify the equation:*lambda* = (*h*v*)/(*m*v^2)*lambda* = *h*/*mv*

## Implications of the de Broglie Equation

What the de Broglie equation does is allow us to find the wavelength of a given particle if we know the mass and its speed. While this equation looks straight-forward enough, de Broglie did not actually perform any tests to prove the validity of his equation. The relationship was not tested until three years later by Clinton Davisson and Lester Germer using a beam of electrons and an x-ray beam. X-rays were known to behave as waves, so they used an x-ray beam as a comparison to the electron beam, which was composed of matter.

Until this point, scientists believe that diffraction, or bending, only occurred with waves and not matter. However, Davisson and Germer discovered that when you shot a beam of electrons at a nickel surface, the electrons scattered off the surface of the nickel were diffracted at a specific angle.In order for the beam of electrons, which are matter, to diffract, they must demonstrate wave-like properties.

Therefore, this simple experiment by Davisson and Germer proved the validity of de Broglie’s assumption of the wave behavior of matter and his equation. This proof also furthered the field of quantum mechanics and allowed scientists of the time to understand many other behaviors at the sub-microscopic level, such as our inability to accurately know the momentum of a particle the better we know its position, which is the Heisenberg Uncertainty principle, and is covered in depth in another lesson.

## Lesson Summary

In review, **Louis de Broglie** was the first physicist to hypothesize on the wave-like behavior of matter. He used both the **Einstein mass-energy relation** and the **Planck equation** to describe how you can derive the wavelength of matter from its mass, speed, and the Planck constant. Although this equation could not be nicely demonstrated, it was based off the assumption that matter behaved like light and had the properties of waves at the nanoscopic level.

The wave-like behavior of light was proved three years later by two scientists who were able to observe that a beam of electrons diffracted when they hit a nickel surface, which is exactly how a beam of x-rays behaves in the same situation. This proof of the wave-like properties of matter validated de Broglie’s equation and allowed for the furtherance of quantum mechanics.