de Broglie Equation — Overview & Calculations

The Waves of Matter

In the early 1900s, Max Planck and Albert Einstein had drawn some comparisons between matter and energy. Then in 1924, Louis de Broglie inferred that light (energy) and matter behave similarly. He proposed that both energy and matter could act as particles or waves. It was well accepted that light or energy moves in waves. But, the idea of matter, specifically electrons, moving in waves was new.

But, de Broglie's work was all theoretical. Eventually, other physical chemists confirmed de Broglie's hypothesis experimentally. How? By using variations on the double-slit experiment and cathode-ray experiment. Scientists have confirmed that de Broglie's work applied to protons, neutrons, electrons, neutral atoms, and even molecules.

So, both energy and matter move in a wavelike motion. This concept became known as the wave-particle duality in quantum mechanics. Let's look at the math!

Combining Scientific Equations

De Broglie suggested a new equation based on equations developed by Einstein and Planck.

Einstein's Equation:

$$E=mc^2$$

Planck's equation:

$$E=\frac{hc}{\lambda}$$

Einstein combined these to create:

$$m=\frac{h}{\lambda c}$$

In this equation, $c$ represents the speed of light for photons. De Broglie developed a similar equation, but instead of photons, he applied it to electrons. He denoted the speed of an electron as $v$.

De Broglie equation:

$$m=\frac{h}{\lambda v} \>\>\> or \>\>\> \lambda=\frac{h}{mv}$$

This equation applies wavelike motions to matter and energy. This phenomenon can only be described mathematically. Why? This wavelike movement only occurs at the atomic level. We are unable to detect it with our eyes or by other means!

So, what's the benefit? It allows us to determine the wavelength ($\lambda$) of electrons. For example, let's consider hydrogen. The electron's mass is $9.11 \times 10^{-31} kg$, and its velocity is $2.2 \times 10^6 m/s$. So, the wavelength of this electron according to the de Broglie equation is 330 picometers.

But, how does this help us? Remember, electrons have specific energy levels. The Bohr model and Planck's quantum theory helped explain atomic spectra and blackbody radiation. We knew electrons absorb and emit set wavelengths of light. But we didn't know why. Suddenly we had a new paradigm. Electrons are waves! De Broglie had set the stage for the quantum mechanical model. Soon Erwin Schrodinger would develop his famous equation. Also, Heisenberg would develop his uncertainty principle. We would live in a world of orbitals defined by probability! How does that explain the Bohr model? The odds of finding an electron between orbitals are minuscule!

(Video) De Broglie Wavelength

By Your Daily Equation

In this video, YourDailyEquation explains the history and implications of the De Broglie wavelength. He begins by revisiting the double-slit experiment. It showed that photons behave like waves. As they pass through the slits, they interfere and create a banded pattern of light and dark. He even plays a video of the actual experimental data being collected! Next, he briefly revisits the photoelectric effect. Albert Einstein showed that photons also behave like particles. So we had a wave-particle duality for light. But, De Broglie reversed the question. He asked, "what if things we think of as particles, like electrons, behave like waves?" The video concludes with De Broglie's insights and the math behind them.