Introduction

In 1924, Louis de Broglie proposed a revolutionary idea that would fundamentally change our understanding of quantum mechanics: matter, like light, can exhibit both wave and particle properties. This concept, known as wave-particle duality, became a cornerstone of modern physics.

De Broglie's Matter Wave Equation

De Broglie proposed that all moving matter possesses wave-like properties, with a wavelength given by:

$$\lambda = \frac{h}{mv}$$

Where:

Standing Waves and Atomic Structure

De Broglie applied his theory to atomic structure, proposing that electrons orbit the nucleus as standing waves. These waves form when two waves of equal frequency and amplitude travel in opposite directions, creating fixed nodes and antinodes.

For an electron orbit to be stable, its circumference must contain an integral number of wavelengths:

$$2\pi r_n = n\lambda$$

Where:

This quantization condition leads to discrete energy levels, explaining Bohr's model of the atom. By combining the matter wave equation with the circumference condition:

$$2\pi r_n = n\frac{h}{mv}$$ $$mvr = n\frac{h}{2\pi}$$ $$L_n = n\frac{h}{2\pi}$$

This result confirms Bohr's quantization of angular momentum, providing theoretical support for his model.

Experimental Evidence

The Davisson-Germer Experiment

In 1927, Clinton Davisson and Lester Germer provided the first experimental confirmation of de Broglie's hypothesis. They observed electron diffraction from a nickel crystal:

The theoretical wavelength calculated using de Broglie's equation was 1.67Å, providing strong support for the matter wave hypothesis.

Double-Slit Experiment

When electrons pass through two closely spaced slits, they create an interference pattern identical to that produced by waves:

Electron Diffraction from Gold Foil

When electrons are directed at a thin gold foil, they produce a circular diffraction pattern:

This pattern provides further confirmation of the wave nature of electrons.